Schrödinger Operators on Graphs and Geometry II. Spectral Estimates for
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-potentials and an Ambartsumian Theorem

Schrödinger Operators on Graphs and Geometry II. Spectral Estimates for...
Boman, Jan; Kurasov, Pavel; Suhr, Rune
2018-05-30 00:00:00
Integr. Equ. Oper. Theory (2018) 90:40 https://doi.org/10.1007/s00020-018-2467-1 Integral Equations Published online May 30, 2018 and Operator Theory c The Author(s) 2018 Schr¨ odinger Operators on Graphs and Geometry II. Spectral Estimates for L -potentials and an Ambartsumian Theorem Jan Boman, Pavel Kurasov and Rune Suhr Abstract. In this paper we study Schr¨ odinger operators with absolutely integrable potentials on metric graphs. Uniform bounds—i.e. depending th only on the graph and the potential—on the diﬀerence between the n eigenvalues of the Laplace and Schr¨ odinger operators are obtained. This in turn allows us to prove an extension of the classical Ambartsumian Theorem which was originally proven for Schr¨ odinger operators with Neumann conditions on an interval. We also extend a previous result relating the spectrum of a Schr¨ odinger operator to the Euler character- istic of the underlying metric graph. Mathematics Subject Classiﬁcation. 34L15, 35R30, 81Q10. Keywords. Quantum graphs, Spectral estimates, Ambartsumian theo- rem. 1. Introduction Quantum graphs—i.e. Schr¨ odinger operators acting on metric graphs—have become an increasingly important branch of mathematical physics in the last 20 years or so. They serve as models of branched thin networks, e.g. nanotubes, and complex molecules. Apart from applications they also serve as a rich source of objects suitable for mathematical inquiry. In particular Schr¨ odinger operators on graphs exhibit spectral properties both of partial diﬀerential and ordinary diﬀerential operators. The aim of this paper is twofold. First we work towards proving a spec- tral estimate, i.e. a comparison between the spectra of the Laplacian and Schr¨ odinger operators acting on the same metric graph. The main motiva- tion for obtaining such estimates is that the spectrum of the Laplacian is P. K. was partially supported by the Swedish Research Council (Grant D0497301) and by the Center for Interdisciplinary Research (ZiF) in Bielefeld in the framework of the cooperation group on Discrete and continuous models in the theory of networks. 40 Page 2 of 24 J. Boman et al. IEOT much easier to calculate. We will prove that just as in the case of a single interval the diﬀerence between the Laplace and Schr¨ odinger eigenvalues is uniformly bounded, provided the potential is just absolutely integrable. The classical proof of this relied heavily on the explicit formula for the resolvent kernel of the Laplacian [7]. The corresponding kernel for metric graphs is not in general given explicitly and we will instead work with general per- turbation theory. The second goal is to extend results—relating the Laplace spectrum to geometric properties of the graph—to Schr¨ odinger operators with L -potentials. In particular we obtain an inverse spectral theorem that generalizes the celebrated theorem of Ambartsumian (see Sect. 6), and show that the Euler characteristic of the underlying graph is reﬂected in the spec- trum of the Schr¨ odinger operator in the case of standard vertex conditions (see (2.1) below). 1.1. Spectral Estimates and Inverse Spectral Theory Inverse spectral theory for the Schr¨ odinger equation in R has classically had as a goal to determine the potential given a spectrum. To solve the inverse problem for a Schr¨ odinger operator on a metric graph completely one has in general to determine not only the potential but the underlying metric graph and vertex conditions. It appears that this complete inverse problem is rather diﬃcult, especially since the set of spectral data is not obvious. Therefore it seems attractive to start investigations assuming that the metric graph and the vertex conditions are ﬁxed. The simplest graph to be considered is the star graph formed by a ﬁnite number of compact edges, and the corresponding inverse problem resembles very much the inverse problem on a single interval, where the potential is determined by two spectra [14, 30, 33–35]. The case of general trees has also been studied and we have a rather good understanding of the problem [2, 8, 9, 31, 37]. The case of graphs with cycles is much more involved—major diﬃculties are related to the reconstruction of the potential on the cycles. To assure the uniqueness of the potential one may either use the dependence of the spectral data on the magnetic ﬂuxes [21, 23], or add extra spectral data like the Dirichlet spectrum [38–41]. The problem to reconstruct the metric graph has not been addressed in full generality. Topological characteristics of the graph may be reconstructed [19, 20]. Assuming that the edge lengths are rationally independent one may even reconstruct the graph using the trace formula [15, 18], but under the condition that the potential is zero. Explicit examples of isospectral graphs have been constructed [4, 5, 15]. Employing the boundary control method one may reconstruct metric trees [2] without assuming zero potential. The prob- lem of reconstructing vertex conditions, as well as the inﬂuence of vertex conditions on the solvability of the inverse problem, is even less understood [3]. For example a metric tree is not always reconstructable, e.g. if the ver- tex conditions are not standard [22]. On the other hand more general vertex conditions may help to solve the inverse problem as it is done in [23, 24]. One result in the inverse spectral theory—the Ambartsumian’s cele- brated theorem from 1929 stating that the spectrum of a (Neumann) Schr¨ o- dinger operator on an interval coincides with the spectrum of the (Neumann) IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 3 of 24 40 Laplacian if and only if the potential is zero—is of great importance (see Theorem 6.1). This theorem is rather special, since in order to reconstruct a non-zero potential, knowledge of two spectra is required. Several authors generalized this theorem for the case of metric graphs. The results can be divided into two categories. Some authors proved that if the spectrum of the standard Schr¨ odinger operator on a ﬁxed metric graph Γ coincides with the spectrum of the standard Laplacian on Γ, then the potential is zero. This result was ﬁrst proved for trees [10, 27, 29, 36] and [30], and then for arbitrary graphs by Davies [12]. It was also noted that the Laplacian on a metric graph Γ is isospectral to the Laplacian on an interval if and only if Γ is formed by a single interval. This can be seen as a geometric version of Ambartsum- ian’s theorem and it is based on the fact that a single interval maximizes the spectral gap for the Laplacian among graphs of ﬁxed total length [13, 25, 28]. The inverse spectral result that we obtain—Theorem 6.5—is that the spectrum of a ﬁnite interval is unique among connected ﬁnite compact quan- tum graphs with standard conditions. More precisely, if the spectrum of a Schr¨ odinger operator on a connected metric graph coincides with the spec- trum of the Laplacian with Neumann conditions on an interval, then the graph is the interval, and the potential of the Schr¨ odinger operator is zero. Here we assume that standard conditions are imposed on the graph. In the case of the graph being a compact interval, standard conditions coincide with Neumann conditions at the end-points. The result can therefore be seen as an extension of Ambartsumian’s theorem. 1.2. Outline of the Paper Section 2 contains some fundamentals on quantum graphs. In particular we give some elementary spectral estimates and prove that normalized eigenfunc- st tions of the Laplacian L are uniformly bounded in the L norm. Section 3 st contains the deﬁnition of the operator L that we associate with the for- st mal expression L + q,for q ∈ L (Γ). The operator is deﬁned via quadratic 0 1 form methods. Section 4 deals with spectral estimates, and we prove that there exists a uniform bound on the diﬀerence of eigenvalues of Laplace and Schr¨ odinger operators on metric graphs. This is done by using the Max– Min and Min–Max principles, along with a Sobolev estimate for functions ψ ∈ W (Γ). Section 5 gives a result on the zeros of trigonometric polynomi- als, namely that if the zeros asymptotically tend to the integers, then all the zeros are in fact exactly the integers. By combining this fact with the spectral estimate of Sect. 4 we are able to prove the inverse spectral theorem given in Sect. 6. Section 7 extends a previous result that the Euler characteristic of a graph is reﬂected in the spectrum of a Schr¨ odinger operator with L potential to the case of L potentials. 2. Preliminaries–Basics on Quantum Graphs A quantum graph is a metric graph equipped with a Schr¨ odinger operator, or more formally, a triple (Γ,L, vc) with Γ a metric graph, L a diﬀerential operator, and vc a set of vertex conditions imposed to connect the edges and 40 Page 4 of 24 J. Boman et al. IEOT ensure the self-adjointness of L. In this paper we limit ourselves to compact ﬁnite graphs (see below). We give a brief overview of these - for a thorough treatment of the theory of quantum graphs, see for example [6] and [26]. Metric Graphs: A compact ﬁnite metric graph is a ﬁnite collection of compact intervals of R glued together at the endpoints. More precisely, let, {E } be n=1 a ﬁnite set of compact intervals, each E considered as a subset of a separate copy of R, E =[x ,x ], 1 ≤ n ≤ N. n 2n−1 2n Let V = {x } = {x ,x } denote the set of endpoints of the intervals. j 2n−1 2n n=1 Fix a partition of V into equivalence classes V = V ∪ ··· ∪ V . 1 M Identifying the endpoints yields a graph with vertices given by the equivalence classes V . We let = x − x denote the length of the edge E and deﬁne the n 2n 2n−1 n total length L of a graph as the sum of its edge lengths: L = . n=1 Diﬀerential Operators on Graphs: L acts as a diﬀerential operator on each 2 2 edge separately, in our case − d /dx + q for some real potential function q—subscripts will denote the potential of the operator, and L means that the potential is identically 0, so that L is the Laplacian. The action is in the Hilbert spaces L (E ) of square integrable functions on the edges E .The 2 i i measure on each edge is given naturally by the identiﬁcation of the edge with an interval. L then acts on L (Γ) = L (E ). 2 2 i i=1 2 2 Note that if q ∈ L (Γ) then the formal expression −d /dx + q should be understood as a sum of quadratic forms (see Sect. 3). Vertex Conditions: We shall solely study quantum graphs with standard conditions—also known as Kirchoﬀ, Neumann, natural or free conditions. st Operators with standard conditions imposed will be written as L . Let us ﬁrst discuss the case of a bounded potential q ∈ L (Γ), where the domain of the Schr¨ odinger operator can be given explicitly. For any ψ ∈ L (Γ), also qψ ∈ 2 2 L (Γ), so L ψ = −(d/dx) ψ + qψ ∈ L (Γ) if and only if ψ ∈ W (Γ \ V ). Let 2 q 2 V = {x ,...,x } and let ψ(x ) denote the limit ψ(x ) = lim ψ(x), j j j j j x→x 1 k i i j where the limit is taken over x inside the interval with x as an end-point. i IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 5 of 24 40 Standard conditions are then given by imposing the following relations at each vertex V ψ(x )= ψ(x )= ··· = ψ(x ) j j j 1 2 k (2.1) ∂ ψ (x )=0, n E j x ∈V n(j) j m where ∂ denotes the normal derivative, i.e. ψ (x ) x left end-point j j ∂ ψ(x )= n j − ψ (x ) x right end-point. j j In other words, ψ is required to be continuous in each vertex V and the sum of normal derivatives should vanish there. This yields a self-adjoint operator st L on Γ. st We will study L for q ∈ L (Γ). This means that in general qψ ∈ L (Γ), 1 2 2 2 so the formal expression L = −d /dx + q lacks an immediate meaning as a sum of operators from L (Γ) to L (Γ). We will however establish that 2 2 the perturbation by q is inﬁnitesimally form-bounded (see Proposition 3.2) with respect to the quadratic form of L . More precisely, the quadratic form st Q (ψ, ψ)=(L ψ, ψ)=(ψ ,ψ ) is deﬁned on the domain of functions from W (Γ \ V ), which are in addition continuous at all vertices. Then the expres- st st sion L + q = L can be assigned a meaning via the KLMN theorem (see 0 q st e.g. [32]), as a self-adjoint operator with the same form domain as L . Note that the domain of the quadratic form only includes the continuity condition from (2.1). The Spectrum: We denote the spectrum of L by σ(L). For ﬁnite compact graphs the spectrum of L is discrete, for any self-adjoint boundary condi- tions, and furthermore satisfy Weyl asymptotics. This also holds for L ,for th st q ∈ L (Γ) (see Sect. 4). The n eigenvalue, counting multiplicities, of L (Γ) st st st st is denoted by λ (L )= λ (L (Γ)). When we write σ(L (Γ )) = σ(L (Γ )) n n 1 2 q q q q 1 2 we mean not only equality as sets, but also that the multiplicities of all eigen- values are equal. For a given ﬁnite compact graph of total length L with N edges and M vertices, the following lower and upper estimates can be proven (see “Appen- dix” for the proof) 2 2 π π 2 st 2 (n − M ) ≤ λ (L ) ≤ (n + N − 1) . (2.2) L L In particular this estimate implies that the multiplicity of the eigenvalues is uniformly bounded by M + N , since (2.2) implies λ ≤ n ≤ λ . n−N +1 n+M The above estimate is enough for our purposes, but it may be improved further taking into account the structure of the graph [17]. Another implication of (2.2) is the well-known Weyl asymptotics: st λ (L ) lim =1. (2.3) n→∞ (πn/L) 40 Page 6 of 24 J. Boman et al. IEOT We shall also need the following uniform bound on the amplitude of eigenfunctions of the Laplacian on ﬁnite metric graphs: ψ ≤ c(Γ) ψ , (2.4) L L ∞ 2 where the constant c(Γ) is determined by the graph Γ but is independent of n. One can ﬁnd explicit proofs of all the above mentioned formulas (2.2), (2.3), and (2.4) in the “Appendix”. 3. Deﬁnition of the Operator The following well-known Sobolev estimate (a special case of Gagliardo- Nirenberg estimate) is valid for any function ψ ∈ W (0,) on an interval of ﬁnite length < ∞ (see proof in the “Appendix”) 2 2 2 ψ ≤ ψ + ψ (3.1) L L L ∞ 2 2 for suﬃciently small > 0. The constants, though suﬃcient for our needs, may be improved if one is willing to sacriﬁce some elegance in the proof. In particular any function ψ ∈ W (Γ \ V ) will satisfy the estimate (3.1) on each edge E of Γ, with = . As the set of edges is ﬁnite there is an n n edge that has minimal length so we obtain a global estimate of |ψ(x)| on Γ: Corollary 3.1. Let Γ be a ﬁnite compact graph and ψ ∈ W (Γ).Then 2 2 2 ψ ≤ ψ + ψ (3.2) L (Γ) L (Γ) L (Γ) ∞ 2 2 for suﬃciently small > 0 and x ∈ Γ. Proposition 3.2. Let q ∈ L (Γ),and let Q be the quadratic form given by 1 q Q (ψ, ψ)= q(x)|ψ(x)| dx, st st and Q the quadratic form associated with L . Then for suﬃciently small L 0 > 0 there exists b() such that |Q (ψ, ψ)|≤ Q st (ψ, ψ)+ b()(ψ, ψ). q L In other words Q is inﬁnitesimally bounded by Q st . Proof. Corollary 3.1 shows that for suﬃciently small > 0 2 2 2 |ψ(x)| ≤ ψ + ψ . L (Γ) L (Γ) 2 2 Multiplying by q and integrating we obtain 2 2 q(x)|ψ(x)| dx ≤ |q(x)||ψ(x)| dx Γ Γ 2 2 ≤ q ψ + q ψ L (Γ) L (Γ) 1 L (Γ) 1 L (Γ) 2 2 = q Q st (ψ, ψ)+ q ψ . L (Γ) L L (Γ) 1 1 L (Γ) 0 2 2 2 Replacing with /q ,wemay chose b()= q . L (Γ) L (Γ) 1 IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 7 of 24 40 The KLMN theorem [32] now lets us conclude that there is a unique bounded from below self-adjoint operator associated with the form Q st +Q : L q st Deﬁnition 3.1. For q ∈ L (Γ) we denote by L (Γ) the operator associated with the form Q st + Q . L q st st We note that the form domains of L and L coincide, and q 0 st st (L ψ, φ)=(L ψ, φ)+ q(x)ψ(x)φ(x)dx, q 0 for all ψ, φ ∈ Dom(Q ) = Dom(Q st ). L L 4. Spectral Estimates We recall the following standard variational theorems, see e.g. [32] for proofs. The lowest eigenvalue will be denoted by λ . Proposition 4.1. (Min-Max). Let A be a self-adjoint, bounded from below, th operator with discrete spectrum, then the n eigenvalue of A is given by λ (A) = min max Q (u, u), n−1 A V u∈V n n u =1 where V ranges over all n-dimensional subspaces of Dom(Q ), the domain n A of the quadratic form Q associated with A. Proposition 4.2. (Max-Min). Let A be a self-adjoint bounded from below, op- th erator with discrete spectrum, then the n eigenvalue of A is given by λ (A) = max min Q (u, u), n−1 A V u⊥V n−1 n−1 u =1 where V ranges over all (n − 1)-dimensional subspaces of Dom(Q ), the n−1 A domain of the quadratic form Q associated with A. In order to apply Propositions 4.1 and 4.2 as we do in the following it st is of course required that the spectrum of L (Γ) is discrete. For Schr¨ odinger operators with L potentials on ﬁnite intervals this is well known, so it is true for ﬁnite metric graphs as well since these are just ﬁnite-rank perturbations— in the resolvent sense—of the Dirichlet Schr¨ odinger operators (deﬁned by Dirichlet conditions at all vertices), which is nothing else than an orthogonal sum of Dirichlet Schr¨ odinger operators on a collection of ﬁnite intervals. We now proceed to prove the spectral estimate for ﬁnite compact graphs, i.e. we show that the diﬀerence between the Laplace and the Schr¨ odinger eigenvalues is uniformly bounded: Theorem 4.3. Let Γ be a ﬁnite compact metric graph, and let q ∈ L (Γ). st st Then the diﬀerence between the eigenvalues λ (L ) and λ (L ) is bounded n n 0 q by a constant, i.e. st st |λ (L ) − λ (L )|≤ C, (4.1) n n 0 q where C = C(Γ, q ) is independent of n. L (Γ) 1 40 Page 8 of 24 J. Boman et al. IEOT Similar questions have been discussed in [11] for the case of equilateral metric graphs. We are going to prove the Theorem using Propositions 4.1 and 4.2.To st illustrate the strategy let us ﬁrst try to derive an upper estimate for λ (L ) using a more naive approach. The quadratic form is 2 2 Q st (u, u)= |u (x)| dx + q(x)|u(x)| dx. Γ Γ It can be estimated from above by 2 2 st st Q (u, u) ≤ Q (u, u)= |u (x)| dx + q (x)|u(x)| dx, (4.2) L L + q q Γ Γ where q is the positive part of the potential q: q(x)= q (x) − q (x),q (x) ≥ 0. (4.3) + − ± This step cannot be improved much, since the new estimate coincides with the original one in the case where q is nonnegative. The idea how to proceed is to choose a particular n-dimensional sub- 0 st space V . Then the Rayleigh quotient gives not an exact value for λ (L ), n−1 n q but an upper estimate when Proposition 4.1 is used Q st (u, u) Q st (u, u) L L q q st λ (L ) = min max ≤ max . n−1 2 2 V u∈V n n u u∈V u L L 2 2 The only reasonable candidate for V we have at hand is the linear span of the Laplacian eigenfunctions corresponding to the n lowest eigenvalues st st st L L L 0 0 0 V = L ψ ,ψ ,...,ψ . (4.4) 0 1 n−1 If q ≡ 0 then this estimate gives the exact value for λ . Therefore it is n−1 natural to split the quadratic form as follows: 2 2 st Q (u, u) L |u (x)| dx q (x)|u(x)| dx st Γ Γ λ (L ) ≤ max ≤ max + max . n−1 2 2 2 0 0 0 u∈V u u∈V u u∈V u n n n L L L 2 2 2 st Then the ﬁrst quotient is equal to λ (L ) and the maximum is attained n−1 on st u = ψ . n−1 If nothing about q is known, then to estimate the second quotient one may use q (x)|u(x)| dx ≤q max |u(x)| . (4.5) + + L (Γ) x∈Γ Γ IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 9 of 24 40 st st L L n−1 2 0 0 We need to estimate |u(x)| , provided u = α ψ . Since max |ψ (x)| j=1 j j ≤ c (formula (2.4)) we obtain with the Schwarz inequality n−1 n−1 st max |u(x)|≤ |α | max |ψ (x)|≤ c |α | j j x∈Γ 0 0 1/2 n−1 √ √ ≤ c n |α | = c nu . j L Hence q (x)|u(x)| dx ≤q c n, + L and st 2 2 Q st (u, u) ≤ λ (L )+ c nq u , (4.6) L n−1 0 + L1 L which implies st st 2 λ (L ) − λ (L ) ≤q c n, (4.7) n−1 n−1 + L q 0 1 i.e. we do not get an estimate uniform in n—the estimate grows linearly with st n. The reason is the splitting of the quadratic form of L into two parts. To obtain the upper bounds we used two intrinsically diﬀerent vectors: the ﬁrst st term is maximized if u = ψ , while to estimate the second term we used n−1 st st st L L L 0 0 0 u = ψ + ψ + ··· + ψ . This is the reason that the estimate (4.7)is 0 1 n−1 not optimal. ProofofTheorem 4.3. We divide the proof into two parts deriving upper and lower estimates separately. Upper Estimate As before we use the estimate 2 2 |u (x)| dx + q (x)|u(x)| dx st Γ Γ λ (L ) ≤ max , (4.8) n−1 u∈V u st n−1 L 0 0 0 where V is deﬁned by (4.4). Every function u = α ψ from V can n j=0 j n be written as a sum u = u + u , where 1 2 st st st L L L 0 0 0 u := α ψ + α ψ + ··· + α ψ , 1 0 1 n−p−1 0 1 n−p−1 st st st L L L 0 0 0 u := α ψ + α ψ + ··· + α ψ . 2 n−p n−p+1 n−1 n−p n−p+1 n−1 Here p is a natural number to be ﬁxed later (independent of n, but depending on Γ and q). Therefore as n increases the ﬁrst function u will contain an increasing number of terms, while the second function will always be given by a sum of p terms. 2 2 2 From the inequality |u + u | dx ≤ 2 |u | dx +2 |u | dx and the 1 2 1 2 fact that q is nonnegative we have 2 2 2 q (x)|u (x)+ u (x)| dx ≤ 2 q (x)|u (x)| dx +2 q (x)|u (x)| dx. + 1 2 + 1 + 2 Γ Γ Γ (4.9) 40 Page 10 of 24 J. Boman et al. IEOT That u ,u are orthogonal is clear, and from this the orthogonality of u and 1 2 u also follows: n−p−1 st st 0 (u ,u )= −(u ,u )= − λ (L )(α ψ u ,u )=0. 2 i i 1 2 1 2 1 0 i=0 Taking this into account we arrive at 2 2 st Q (u, u) ≤ |u (x)| dx +2 q (x)|u (x)| dx L + 1 Γ Γ =: Q st (u ,u ) 1 1 2q (4.10) 2 2 + |u (x)| dx +2 q (x)|u (x)| dx . + 2 Γ Γ =: Q st (u ,u ) L 2 2 2q To estimate the ﬁrst form we use (4.5) and the Sobolev estimate (3.2)for max |u(x)|.Weget 2 2 st Q (u ,u )= |u (x)| dx +2 q (x)|u (x)| dx 1 1 + 1 L 1 Γ Γ 2q 2 2 ≤u +2q max |u (x)| + L x∈Γ 1 1 L 1 2 2 2 2 ≤u +2q (u + u ) + L 1 1 L 1 1 L L 2 2 2 2 4 2 =(1 + 2q )u + q u . + L + L 1 1 1 L 1 L 2 2 Using n−p−1 2 st u =(u ,u )= −(u ,u )= λ (L )(α ψ ,α ψ ) 1 j j j j j 1 L 1 1 1 0 j=0 (4.11) n−p−1 n−p−1 st 2 st 2 st 2 = λ (L )|α | ≤ λ (L ) |α | = λ (L )u , j 0 j n−p−1 0 j n−p−1 0 1 L j=0 j=0 we get 2 2 Q st (u ,u ) ≤ (1 + 2q )u + q u L 1 1 + L + L 1 1 1 L 1 L 2q 2 2 st 4 2 ≤ (1 + 2q )λ (L )+ q u . + L n−p−1 + L 1 1 0 1 L The key point is that and p canbechoseninsuchaway that st st (1 + 2q )λ (L )+ q <λ (L ) + L n−p−1 + L n−1 1 0 1 0 holds (see (4.14) below). On the other hand, our naive approach (4.6) can be applied to the second form with the only diﬀerence being that the number of eigenfunctions involved is p,not n st 2 2 Q st (u ,u ) ≤ λ (L )+2c pq u . (4.12) L 2 2 n−1 + L 2 0 1 L 2q 2 + IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 11 of 24 40 Putting together the obtained estimates in (4.10) and using that u = 2 2 u −u we get L L 2 2 st 4 2 Q st (u, u) ≤ (1 + 2q )λ (L )+ q u + L n−p−1 + L 1 L 1 0 1 q 2 st 2 2 + λ (L )+2c pq u n−1 + L 2 0 1 L st 2 2 2 ≤ λ (L )u +2c pq pu n−1 + L 0 L 1 L 2 2 st st 2 − λ (L ) − (1 + 2q )λ (L ) − q u . n−1 + L n−p−1 + L 1 0 1 0 1 L We would get the desired estimate st Q (u, u) st st λ (L ) ≤ max ≤ λ (L )+ C (4.13) n−1 n−1 q 0 u∈V u with C =2c pq if we manage to prove that + L st st λ (L ) − (1 + 2q )λ (L ) − q > 0 (4.14) n−1 + L n−p−1 + L 0 1 0 1 for a certain that may depend on n and p. We use the estimate for Laplacian eigenvalues given in (2.2): 2 2 π π 2 st 2 (n − M ) ≤ λ (L ) ≤ (n + N − 1) . (4.15) L L st st Substituting λ (L ) with the lower bound and λ (L ) with the n−1 n−p−1 0 0 upper and setting =1/n, we get the following inequality for the left-hand side of (4.14) st st λ (L ) − (1 + 2/nq )λ (L ) − 4nq n−1 + L n−p−1 + L 0 1 0 1 ≥ (n − 1 − M ) − (1 + 2/nq ) + L × (n − 1 − p + N − 1) − 4nq + L 2 2 π L =2n p − M − N +1 − 1+2 q + O(1). + L L π We see that for any integer p>M + N − 1+(1 +2(L/π) )q the + L expression is positive for suﬃciently large n and the diﬀerence between the eigenvalues possesses the uniform upper estimate: st st λ (L ) − λ (L ) ≤ C. (4.16) n n q 0 If one is interested in the diﬀerence between the eigenvalues for large n only, then the constant C can be taken equal to 2 2 C =2c q (M + N − 1+(1 +2(L/π) )q ), L L 1 1 but this value of C may be too small in order to ensure that (4.16) holds for all n, since proving (4.14) we assumed that n is suﬃciently large. The latter assumption does not aﬀect the ﬁnal result, since for a ﬁnite number of eigenvalues (4.16) is always satisﬁed, but the value of the constant C may be aﬀected. 40 Page 12 of 24 J. Boman et al. IEOT Lower Estimate To obtain a lower estimate we are going to use the Max-Min principle (Propo- sition 4.2). The ﬁrst step is to notice that 2 2 Q st (u, u) ≥ |u (x)| dx − q (x)|u(x)| dx. (4.17) L − Γ Γ Using the subspace V deﬁned in (4.4)weget n−1 st Q (u, u) st λ (L ) ≥ min . n−1 u⊥V u n−1 Since u is orthogonal to V it possesses the representation n−1 st u = α ψ . j=n−1 As before let us split the function u = u + u 1 2 st st st L L L 0 0 0 u := α ψ + α ψ + ··· + α ψ , 1 n−1 n n n+p−2 n−1 n+p−2 st st L L 0 0 u := α ψ + α ψ + .... 2 n+p−1 n+p n+p−1 n+p Note two important diﬀerences: • the function u is given by the sum of p terms, where the number p in- dependent of n will be chosen later, so the functions u and u exchange 1 2 roles compared with the proof of the upper estimate; • the function u is given by an inﬁnite series, not by an increasing number of terms as the function u in the proof of upper estimate. Using the fact that q is nonnegative we may split the quadratic form (com- pare (4.10)) 2 2 Q st (u, u) ≥ |u (x)| dx − 2 q (x)|u (x)| dx L − 1 Γ Γ st =: Q (u ,u ) 1 1 −2q 2 2 + |u (x)| dx − 2 q (x)|u (x)| dx . (4.18) − 2 Γ Γ =: Q st (u ,u ) L 2 2 −2q Now the function u is given by a ﬁnite number of terms and we may similarly to (4.12) estimate st 2 2 Q st (u ,u ) ≥ λ (L ) − 2c pq u . (4.19) L 1 1 n−1 − L (Γ) 1 0 1 L −2q 2 To estimate the second form we use (4.5) and the Sobolev estimate (3.2)for max |u(x)| .Weget 2 2 Q st (u ,u ) ≥u − 2q max |u (x)| L 2 2 − L 2 2 L 1 −2q 2 2 2 2 ≥u − 2q u + u − L 2 2 L 1 2 L L 2 2 2 4q − L 2 2 =(1 − 2q )u − u . − L 2 1 2 L L 2 2 IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 13 of 24 40 Taking into account 2 st u =(u ,u )= −(u ,u )= λ (L )(α ψ ,α ψ ) 2 j j j j j 2 L 2 2 2 0 j=n+p−1 ∞ ∞ (4.20) st 2 st 2 = λ (L )|α | ≥ λ (L ) |α | j j n+p−1 j 0 0 j=n+p−1 j=n+p−1 st 2 = λ (L )u , n+p−1 2 0 L we arrive at Q st (u ,u ) L 2 2 −2q 4q − L (Γ) st 1 2 ≥ (1 − 2q )λ (L ) − u . (4.21) − L (Γ) n+p−1 2 1 0 L Summing the estimates (4.19) and (4.21) and taking into account that 2 2 2 u = u −u we get 2 1 L L L 2 2 2 st 2 2 st Q (u, u) ≥ λ (L ) − 2c pq u L n−1 − L 1 0 1 L q 2 4q − L st 1 2 + (1 − 2q )λ (L ) − u − L n+p−1 2 1 0 L st 2 2 2 ≥ λ (L )u − 2c pq u n−1 − L 0 L 1 L 2 2 4q − L st 1 st 2 + (1 − 2q )λ (L ) − − λ (L ) u . − L n+p−1 n−1 2 1 0 0 L As before, to prove the desired uniform estimate it is suﬃcient to show that for large enough n the following expression can be made positive by choosing an appropriate : 4q − L st 1 st (1 − 2q )λ (L ) − − λ (L ) > 0. (4.22) − L n+p−1 n−1 1 0 0 st Again we use (4.15): we substitute λ (L ) with the lower bound and n+p−1 st λ (L ) with the upper. As before we choose =1/n, so the left-hand side n−1 of (4.22) becomes st st (1 − 2q )λ (L ) − q − λ (L ) − L n+p−1 − L n−1 1 0 1 0 ≥ (1 − 2q /n) (n + p − 1 − M ) − L −4nq − (n − 1+ N − 1) − L π L =2n p − M − N +1 − 1+2 q + O(1). − L L π If p>M + N − 1+(1 +2(L/π) )q , then for suﬃciently large n the − L expression is positive, hence the following lower estimate holds st st λ (L ) − λ (L ) ≥ C, (4.23) n n q 0 where the exact value of C is determined by the diﬀerence between the ﬁrst few eigenvalues as described above. 40 Page 14 of 24 J. Boman et al. IEOT Theorem 4.3 allows us to conclude that the spectra of Schr¨ odinger op- erators satisfy Weyl asymptotics as well: st Corollary 4.4. Let q ∈ L (Γ), then λ (L (Γ)) satisﬁes Weyl asymptotics. 1 n Proof. This is an immediate consequence of (2.3) and Theorem 4.3. More importantly we may now show that the eﬀect of an L -perturbation on the eigenvalues will tend to zero in n in the scale of square roots. This step is critical in the proof of Theorem 6.5. st 2 st 2 st st Corollary 4.5. Let λ (L )= k and λ (L )= k .If |λ (L )−λ (L )|≤ n n n n q n,q 0 n,0 q 0 C ∈ R then |k − k |≤ , for some constant C ∈ R. In particular, n,q n,0 0 |k − k |→ 0,n →∞. (4.24) n,q n,0 Proof. Since the eigenvalues satisfy Weyl asymptotics that depend only on the length of Γ we have that k ,k ≥ nD for some constant D and suﬃ- n,q n,0 ciently large n.Wehave 2 2 k −k (k +k )(k −k ) n,q n,0 n,q n,0 n,q n,0 |k − k | = = n,q n,0 k +k k +k n,q n,0 n,q n,0 st st λ (L )−λ (L ) n n q 0 C C 0 = ≤ ≤ , k +k |k +k | n n,q n,0 n,q n,0 for some constant C ∈ R. 5. On the Zeros of Trigonometric Polynomials st In this section we recall the secular equation for the spectrum σ(L (Γ)): it is given as the squares of the zeros of a trigonometric polynomial. We then prove that if the zeros k of such a ﬁnite trigonometric polynomial with constant coeﬃcients are close to a certain equispaced sequence, i.e. satisfy |k − mπ/L| → 0 then in fact k = mπ/L for all m (Theorem 5.2). From m m this we then prove the geometric Ambartsumian Theorem 6.5. st Theorem 5.1. Let Γ be a ﬁnite compact metric graph. The eigenvalues λ (L ) are given by the squares of the zeros of a certain trigonometric polynomial iω k p(k)= a e n=1 st 2 with k-independent coeﬃcients a ∈ C and ω ∈ R: λ (L )= k if and n n n 0 n,0 only if p(k )=0. n,0 Proof. We sketch the proof: for each non-zero eigenvalue the corresponding eigenfunction is edge-wise just a sum of sine and cosine functions: ψ(x)= a cos(kx)+ b sin(kx),x ∈ E . The solutions have to satisfy the vertex j j j conditions. Continuity can at each V be written as a cos kx + b sin kx = a cos ky + b sin ky, i i j j IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 15 of 24 40 if x ∈ E , y ∈ E ,and x, y ∈ V . This yields 2N − M equations, where N is i j the number of edges and M is the number of vertices. The conditions on the normal derivatives can at each vertex V be written as (−1) −a sin kx + b cos kx =0, [(j+1)/2] j [(j+1)/2] j x ∈V where k has been factored out. This yields an additional M equations, so that we in total have 2N equations, which may be written in the form: T (k) c =0, with c a vector of the coeﬃcients a ,b and T (k) a matrix with trigonometric i i 2 st entries depending on k.Areal number λ = k is an eigenvalue of L if and only if k is a root of the trigonometric polynomial det T (k)=0. We refer to [6, 18] for details. See also eg. [15] and [26]. Note that it is crucial that the vertex conditions were standard: in the case of more general vertex conditions the secular equation is given by a quasipolynomial instead of the trigonometric polynomial. Theorem 5.2. Let f be the trigonometric polynomial iω k p(k)= a e (5.1) j=1 with all ω ∈ R, a ∈ C. If the zeros k of f satisfy j j m lim (k − m)=0 m→∞ then k = m for all m. First we need a Lemma: Lemma 5.3. Given ω ,...,ω ∈ R there exists a subsequence {m } of the 1 J n natural numbers such that, for each ω , iω m j n e → 1. Proof. For ω := (ω ,...,ω ) ∈ R let [ω ]:=([ω ],..., [ω ]) denote the 1 J 1 J J J image of ω under the standard projection to the J -torus: R → (R/2πZ) . The statement of the Lemma is equivalent to the existence of an increasing sequence of integers m such that [m ω] → 0:=(0,..., 0) ∈ (R/2πZ) . n n Consider the set of points m ω, with m ∈ N and its projection [m ω]= ([mω ],..., [mω ]). Since the J -torus is compact this set has a limit point 1 J z and an increasing subsequence (m ) ⊂ N such that [m ω ] → z. This is i i a Cauchy sequence so for any > 0 there exists I() such that for any i ,i ≥ I() 1 2 d([n ω ], [n ω ]) <, i i 1 2 where d(·, ·) denotes the metric on the J -torus. Taking a sequence → 0 we may chose i ( )= I( ) and in each step i ( ) > I( ) so large that the 1 i i 2 i i diﬀerence m := n − n , i i ( ) i ( ) 2 i 1 i 40 Page 16 of 24 J. Boman et al. IEOT is an increasing sequence. It follows that [m ω ]=[(n − n )ω ] → 0. i i ( ) i ( ) 2 i 1 i Remark 5.1. Note that in the special case where 2π, ω ,...,ω are rationally 1 J independent the classical theorem of Kronecker (see e.g. [16]) can be used. ProofofTheorem 5.2. Consider the trigonometric polynomial p(k)in(5.1). Denote k − m =: γ so that γ tends to zero as m →∞.Wehave k = m m m m m + γ so for each ω : m j iω k iω m iω γ j m j j m e = e e . Choose a subsequence {m } as described in Lemma 5.3 and pass, for an arbitrary r ∈ N,tothe (r + m ):th zero of p.Wehave iω (r+m ) iω γ j n j (r+m ) 0= p(k )= a e e r+m j j=1 J J iω r iω m iω γ iω r j j n j (r+m ) j = a e e e → a e = p(r), j j j=1 j=1 as n →∞. The limit follows from the choice of m and the fact that γ n (r+m ) tends to 0. The above calculation shows that p(r)=0. But p(r)=0and k − r → 0 together imply that k = r, since even a single extra zero would make the asymptotic behaviour impossible. Remark 5.2. It is not important in the above theorem that k tends to the integers. A scaling argument allows one to extend it to the case where k are close to integer-multiples of an arbitrary real number. 6. An Ambartsumian Theorem With the result of the previous two sections we are now in a position to prove an inverse spectral theorem that may be seen as a generalization of Ambartsumian’s classical theorem. For the proof we recall the classical result as well as its geometric version for Laplacians. The following theorem has been a source of inspiration for researchers in inverse problems for almost a century. In the original article [1]itwas assumed that the potential is continuous, but the result holds even if q ∈ L . We adjusted the formulation to our notations. Proposition 6.1. (Ambartsumian’s theorem [1]) Let q be a real-valued abso- lutely integrable function on an interval I. Then the spectrum of the stan- st dard Schr¨ odinger operator L (I) coincides with the spectrum of the standard st Laplacian L (I) if and only if the potential q is identically equal to zero. Standard conditions on a single compact interval is of course just the classical Neumann conditions at both end-points. It appears that the theorem is still valid if instead of the interval I we have arbitrary connected ﬁnite com- pact metric graph Γ. This result was proven step-by-step by several authors IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 17 of 24 40 [10, 27, 29, 36] and [30], but the most general version was given by E.B. Davies [12] (Davies proved it for q ∈ L (Γ) but noted that this condition surely can be weakened). st st Proposition 6.2. (following [12]) Let L (Γ) and L (Γ) be the standard Schr¨ o- q 0 dinger and Laplace operators on a connected ﬁnite compact metric graph Γ. Assume that the potential q is absolutely integrable. Then the eigenvalues of the two operators coincide if and only if the potential q is equaltozeroalmost everywhere. The second theorem is a geometric version of Ambartsumian theorem for standard Laplacians. We start by recalling the result that among graphs with ﬁxed total length the spectral gap is minimized by the single interval [13, 28] and [25]: st Proposition 6.3. (Theorem 3 from [25]) Let L (Γ) be the standard Laplace operator on a connected ﬁnite compact metric graph Γ of total length L(Γ). st Assume that the ﬁrst (nonzero) eigenvalue of L (Γ) coincides with the ﬁrst (nonzero) eigenvalue of the Laplacian on the interval I of length L(Γ) st st λ (L (Γ)) = λ (L (I)); 1 1 0 0 then the graph Γ coincides with the interval I. The multiplicity of the eigenvalue zero for the standard Laplacian is equal to the number of connected components in the metric graph and the asymptotics of the spectrum determines the total length of the graph. Hence the above proposition implies: Theorem 6.4. Let Γ be a ﬁnite compact metric graph. The spectrum of the standard Laplacian on Γ coincides with the spectrum of the standard Lapla- cian on the interval I st st λ (L (Γ)) = λ (L (I)),j =0, 1, 2,... , (6.1) j j 0 0 if and only if the graph Γ coincides with the interval I. The assumptions of the theorem can be weakened, since to ensure that Γand I have the same total length it is enough to check the asymptiotics. Our goal is to prove that if the spectrum of a Schr¨ odinger operator on a metric graph coincides with the spectrum of the Laplacian on an interval then the graph coincides with the interval and the potential is zero. This statement cannot be proven as a simple combination of the above mentioned results (Proposition 6.2 and Theorem 6.4). The main diﬃculty is to show that the graph coincides with the interval. Theorem 6.4 cannot be applied directly, since it requires q ≡ 0. Theorem 6.5. Let Γ be a ﬁnite compact metric graph and q ∈ L (Γ).The st spectrum of the standard Schr¨ odinger operator L (Γ) coincides with the spec- trum of the standard (i.e. Neumann) Laplacian on an interval st st λ (L (Γ)) = λ (L (I)), (6.2) j j q 0 if and only if Γ= I and q ≡ 0. 40 Page 18 of 24 J. Boman et al. IEOT Proof. Equation (6.2) implies that the total length of the graph Γ coincides with the length of the interval I. To see this it is enough to compare the corresponding asymptotics. Then the spectrum of the Laplacian on I is πn st λ (L (I)) = ,n ∈ N. The spectral estimate (4.1) implies that st st λ (L (Γ)) −λ (L (Γ) = O(1). n n q 0 πn st Hence the square roots of the eigenvalues of L (Γ) satisfy πn st k (L (Γ)) − → 0, as n →∞. (6.3) But k are given as the zeros of a trigonometric polynomial (Theorem 5.1). Hence the spectrum of the Laplacian on Γ given by zeroes of the trigono- metric polynomial is asymptotically close to a set of equidistant points and st Theorem 5.2 can be applied. We conclude that in fact k (L (Γ)) = πn/L,for all n. This means that the spectrum of the Laplacian on Γ coincides with the spectrum of the Laplacian on an intreval. We may ﬁnally apply the geometric version of Ambartsumian theorem for Laplacians (Theorem 6.4) to conclude that the graph Γ coincides with the interval I. But then Ambartsumian’s classical theorem (Proposition 6.1) implies that the potential is identically equal to zero q ≡ 0. Proving our main theorem we have shown that in order to ensure that the graph Γ coincides with the interval, it is enough to require that the spectrum of the Schr¨ odinger operator is close to the spectrum of the Laplacian on an interval. In fact we proved the following theorem: st Theorem 6.6. Let L (Γ) be standard Schr¨ odinger operator with L -potential on a ﬁnite compact connected metric graph Γ.Then Γ= I if st st λ (L (Γ)) − λ (L (I)) → 0,n →∞. (6.4) n n q 0 7. Euler Characteristic The spectral estimate (4.1) allows us to extend a previous result of one of the authors regarding the Euler characteristic χ = M − N . Theorem 7.1. Let Γ be a ﬁnite compact metric graph and q ∈ L (Γ). Then the st Euler characteristic χ(Γ) is uniquely determined by the spectrum σ(L (Γ)), and can be calculated as the limit ⎛ ⎞ st sin λ (L )/2t st ⎝ ⎠ χ(Γ) = 2 lim cos λ (L )/t , (7.1) t→∞ st λ (L )/2t n=0 n q IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 19 of 24 40 with the convention that λ =0 implies st sin λ (L )/2t =1. (7.2) st λ (L )/2t Proof. In [19] formula (7.1) was proven for the case of zero potential. In other words formula (7.1) gives the Euler characteristics if one substitutes the st st eigenvalues λ (L ). We have shown that the diﬀerence between λ (L (Γ)) n n 0 q st and λ (L (Γ)) is uniformly bounded and therefore |k − k | = O . (7.3) Taking into account Weyls asymptotics k = n + O(1), (7.4) one may apply Lemma 2 from [20] to conclude that formula (7.1) gives the same result independently of whether the eigenvalues of the standard Lapla- st st cian L (Γ) or of the standard Schr¨ odinger operator L (Γ) are 0 q substituted. Note that proving the theorem we assumed that the vertex conditions are standard; this assumption in general cannot be removed. 8. Discussion It would be interesting to understand under which assumptions a similar result holds for other than the standard vertex conditions that are assumed throughout in this paper. The cases, where the vertex conditions are such that the coeﬃcients of the trigonometric polynomial given by Theorem 5.1 are k independent, could potentially be dealt with in a similar way as in this paper. The bounds on the diﬀerences of the eigenvalues obtained in Theorem 4.3 can be established for other choices of vertex conditions, but Theorem 6.4 in general may fail. Acknowledgements The authors would like to thank G. Berkolaiko for pointing out an explicit improvement leading to estimate (A.4). Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4. 0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, pro- vide a link to the Creative Commons license, and indicate if changes were made. 40 Page 20 of 24 J. Boman et al. IEOT Appendix A: Elementary spectral properties In this appendix we provide proofs for the elementary spectral properties of quantum graphs already mentioned in Sect. 2. Most of these results are well-known, but we need them for the sake of completeness. Proofofformula (2.2). For a bounded from below self-adjoint operator A with discrete spectrum, deﬁne the eigenvalue counting function E : R → N, by E (λ)=#{λ ∈ σ(A) | λ ≤ λ}. A j j The standard Laplacian is positive (hence the lower estimate in (2.2)isin- teresting only if n>M ) therefore when calculating the eigenvalue counting function we assume that λ ≥ 0. Consider ﬁrst the Laplace operator L on a single interval I of length πn with Dirichlet conditions at the end-points. The eigenvalues are λ = , n =1, 2,... So the eigenvalue counting function for λ ≥ 0 is in fact given by E D (λ)= , L (I) where square-brackets denote the integer-part of the argument. Returning to Γ we note that if we impose Dirichlet conditions on the vertices of Γ—denote the operator by L (Γ)—then we really just have a decoupled set of intervals and therefore the set of eigenvalues is just the union of the eigenvalues for each interval (counting multiplicities). Therefore the corresponding counting function E D is given by L (Γ) E D (λ)= E D (λ) L (Γ) L (E ) 0 0 n n=1 √ √ √ √ λ λ λ λ = + + ··· + ≤ L , (A.1) 1 2 N π π π π since taking integer parts may only decrease the value. First adding and then taking integer parts may—compared to adding the integer parts—at most raise the value by the number of terms −1, i.e. the number of edges −1, so conversely we also have L − N +1 ≤ E D (λ). (A.2) L (Γ) Formulas (A.1) and (A.2) give eﬀective two-sided bounds for the eigenvalues of the Dirichlet Laplacian on Γ. D −1 st −1 We now show that (L (Γ) − λ) − (L (Γ) − λ) is of ﬁnite rank. Take 0 0 λ = 0, and suppose that D st (L − λ)u = f, (L − λ)u = f. D st 0 0 2 2 Then for the diﬀerential operator −d /dx we have that − − λ (u − u )=0. D st dx IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 21 of 24 40 D st Since functions in the domain of L and L are continuous, to determine the 0 0 rank of the resolvent diﬀerence, we need to determine dim ker − − λ on dx continuous functions. Prescribing values u at each vertex V in Γ, we see j j that a unique solution to this boundary value problem is given as follows: on an edge E =[x ,x ] between V and V ,set n 2n−1 2n i j sin k(x − x ) sin k(x − x ) 2n 2n−1 u(x):= u + u i j sin k(x − x ) sin k(x − x ) 2n−1 2n 2n 2n−1 2 2 2 for k = λ. Then u is continous on Γ and solves (−d /dx − λ)u =0, so 2 2 dim ker(−d /dx − λ) ≤ M . Therefore we have E st (λ) ≤ E D (λ)+ M ≤ L + M. (A.3) L (Γ) L (Γ) The lower estimate estimate (A.2) can be modiﬁed in a similar way, but st D instead we shall take into account that L (Γ) ≤ L (Γ). Really Dirichlet con- 0 0 ditions in particular imply the continuity of functions in the domain of the quadratic form. Passing to standard conditions means weakening the con- ditions on functions in the domain of the quadratic form, since now only continuity is required at the vertices. Therefore the domain of the quadratic form Q st is larger than that of Q D , so by the Min-Max principle (see Propo- 0 0 sition 4.1) eigenvalues can only go up when imposing Dirichlet conditions. In particular, the lower bound (A.2) on the eigenvalue counting function E D L (Γ) is also valid for E st . L (Γ) Putting the lower and upper estimates together we have √ √ λ λ st L − N +1 ≤ E (λ) ≤ L + M (A.4) L (Γ) π π Setting λ = n we obtain n − N +1 ≤ E st n ≤ n + M, L (Γ) 0 2 so λ ≤ n ≤ λ . n−N +1 n+M π 2 Setting n = n + M we get λ ≥ (n − M ) and similarly we ﬁnd λ ≤ n 2 n π 2 (n + N − 1) , which proves the theorem. Proofofformula (2.3). The Weyl asymptotics follow from the relation 2 2 π π 2 st 2 (n − M ) ≤ λ (L ) ≤ (n + N − 1) . L L Proofofformula (2.4). For ψ corresponding to λ = k we have 1 k 2π 2 2 ψ ≥ |ψ(x)| dx ≥ max |ψ(x)| , L (Γ) x∈E n 2 2π k where [·] denotes the integer part of the argument. [ k/2π] may be equal to zero for only ﬁnitely many k since the eigenvalues satisfy Weyl asymptotics. 40 Page 22 of 24 J. Boman et al. IEOT Since [ k/2π]/k is bounded for k ∈ R this implies the existence of a k- independent bound c(Γ). Proofofformula (3.1). Let x denote a global minimum for ψ. Then in min 2 2 particular |ψ(x )| ≤ψ /. We then have min 2 2 |ψ(x)| = |ψ(x )| +2 ψ(y)ψ (y)dy min min ≤|ψ(x )| +2 |ψ(y)ψ (y)|dy min 2 2 2 ≤|ψ(x )| + |ψ (y)| dy + |ψ(y)| dy min 0 0 2 2 2 ≤ψ / + ψ + ψ 2 2 2 1 1 2 2 = ψ + + ψ . 2 2 For < we have 1/ > 1/ and the claim follows. References [1] Ambartsumian, V.: Uber eine Frage der Eigenwerttheorie. Z. Phys. 53, 690–695 (1929) [2] Avdonin, S., Kurasov, P.: Inverse problems for quantum trees. Inverse Probl. Imaging 2, 1–21 (2008) [3] Avdonin, S., Kurasov, P., Nowaczyk, M.: Inverse problems for quantum trees II: recovering matching conditions for star graphs. Inverse Probl. 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Theory 5, 363–380 (2015) [34] Visco-Comandini, F., Mirrahimi, M., Sorine, M.: Some inverse scattering prob- lems on star-shaped graphs. J. Math. Anal. Appl. 378, 343–358 (2011) [35] Yang, C.F.: Inverse spectral problems for the Sturm-Liouville operator on a d-star graph. J. Math. Anal. Appl. 365, 742–749 (2010) 40 Page 24 of 24 J. Boman et al. IEOT [36] Yang, C.F., Huang, Z.Y., Yang, X.P.: Ambarzumyan-type theorems for the Sturm-Liouville equation on a graph. Rocky Mountain J. Math. 39, 1353–1372 (2009) [37] Yurko, V.: Inverse spectral problems for Sturm-Liouville operators on graphs. Inverse Probl. 21, 1075–1086 (2005) [38] Yurko, V.: Inverse problems for Sturm-Liouville operators on graphs with a cycle’. Oper. Matrices 2, 543–553 (2008) [39] Yurko, V.: Reconstruction of Sturm-Liouville operators from the spectra on a graph with a cycle’ (Russian), Mat. Sb. 200 (2009) 147–160; translation in Sb. Math. 200(2009), 1403–1415 [40] Yurko, V.: Uniqueness of recovering Sturm-Liouville operators on a-graphs from spectra. Results Math. 55, 199–207 (2009) [41] Yurko, V.: An inverse problem for Sturm-Liouville diﬀerential operators on A-graphs. Appl. Math. Lett. 23, 875–879 (2010) Jan Boman, Pavel Kurasov ( ) and Rune Suhr Department of Mathematics Stockholm University 106 91 Stockholm Sweden e-mail: kurasov@math.su.se Jan Boman e-mail: jabo@math.su.se Rune Suhr e-mail: suhr@math.su.se Received: January 19, 2018. Revised: April 23, 2018.
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Schrödinger Operators on Graphs and Geometry II. Spectral Estimates for
$${\varvec{L}}_\mathbf{1}$$
L
1
-potentials and an Ambartsumian Theorem

Integr. Equ. Oper. Theory (2018) 90:40 https://doi.org/10.1007/s00020-018-2467-1 Integral Equations Published online May 30, 2018 and Operator Theory c The Author(s) 2018 Schr¨ odinger Operators on Graphs and Geometry II. Spectral Estimates for L -potentials and an Ambartsumian Theorem Jan Boman, Pavel Kurasov and Rune Suhr Abstract. In this paper we study Schr¨ odinger operators with absolutely integrable potentials on metric graphs. Uniform bounds—i.e. depending th only on the graph and the potential—on the diﬀerence between the n eigenvalues of the Laplace and Schr¨ odinger operators are obtained. This in turn allows us to prove an extension of the classical Ambartsumian Theorem which was originally proven for Schr¨ odinger operators with Neumann conditions on an interval. We also extend a previous result relating the spectrum of a Schr¨ odinger operator to the Euler character- istic of the underlying metric graph. Mathematics Subject Classiﬁcation. 34L15, 35R30, 81Q10. Keywords. Quantum graphs, Spectral estimates, Ambartsumian theo- rem. 1. Introduction Quantum graphs—i.e. Schr¨ odinger operators acting on metric graphs—have become an increasingly important branch of mathematical physics in the last 20 years or so. They serve as models of branched thin networks, e.g. nanotubes, and complex molecules. Apart from applications they also serve as a rich source of objects suitable for mathematical inquiry. In particular Schr¨ odinger operators on graphs exhibit spectral properties both of partial diﬀerential and ordinary diﬀerential operators. The aim of this paper is twofold. First we work towards proving a spec- tral estimate, i.e. a comparison between the spectra of the Laplacian and Schr¨ odinger operators acting on the same metric graph. The main motiva- tion for obtaining such estimates is that the spectrum of the Laplacian is P. K. was partially supported by the Swedish Research Council (Grant D0497301) and by the Center for Interdisciplinary Research (ZiF) in Bielefeld in the framework of the cooperation group on Discrete and continuous models in the theory of networks. 40 Page 2 of 24 J. Boman et al. IEOT much easier to calculate. We will prove that just as in the case of a single interval the diﬀerence between the Laplace and Schr¨ odinger eigenvalues is uniformly bounded, provided the potential is just absolutely integrable. The classical proof of this relied heavily on the explicit formula for the resolvent kernel of the Laplacian [7]. The corresponding kernel for metric graphs is not in general given explicitly and we will instead work with general per- turbation theory. The second goal is to extend results—relating the Laplace spectrum to geometric properties of the graph—to Schr¨ odinger operators with L -potentials. In particular we obtain an inverse spectral theorem that generalizes the celebrated theorem of Ambartsumian (see Sect. 6), and show that the Euler characteristic of the underlying graph is reﬂected in the spec- trum of the Schr¨ odinger operator in the case of standard vertex conditions (see (2.1) below). 1.1. Spectral Estimates and Inverse Spectral Theory Inverse spectral theory for the Schr¨ odinger equation in R has classically had as a goal to determine the potential given a spectrum. To solve the inverse problem for a Schr¨ odinger operator on a metric graph completely one has in general to determine not only the potential but the underlying metric graph and vertex conditions. It appears that this complete inverse problem is rather diﬃcult, especially since the set of spectral data is not obvious. Therefore it seems attractive to start investigations assuming that the metric graph and the vertex conditions are ﬁxed. The simplest graph to be considered is the star graph formed by a ﬁnite number of compact edges, and the corresponding inverse problem resembles very much the inverse problem on a single interval, where the potential is determined by two spectra [14, 30, 33–35]. The case of general trees has also been studied and we have a rather good understanding of the problem [2, 8, 9, 31, 37]. The case of graphs with cycles is much more involved—major diﬃculties are related to the reconstruction of the potential on the cycles. To assure the uniqueness of the potential one may either use the dependence of the spectral data on the magnetic ﬂuxes [21, 23], or add extra spectral data like the Dirichlet spectrum [38–41]. The problem to reconstruct the metric graph has not been addressed in full generality. Topological characteristics of the graph may be reconstructed [19, 20]. Assuming that the edge lengths are rationally independent one may even reconstruct the graph using the trace formula [15, 18], but under the condition that the potential is zero. Explicit examples of isospectral graphs have been constructed [4, 5, 15]. Employing the boundary control method one may reconstruct metric trees [2] without assuming zero potential. The prob- lem of reconstructing vertex conditions, as well as the inﬂuence of vertex conditions on the solvability of the inverse problem, is even less understood [3]. For example a metric tree is not always reconstructable, e.g. if the ver- tex conditions are not standard [22]. On the other hand more general vertex conditions may help to solve the inverse problem as it is done in [23, 24]. One result in the inverse spectral theory—the Ambartsumian’s cele- brated theorem from 1929 stating that the spectrum of a (Neumann) Schr¨ o- dinger operator on an interval coincides with the spectrum of the (Neumann) IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 3 of 24 40 Laplacian if and only if the potential is zero—is of great importance (see Theorem 6.1). This theorem is rather special, since in order to reconstruct a non-zero potential, knowledge of two spectra is required. Several authors generalized this theorem for the case of metric graphs. The results can be divided into two categories. Some authors proved that if the spectrum of the standard Schr¨ odinger operator on a ﬁxed metric graph Γ coincides with the spectrum of the standard Laplacian on Γ, then the potential is zero. This result was ﬁrst proved for trees [10, 27, 29, 36] and [30], and then for arbitrary graphs by Davies [12]. It was also noted that the Laplacian on a metric graph Γ is isospectral to the Laplacian on an interval if and only if Γ is formed by a single interval. This can be seen as a geometric version of Ambartsum- ian’s theorem and it is based on the fact that a single interval maximizes the spectral gap for the Laplacian among graphs of ﬁxed total length [13, 25, 28]. The inverse spectral result that we obtain—Theorem 6.5—is that the spectrum of a ﬁnite interval is unique among connected ﬁnite compact quan- tum graphs with standard conditions. More precisely, if the spectrum of a Schr¨ odinger operator on a connected metric graph coincides with the spec- trum of the Laplacian with Neumann conditions on an interval, then the graph is the interval, and the potential of the Schr¨ odinger operator is zero. Here we assume that standard conditions are imposed on the graph. In the case of the graph being a compact interval, standard conditions coincide with Neumann conditions at the end-points. The result can therefore be seen as an extension of Ambartsumian’s theorem. 1.2. Outline of the Paper Section 2 contains some fundamentals on quantum graphs. In particular we give some elementary spectral estimates and prove that normalized eigenfunc- st tions of the Laplacian L are uniformly bounded in the L norm. Section 3 st contains the deﬁnition of the operator L that we associate with the for- st mal expression L + q,for q ∈ L (Γ). The operator is deﬁned via quadratic 0 1 form methods. Section 4 deals with spectral estimates, and we prove that there exists a uniform bound on the diﬀerence of eigenvalues of Laplace and Schr¨ odinger operators on metric graphs. This is done by using the Max– Min and Min–Max principles, along with a Sobolev estimate for functions ψ ∈ W (Γ). Section 5 gives a result on the zeros of trigonometric polynomi- als, namely that if the zeros asymptotically tend to the integers, then all the zeros are in fact exactly the integers. By combining this fact with the spectral estimate of Sect. 4 we are able to prove the inverse spectral theorem given in Sect. 6. Section 7 extends a previous result that the Euler characteristic of a graph is reﬂected in the spectrum of a Schr¨ odinger operator with L potential to the case of L potentials. 2. Preliminaries–Basics on Quantum Graphs A quantum graph is a metric graph equipped with a Schr¨ odinger operator, or more formally, a triple (Γ,L, vc) with Γ a metric graph, L a diﬀerential operator, and vc a set of vertex conditions imposed to connect the edges and 40 Page 4 of 24 J. Boman et al. IEOT ensure the self-adjointness of L. In this paper we limit ourselves to compact ﬁnite graphs (see below). We give a brief overview of these - for a thorough treatment of the theory of quantum graphs, see for example [6] and [26]. Metric Graphs: A compact ﬁnite metric graph is a ﬁnite collection of compact intervals of R glued together at the endpoints. More precisely, let, {E } be n=1 a ﬁnite set of compact intervals, each E considered as a subset of a separate copy of R, E =[x ,x ], 1 ≤ n ≤ N. n 2n−1 2n Let V = {x } = {x ,x } denote the set of endpoints of the intervals. j 2n−1 2n n=1 Fix a partition of V into equivalence classes V = V ∪ ··· ∪ V . 1 M Identifying the endpoints yields a graph with vertices given by the equivalence classes V . We let = x − x denote the length of the edge E and deﬁne the n 2n 2n−1 n total length L of a graph as the sum of its edge lengths: L = . n=1 Diﬀerential Operators on Graphs: L acts as a diﬀerential operator on each 2 2 edge separately, in our case − d /dx + q for some real potential function q—subscripts will denote the potential of the operator, and L means that the potential is identically 0, so that L is the Laplacian. The action is in the Hilbert spaces L (E ) of square integrable functions on the edges E .The 2 i i measure on each edge is given naturally by the identiﬁcation of the edge with an interval. L then acts on L (Γ) = L (E ). 2 2 i i=1 2 2 Note that if q ∈ L (Γ) then the formal expression −d /dx + q should be understood as a sum of quadratic forms (see Sect. 3). Vertex Conditions: We shall solely study quantum graphs with standard conditions—also known as Kirchoﬀ, Neumann, natural or free conditions. st Operators with standard conditions imposed will be written as L . Let us ﬁrst discuss the case of a bounded potential q ∈ L (Γ), where the domain of the Schr¨ odinger operator can be given explicitly. For any ψ ∈ L (Γ), also qψ ∈ 2 2 L (Γ), so L ψ = −(d/dx) ψ + qψ ∈ L (Γ) if and only if ψ ∈ W (Γ \ V ). Let 2 q 2 V = {x ,...,x } and let ψ(x ) denote the limit ψ(x ) = lim ψ(x), j j j j j x→x 1 k i i j where the limit is taken over x inside the interval with x as an end-point. i IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 5 of 24 40 Standard conditions are then given by imposing the following relations at each vertex V ψ(x )= ψ(x )= ··· = ψ(x ) j j j 1 2 k (2.1) ∂ ψ (x )=0, n E j x ∈V n(j) j m where ∂ denotes the normal derivative, i.e. ψ (x ) x left end-point j j ∂ ψ(x )= n j − ψ (x ) x right end-point. j j In other words, ψ is required to be continuous in each vertex V and the sum of normal derivatives should vanish there. This yields a self-adjoint operator st L on Γ. st We will study L for q ∈ L (Γ). This means that in general qψ ∈ L (Γ), 1 2 2 2 so the formal expression L = −d /dx + q lacks an immediate meaning as a sum of operators from L (Γ) to L (Γ). We will however establish that 2 2 the perturbation by q is inﬁnitesimally form-bounded (see Proposition 3.2) with respect to the quadratic form of L . More precisely, the quadratic form st Q (ψ, ψ)=(L ψ, ψ)=(ψ ,ψ ) is deﬁned on the domain of functions from W (Γ \ V ), which are in addition continuous at all vertices. Then the expres- st st sion L + q = L can be assigned a meaning via the KLMN theorem (see 0 q st e.g. [32]), as a self-adjoint operator with the same form domain as L . Note that the domain of the quadratic form only includes the continuity condition from (2.1). The Spectrum: We denote the spectrum of L by σ(L). For ﬁnite compact graphs the spectrum of L is discrete, for any self-adjoint boundary condi- tions, and furthermore satisfy Weyl asymptotics. This also holds for L ,for th st q ∈ L (Γ) (see Sect. 4). The n eigenvalue, counting multiplicities, of L (Γ) st st st st is denoted by λ (L )= λ (L (Γ)). When we write σ(L (Γ )) = σ(L (Γ )) n n 1 2 q q q q 1 2 we mean not only equality as sets, but also that the multiplicities of all eigen- values are equal. For a given ﬁnite compact graph of total length L with N edges and M vertices, the following lower and upper estimates can be proven (see “Appen- dix” for the proof) 2 2 π π 2 st 2 (n − M ) ≤ λ (L ) ≤ (n + N − 1) . (2.2) L L In particular this estimate implies that the multiplicity of the eigenvalues is uniformly bounded by M + N , since (2.2) implies λ ≤ n ≤ λ . n−N +1 n+M The above estimate is enough for our purposes, but it may be improved further taking into account the structure of the graph [17]. Another implication of (2.2) is the well-known Weyl asymptotics: st λ (L ) lim =1. (2.3) n→∞ (πn/L) 40 Page 6 of 24 J. Boman et al. IEOT We shall also need the following uniform bound on the amplitude of eigenfunctions of the Laplacian on ﬁnite metric graphs: ψ ≤ c(Γ) ψ , (2.4) L L ∞ 2 where the constant c(Γ) is determined by the graph Γ but is independent of n. One can ﬁnd explicit proofs of all the above mentioned formulas (2.2), (2.3), and (2.4) in the “Appendix”. 3. Deﬁnition of the Operator The following well-known Sobolev estimate (a special case of Gagliardo- Nirenberg estimate) is valid for any function ψ ∈ W (0,) on an interval of ﬁnite length < ∞ (see proof in the “Appendix”) 2 2 2 ψ ≤ ψ + ψ (3.1) L L L ∞ 2 2 for suﬃciently small > 0. The constants, though suﬃcient for our needs, may be improved if one is willing to sacriﬁce some elegance in the proof. In particular any function ψ ∈ W (Γ \ V ) will satisfy the estimate (3.1) on each edge E of Γ, with = . As the set of edges is ﬁnite there is an n n edge that has minimal length so we obtain a global estimate of |ψ(x)| on Γ: Corollary 3.1. Let Γ be a ﬁnite compact graph and ψ ∈ W (Γ).Then 2 2 2 ψ ≤ ψ + ψ (3.2) L (Γ) L (Γ) L (Γ) ∞ 2 2 for suﬃciently small > 0 and x ∈ Γ. Proposition 3.2. Let q ∈ L (Γ),and let Q be the quadratic form given by 1 q Q (ψ, ψ)= q(x)|ψ(x)| dx, st st and Q the quadratic form associated with L . Then for suﬃciently small L 0 > 0 there exists b() such that |Q (ψ, ψ)|≤ Q st (ψ, ψ)+ b()(ψ, ψ). q L In other words Q is inﬁnitesimally bounded by Q st . Proof. Corollary 3.1 shows that for suﬃciently small > 0 2 2 2 |ψ(x)| ≤ ψ + ψ . L (Γ) L (Γ) 2 2 Multiplying by q and integrating we obtain 2 2 q(x)|ψ(x)| dx ≤ |q(x)||ψ(x)| dx Γ Γ 2 2 ≤ q ψ + q ψ L (Γ) L (Γ) 1 L (Γ) 1 L (Γ) 2 2 = q Q st (ψ, ψ)+ q ψ . L (Γ) L L (Γ) 1 1 L (Γ) 0 2 2 2 Replacing with /q ,wemay chose b()= q . L (Γ) L (Γ) 1 IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 7 of 24 40 The KLMN theorem [32] now lets us conclude that there is a unique bounded from below self-adjoint operator associated with the form Q st +Q : L q st Deﬁnition 3.1. For q ∈ L (Γ) we denote by L (Γ) the operator associated with the form Q st + Q . L q st st We note that the form domains of L and L coincide, and q 0 st st (L ψ, φ)=(L ψ, φ)+ q(x)ψ(x)φ(x)dx, q 0 for all ψ, φ ∈ Dom(Q ) = Dom(Q st ). L L 4. Spectral Estimates We recall the following standard variational theorems, see e.g. [32] for proofs. The lowest eigenvalue will be denoted by λ . Proposition 4.1. (Min-Max). Let A be a self-adjoint, bounded from below, th operator with discrete spectrum, then the n eigenvalue of A is given by λ (A) = min max Q (u, u), n−1 A V u∈V n n u =1 where V ranges over all n-dimensional subspaces of Dom(Q ), the domain n A of the quadratic form Q associated with A. Proposition 4.2. (Max-Min). Let A be a self-adjoint bounded from below, op- th erator with discrete spectrum, then the n eigenvalue of A is given by λ (A) = max min Q (u, u), n−1 A V u⊥V n−1 n−1 u =1 where V ranges over all (n − 1)-dimensional subspaces of Dom(Q ), the n−1 A domain of the quadratic form Q associated with A. In order to apply Propositions 4.1 and 4.2 as we do in the following it st is of course required that the spectrum of L (Γ) is discrete. For Schr¨ odinger operators with L potentials on ﬁnite intervals this is well known, so it is true for ﬁnite metric graphs as well since these are just ﬁnite-rank perturbations— in the resolvent sense—of the Dirichlet Schr¨ odinger operators (deﬁned by Dirichlet conditions at all vertices), which is nothing else than an orthogonal sum of Dirichlet Schr¨ odinger operators on a collection of ﬁnite intervals. We now proceed to prove the spectral estimate for ﬁnite compact graphs, i.e. we show that the diﬀerence between the Laplace and the Schr¨ odinger eigenvalues is uniformly bounded: Theorem 4.3. Let Γ be a ﬁnite compact metric graph, and let q ∈ L (Γ). st st Then the diﬀerence between the eigenvalues λ (L ) and λ (L ) is bounded n n 0 q by a constant, i.e. st st |λ (L ) − λ (L )|≤ C, (4.1) n n 0 q where C = C(Γ, q ) is independent of n. L (Γ) 1 40 Page 8 of 24 J. Boman et al. IEOT Similar questions have been discussed in [11] for the case of equilateral metric graphs. We are going to prove the Theorem using Propositions 4.1 and 4.2.To st illustrate the strategy let us ﬁrst try to derive an upper estimate for λ (L ) using a more naive approach. The quadratic form is 2 2 Q st (u, u)= |u (x)| dx + q(x)|u(x)| dx. Γ Γ It can be estimated from above by 2 2 st st Q (u, u) ≤ Q (u, u)= |u (x)| dx + q (x)|u(x)| dx, (4.2) L L + q q Γ Γ where q is the positive part of the potential q: q(x)= q (x) − q (x),q (x) ≥ 0. (4.3) + − ± This step cannot be improved much, since the new estimate coincides with the original one in the case where q is nonnegative. The idea how to proceed is to choose a particular n-dimensional sub- 0 st space V . Then the Rayleigh quotient gives not an exact value for λ (L ), n−1 n q but an upper estimate when Proposition 4.1 is used Q st (u, u) Q st (u, u) L L q q st λ (L ) = min max ≤ max . n−1 2 2 V u∈V n n u u∈V u L L 2 2 The only reasonable candidate for V we have at hand is the linear span of the Laplacian eigenfunctions corresponding to the n lowest eigenvalues st st st L L L 0 0 0 V = L ψ ,ψ ,...,ψ . (4.4) 0 1 n−1 If q ≡ 0 then this estimate gives the exact value for λ . Therefore it is n−1 natural to split the quadratic form as follows: 2 2 st Q (u, u) L |u (x)| dx q (x)|u(x)| dx st Γ Γ λ (L ) ≤ max ≤ max + max . n−1 2 2 2 0 0 0 u∈V u u∈V u u∈V u n n n L L L 2 2 2 st Then the ﬁrst quotient is equal to λ (L ) and the maximum is attained n−1 on st u = ψ . n−1 If nothing about q is known, then to estimate the second quotient one may use q (x)|u(x)| dx ≤q max |u(x)| . (4.5) + + L (Γ) x∈Γ Γ IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 9 of 24 40 st st L L n−1 2 0 0 We need to estimate |u(x)| , provided u = α ψ . Since max |ψ (x)| j=1 j j ≤ c (formula (2.4)) we obtain with the Schwarz inequality n−1 n−1 st max |u(x)|≤ |α | max |ψ (x)|≤ c |α | j j x∈Γ 0 0 1/2 n−1 √ √ ≤ c n |α | = c nu . j L Hence q (x)|u(x)| dx ≤q c n, + L and st 2 2 Q st (u, u) ≤ λ (L )+ c nq u , (4.6) L n−1 0 + L1 L which implies st st 2 λ (L ) − λ (L ) ≤q c n, (4.7) n−1 n−1 + L q 0 1 i.e. we do not get an estimate uniform in n—the estimate grows linearly with st n. The reason is the splitting of the quadratic form of L into two parts. To obtain the upper bounds we used two intrinsically diﬀerent vectors: the ﬁrst st term is maximized if u = ψ , while to estimate the second term we used n−1 st st st L L L 0 0 0 u = ψ + ψ + ··· + ψ . This is the reason that the estimate (4.7)is 0 1 n−1 not optimal. ProofofTheorem 4.3. We divide the proof into two parts deriving upper and lower estimates separately. Upper Estimate As before we use the estimate 2 2 |u (x)| dx + q (x)|u(x)| dx st Γ Γ λ (L ) ≤ max , (4.8) n−1 u∈V u st n−1 L 0 0 0 where V is deﬁned by (4.4). Every function u = α ψ from V can n j=0 j n be written as a sum u = u + u , where 1 2 st st st L L L 0 0 0 u := α ψ + α ψ + ··· + α ψ , 1 0 1 n−p−1 0 1 n−p−1 st st st L L L 0 0 0 u := α ψ + α ψ + ··· + α ψ . 2 n−p n−p+1 n−1 n−p n−p+1 n−1 Here p is a natural number to be ﬁxed later (independent of n, but depending on Γ and q). Therefore as n increases the ﬁrst function u will contain an increasing number of terms, while the second function will always be given by a sum of p terms. 2 2 2 From the inequality |u + u | dx ≤ 2 |u | dx +2 |u | dx and the 1 2 1 2 fact that q is nonnegative we have 2 2 2 q (x)|u (x)+ u (x)| dx ≤ 2 q (x)|u (x)| dx +2 q (x)|u (x)| dx. + 1 2 + 1 + 2 Γ Γ Γ (4.9) 40 Page 10 of 24 J. Boman et al. IEOT That u ,u are orthogonal is clear, and from this the orthogonality of u and 1 2 u also follows: n−p−1 st st 0 (u ,u )= −(u ,u )= − λ (L )(α ψ u ,u )=0. 2 i i 1 2 1 2 1 0 i=0 Taking this into account we arrive at 2 2 st Q (u, u) ≤ |u (x)| dx +2 q (x)|u (x)| dx L + 1 Γ Γ =: Q st (u ,u ) 1 1 2q (4.10) 2 2 + |u (x)| dx +2 q (x)|u (x)| dx . + 2 Γ Γ =: Q st (u ,u ) L 2 2 2q To estimate the ﬁrst form we use (4.5) and the Sobolev estimate (3.2)for max |u(x)|.Weget 2 2 st Q (u ,u )= |u (x)| dx +2 q (x)|u (x)| dx 1 1 + 1 L 1 Γ Γ 2q 2 2 ≤u +2q max |u (x)| + L x∈Γ 1 1 L 1 2 2 2 2 ≤u +2q (u + u ) + L 1 1 L 1 1 L L 2 2 2 2 4 2 =(1 + 2q )u + q u . + L + L 1 1 1 L 1 L 2 2 Using n−p−1 2 st u =(u ,u )= −(u ,u )= λ (L )(α ψ ,α ψ ) 1 j j j j j 1 L 1 1 1 0 j=0 (4.11) n−p−1 n−p−1 st 2 st 2 st 2 = λ (L )|α | ≤ λ (L ) |α | = λ (L )u , j 0 j n−p−1 0 j n−p−1 0 1 L j=0 j=0 we get 2 2 Q st (u ,u ) ≤ (1 + 2q )u + q u L 1 1 + L + L 1 1 1 L 1 L 2q 2 2 st 4 2 ≤ (1 + 2q )λ (L )+ q u . + L n−p−1 + L 1 1 0 1 L The key point is that and p canbechoseninsuchaway that st st (1 + 2q )λ (L )+ q <λ (L ) + L n−p−1 + L n−1 1 0 1 0 holds (see (4.14) below). On the other hand, our naive approach (4.6) can be applied to the second form with the only diﬀerence being that the number of eigenfunctions involved is p,not n st 2 2 Q st (u ,u ) ≤ λ (L )+2c pq u . (4.12) L 2 2 n−1 + L 2 0 1 L 2q 2 + IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 11 of 24 40 Putting together the obtained estimates in (4.10) and using that u = 2 2 u −u we get L L 2 2 st 4 2 Q st (u, u) ≤ (1 + 2q )λ (L )+ q u + L n−p−1 + L 1 L 1 0 1 q 2 st 2 2 + λ (L )+2c pq u n−1 + L 2 0 1 L st 2 2 2 ≤ λ (L )u +2c pq pu n−1 + L 0 L 1 L 2 2 st st 2 − λ (L ) − (1 + 2q )λ (L ) − q u . n−1 + L n−p−1 + L 1 0 1 0 1 L We would get the desired estimate st Q (u, u) st st λ (L ) ≤ max ≤ λ (L )+ C (4.13) n−1 n−1 q 0 u∈V u with C =2c pq if we manage to prove that + L st st λ (L ) − (1 + 2q )λ (L ) − q > 0 (4.14) n−1 + L n−p−1 + L 0 1 0 1 for a certain that may depend on n and p. We use the estimate for Laplacian eigenvalues given in (2.2): 2 2 π π 2 st 2 (n − M ) ≤ λ (L ) ≤ (n + N − 1) . (4.15) L L st st Substituting λ (L ) with the lower bound and λ (L ) with the n−1 n−p−1 0 0 upper and setting =1/n, we get the following inequality for the left-hand side of (4.14) st st λ (L ) − (1 + 2/nq )λ (L ) − 4nq n−1 + L n−p−1 + L 0 1 0 1 ≥ (n − 1 − M ) − (1 + 2/nq ) + L × (n − 1 − p + N − 1) − 4nq + L 2 2 π L =2n p − M − N +1 − 1+2 q + O(1). + L L π We see that for any integer p>M + N − 1+(1 +2(L/π) )q the + L expression is positive for suﬃciently large n and the diﬀerence between the eigenvalues possesses the uniform upper estimate: st st λ (L ) − λ (L ) ≤ C. (4.16) n n q 0 If one is interested in the diﬀerence between the eigenvalues for large n only, then the constant C can be taken equal to 2 2 C =2c q (M + N − 1+(1 +2(L/π) )q ), L L 1 1 but this value of C may be too small in order to ensure that (4.16) holds for all n, since proving (4.14) we assumed that n is suﬃciently large. The latter assumption does not aﬀect the ﬁnal result, since for a ﬁnite number of eigenvalues (4.16) is always satisﬁed, but the value of the constant C may be aﬀected. 40 Page 12 of 24 J. Boman et al. IEOT Lower Estimate To obtain a lower estimate we are going to use the Max-Min principle (Propo- sition 4.2). The ﬁrst step is to notice that 2 2 Q st (u, u) ≥ |u (x)| dx − q (x)|u(x)| dx. (4.17) L − Γ Γ Using the subspace V deﬁned in (4.4)weget n−1 st Q (u, u) st λ (L ) ≥ min . n−1 u⊥V u n−1 Since u is orthogonal to V it possesses the representation n−1 st u = α ψ . j=n−1 As before let us split the function u = u + u 1 2 st st st L L L 0 0 0 u := α ψ + α ψ + ··· + α ψ , 1 n−1 n n n+p−2 n−1 n+p−2 st st L L 0 0 u := α ψ + α ψ + .... 2 n+p−1 n+p n+p−1 n+p Note two important diﬀerences: • the function u is given by the sum of p terms, where the number p in- dependent of n will be chosen later, so the functions u and u exchange 1 2 roles compared with the proof of the upper estimate; • the function u is given by an inﬁnite series, not by an increasing number of terms as the function u in the proof of upper estimate. Using the fact that q is nonnegative we may split the quadratic form (com- pare (4.10)) 2 2 Q st (u, u) ≥ |u (x)| dx − 2 q (x)|u (x)| dx L − 1 Γ Γ st =: Q (u ,u ) 1 1 −2q 2 2 + |u (x)| dx − 2 q (x)|u (x)| dx . (4.18) − 2 Γ Γ =: Q st (u ,u ) L 2 2 −2q Now the function u is given by a ﬁnite number of terms and we may similarly to (4.12) estimate st 2 2 Q st (u ,u ) ≥ λ (L ) − 2c pq u . (4.19) L 1 1 n−1 − L (Γ) 1 0 1 L −2q 2 To estimate the second form we use (4.5) and the Sobolev estimate (3.2)for max |u(x)| .Weget 2 2 Q st (u ,u ) ≥u − 2q max |u (x)| L 2 2 − L 2 2 L 1 −2q 2 2 2 2 ≥u − 2q u + u − L 2 2 L 1 2 L L 2 2 2 4q − L 2 2 =(1 − 2q )u − u . − L 2 1 2 L L 2 2 IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 13 of 24 40 Taking into account 2 st u =(u ,u )= −(u ,u )= λ (L )(α ψ ,α ψ ) 2 j j j j j 2 L 2 2 2 0 j=n+p−1 ∞ ∞ (4.20) st 2 st 2 = λ (L )|α | ≥ λ (L ) |α | j j n+p−1 j 0 0 j=n+p−1 j=n+p−1 st 2 = λ (L )u , n+p−1 2 0 L we arrive at Q st (u ,u ) L 2 2 −2q 4q − L (Γ) st 1 2 ≥ (1 − 2q )λ (L ) − u . (4.21) − L (Γ) n+p−1 2 1 0 L Summing the estimates (4.19) and (4.21) and taking into account that 2 2 2 u = u −u we get 2 1 L L L 2 2 2 st 2 2 st Q (u, u) ≥ λ (L ) − 2c pq u L n−1 − L 1 0 1 L q 2 4q − L st 1 2 + (1 − 2q )λ (L ) − u − L n+p−1 2 1 0 L st 2 2 2 ≥ λ (L )u − 2c pq u n−1 − L 0 L 1 L 2 2 4q − L st 1 st 2 + (1 − 2q )λ (L ) − − λ (L ) u . − L n+p−1 n−1 2 1 0 0 L As before, to prove the desired uniform estimate it is suﬃcient to show that for large enough n the following expression can be made positive by choosing an appropriate : 4q − L st 1 st (1 − 2q )λ (L ) − − λ (L ) > 0. (4.22) − L n+p−1 n−1 1 0 0 st Again we use (4.15): we substitute λ (L ) with the lower bound and n+p−1 st λ (L ) with the upper. As before we choose =1/n, so the left-hand side n−1 of (4.22) becomes st st (1 − 2q )λ (L ) − q − λ (L ) − L n+p−1 − L n−1 1 0 1 0 ≥ (1 − 2q /n) (n + p − 1 − M ) − L −4nq − (n − 1+ N − 1) − L π L =2n p − M − N +1 − 1+2 q + O(1). − L L π If p>M + N − 1+(1 +2(L/π) )q , then for suﬃciently large n the − L expression is positive, hence the following lower estimate holds st st λ (L ) − λ (L ) ≥ C, (4.23) n n q 0 where the exact value of C is determined by the diﬀerence between the ﬁrst few eigenvalues as described above. 40 Page 14 of 24 J. Boman et al. IEOT Theorem 4.3 allows us to conclude that the spectra of Schr¨ odinger op- erators satisfy Weyl asymptotics as well: st Corollary 4.4. Let q ∈ L (Γ), then λ (L (Γ)) satisﬁes Weyl asymptotics. 1 n Proof. This is an immediate consequence of (2.3) and Theorem 4.3. More importantly we may now show that the eﬀect of an L -perturbation on the eigenvalues will tend to zero in n in the scale of square roots. This step is critical in the proof of Theorem 6.5. st 2 st 2 st st Corollary 4.5. Let λ (L )= k and λ (L )= k .If |λ (L )−λ (L )|≤ n n n n q n,q 0 n,0 q 0 C ∈ R then |k − k |≤ , for some constant C ∈ R. In particular, n,q n,0 0 |k − k |→ 0,n →∞. (4.24) n,q n,0 Proof. Since the eigenvalues satisfy Weyl asymptotics that depend only on the length of Γ we have that k ,k ≥ nD for some constant D and suﬃ- n,q n,0 ciently large n.Wehave 2 2 k −k (k +k )(k −k ) n,q n,0 n,q n,0 n,q n,0 |k − k | = = n,q n,0 k +k k +k n,q n,0 n,q n,0 st st λ (L )−λ (L ) n n q 0 C C 0 = ≤ ≤ , k +k |k +k | n n,q n,0 n,q n,0 for some constant C ∈ R. 5. On the Zeros of Trigonometric Polynomials st In this section we recall the secular equation for the spectrum σ(L (Γ)): it is given as the squares of the zeros of a trigonometric polynomial. We then prove that if the zeros k of such a ﬁnite trigonometric polynomial with constant coeﬃcients are close to a certain equispaced sequence, i.e. satisfy |k − mπ/L| → 0 then in fact k = mπ/L for all m (Theorem 5.2). From m m this we then prove the geometric Ambartsumian Theorem 6.5. st Theorem 5.1. Let Γ be a ﬁnite compact metric graph. The eigenvalues λ (L ) are given by the squares of the zeros of a certain trigonometric polynomial iω k p(k)= a e n=1 st 2 with k-independent coeﬃcients a ∈ C and ω ∈ R: λ (L )= k if and n n n 0 n,0 only if p(k )=0. n,0 Proof. We sketch the proof: for each non-zero eigenvalue the corresponding eigenfunction is edge-wise just a sum of sine and cosine functions: ψ(x)= a cos(kx)+ b sin(kx),x ∈ E . The solutions have to satisfy the vertex j j j conditions. Continuity can at each V be written as a cos kx + b sin kx = a cos ky + b sin ky, i i j j IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 15 of 24 40 if x ∈ E , y ∈ E ,and x, y ∈ V . This yields 2N − M equations, where N is i j the number of edges and M is the number of vertices. The conditions on the normal derivatives can at each vertex V be written as (−1) −a sin kx + b cos kx =0, [(j+1)/2] j [(j+1)/2] j x ∈V where k has been factored out. This yields an additional M equations, so that we in total have 2N equations, which may be written in the form: T (k) c =0, with c a vector of the coeﬃcients a ,b and T (k) a matrix with trigonometric i i 2 st entries depending on k.Areal number λ = k is an eigenvalue of L if and only if k is a root of the trigonometric polynomial det T (k)=0. We refer to [6, 18] for details. See also eg. [15] and [26]. Note that it is crucial that the vertex conditions were standard: in the case of more general vertex conditions the secular equation is given by a quasipolynomial instead of the trigonometric polynomial. Theorem 5.2. Let f be the trigonometric polynomial iω k p(k)= a e (5.1) j=1 with all ω ∈ R, a ∈ C. If the zeros k of f satisfy j j m lim (k − m)=0 m→∞ then k = m for all m. First we need a Lemma: Lemma 5.3. Given ω ,...,ω ∈ R there exists a subsequence {m } of the 1 J n natural numbers such that, for each ω , iω m j n e → 1. Proof. For ω := (ω ,...,ω ) ∈ R let [ω ]:=([ω ],..., [ω ]) denote the 1 J 1 J J J image of ω under the standard projection to the J -torus: R → (R/2πZ) . The statement of the Lemma is equivalent to the existence of an increasing sequence of integers m such that [m ω] → 0:=(0,..., 0) ∈ (R/2πZ) . n n Consider the set of points m ω, with m ∈ N and its projection [m ω]= ([mω ],..., [mω ]). Since the J -torus is compact this set has a limit point 1 J z and an increasing subsequence (m ) ⊂ N such that [m ω ] → z. This is i i a Cauchy sequence so for any > 0 there exists I() such that for any i ,i ≥ I() 1 2 d([n ω ], [n ω ]) <, i i 1 2 where d(·, ·) denotes the metric on the J -torus. Taking a sequence → 0 we may chose i ( )= I( ) and in each step i ( ) > I( ) so large that the 1 i i 2 i i diﬀerence m := n − n , i i ( ) i ( ) 2 i 1 i 40 Page 16 of 24 J. Boman et al. IEOT is an increasing sequence. It follows that [m ω ]=[(n − n )ω ] → 0. i i ( ) i ( ) 2 i 1 i Remark 5.1. Note that in the special case where 2π, ω ,...,ω are rationally 1 J independent the classical theorem of Kronecker (see e.g. [16]) can be used. ProofofTheorem 5.2. Consider the trigonometric polynomial p(k)in(5.1). Denote k − m =: γ so that γ tends to zero as m →∞.Wehave k = m m m m m + γ so for each ω : m j iω k iω m iω γ j m j j m e = e e . Choose a subsequence {m } as described in Lemma 5.3 and pass, for an arbitrary r ∈ N,tothe (r + m ):th zero of p.Wehave iω (r+m ) iω γ j n j (r+m ) 0= p(k )= a e e r+m j j=1 J J iω r iω m iω γ iω r j j n j (r+m ) j = a e e e → a e = p(r), j j j=1 j=1 as n →∞. The limit follows from the choice of m and the fact that γ n (r+m ) tends to 0. The above calculation shows that p(r)=0. But p(r)=0and k − r → 0 together imply that k = r, since even a single extra zero would make the asymptotic behaviour impossible. Remark 5.2. It is not important in the above theorem that k tends to the integers. A scaling argument allows one to extend it to the case where k are close to integer-multiples of an arbitrary real number. 6. An Ambartsumian Theorem With the result of the previous two sections we are now in a position to prove an inverse spectral theorem that may be seen as a generalization of Ambartsumian’s classical theorem. For the proof we recall the classical result as well as its geometric version for Laplacians. The following theorem has been a source of inspiration for researchers in inverse problems for almost a century. In the original article [1]itwas assumed that the potential is continuous, but the result holds even if q ∈ L . We adjusted the formulation to our notations. Proposition 6.1. (Ambartsumian’s theorem [1]) Let q be a real-valued abso- lutely integrable function on an interval I. Then the spectrum of the stan- st dard Schr¨ odinger operator L (I) coincides with the spectrum of the standard st Laplacian L (I) if and only if the potential q is identically equal to zero. Standard conditions on a single compact interval is of course just the classical Neumann conditions at both end-points. It appears that the theorem is still valid if instead of the interval I we have arbitrary connected ﬁnite com- pact metric graph Γ. This result was proven step-by-step by several authors IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 17 of 24 40 [10, 27, 29, 36] and [30], but the most general version was given by E.B. Davies [12] (Davies proved it for q ∈ L (Γ) but noted that this condition surely can be weakened). st st Proposition 6.2. (following [12]) Let L (Γ) and L (Γ) be the standard Schr¨ o- q 0 dinger and Laplace operators on a connected ﬁnite compact metric graph Γ. Assume that the potential q is absolutely integrable. Then the eigenvalues of the two operators coincide if and only if the potential q is equaltozeroalmost everywhere. The second theorem is a geometric version of Ambartsumian theorem for standard Laplacians. We start by recalling the result that among graphs with ﬁxed total length the spectral gap is minimized by the single interval [13, 28] and [25]: st Proposition 6.3. (Theorem 3 from [25]) Let L (Γ) be the standard Laplace operator on a connected ﬁnite compact metric graph Γ of total length L(Γ). st Assume that the ﬁrst (nonzero) eigenvalue of L (Γ) coincides with the ﬁrst (nonzero) eigenvalue of the Laplacian on the interval I of length L(Γ) st st λ (L (Γ)) = λ (L (I)); 1 1 0 0 then the graph Γ coincides with the interval I. The multiplicity of the eigenvalue zero for the standard Laplacian is equal to the number of connected components in the metric graph and the asymptotics of the spectrum determines the total length of the graph. Hence the above proposition implies: Theorem 6.4. Let Γ be a ﬁnite compact metric graph. The spectrum of the standard Laplacian on Γ coincides with the spectrum of the standard Lapla- cian on the interval I st st λ (L (Γ)) = λ (L (I)),j =0, 1, 2,... , (6.1) j j 0 0 if and only if the graph Γ coincides with the interval I. The assumptions of the theorem can be weakened, since to ensure that Γand I have the same total length it is enough to check the asymptiotics. Our goal is to prove that if the spectrum of a Schr¨ odinger operator on a metric graph coincides with the spectrum of the Laplacian on an interval then the graph coincides with the interval and the potential is zero. This statement cannot be proven as a simple combination of the above mentioned results (Proposition 6.2 and Theorem 6.4). The main diﬃculty is to show that the graph coincides with the interval. Theorem 6.4 cannot be applied directly, since it requires q ≡ 0. Theorem 6.5. Let Γ be a ﬁnite compact metric graph and q ∈ L (Γ).The st spectrum of the standard Schr¨ odinger operator L (Γ) coincides with the spec- trum of the standard (i.e. Neumann) Laplacian on an interval st st λ (L (Γ)) = λ (L (I)), (6.2) j j q 0 if and only if Γ= I and q ≡ 0. 40 Page 18 of 24 J. Boman et al. IEOT Proof. Equation (6.2) implies that the total length of the graph Γ coincides with the length of the interval I. To see this it is enough to compare the corresponding asymptotics. Then the spectrum of the Laplacian on I is πn st λ (L (I)) = ,n ∈ N. The spectral estimate (4.1) implies that st st λ (L (Γ)) −λ (L (Γ) = O(1). n n q 0 πn st Hence the square roots of the eigenvalues of L (Γ) satisfy πn st k (L (Γ)) − → 0, as n →∞. (6.3) But k are given as the zeros of a trigonometric polynomial (Theorem 5.1). Hence the spectrum of the Laplacian on Γ given by zeroes of the trigono- metric polynomial is asymptotically close to a set of equidistant points and st Theorem 5.2 can be applied. We conclude that in fact k (L (Γ)) = πn/L,for all n. This means that the spectrum of the Laplacian on Γ coincides with the spectrum of the Laplacian on an intreval. We may ﬁnally apply the geometric version of Ambartsumian theorem for Laplacians (Theorem 6.4) to conclude that the graph Γ coincides with the interval I. But then Ambartsumian’s classical theorem (Proposition 6.1) implies that the potential is identically equal to zero q ≡ 0. Proving our main theorem we have shown that in order to ensure that the graph Γ coincides with the interval, it is enough to require that the spectrum of the Schr¨ odinger operator is close to the spectrum of the Laplacian on an interval. In fact we proved the following theorem: st Theorem 6.6. Let L (Γ) be standard Schr¨ odinger operator with L -potential on a ﬁnite compact connected metric graph Γ.Then Γ= I if st st λ (L (Γ)) − λ (L (I)) → 0,n →∞. (6.4) n n q 0 7. Euler Characteristic The spectral estimate (4.1) allows us to extend a previous result of one of the authors regarding the Euler characteristic χ = M − N . Theorem 7.1. Let Γ be a ﬁnite compact metric graph and q ∈ L (Γ). Then the st Euler characteristic χ(Γ) is uniquely determined by the spectrum σ(L (Γ)), and can be calculated as the limit ⎛ ⎞ st sin λ (L )/2t st ⎝ ⎠ χ(Γ) = 2 lim cos λ (L )/t , (7.1) t→∞ st λ (L )/2t n=0 n q IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 19 of 24 40 with the convention that λ =0 implies st sin λ (L )/2t =1. (7.2) st λ (L )/2t Proof. In [19] formula (7.1) was proven for the case of zero potential. In other words formula (7.1) gives the Euler characteristics if one substitutes the st st eigenvalues λ (L ). We have shown that the diﬀerence between λ (L (Γ)) n n 0 q st and λ (L (Γ)) is uniformly bounded and therefore |k − k | = O . (7.3) Taking into account Weyls asymptotics k = n + O(1), (7.4) one may apply Lemma 2 from [20] to conclude that formula (7.1) gives the same result independently of whether the eigenvalues of the standard Lapla- st st cian L (Γ) or of the standard Schr¨ odinger operator L (Γ) are 0 q substituted. Note that proving the theorem we assumed that the vertex conditions are standard; this assumption in general cannot be removed. 8. Discussion It would be interesting to understand under which assumptions a similar result holds for other than the standard vertex conditions that are assumed throughout in this paper. The cases, where the vertex conditions are such that the coeﬃcients of the trigonometric polynomial given by Theorem 5.1 are k independent, could potentially be dealt with in a similar way as in this paper. The bounds on the diﬀerences of the eigenvalues obtained in Theorem 4.3 can be established for other choices of vertex conditions, but Theorem 6.4 in general may fail. Acknowledgements The authors would like to thank G. Berkolaiko for pointing out an explicit improvement leading to estimate (A.4). Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4. 0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, pro- vide a link to the Creative Commons license, and indicate if changes were made. 40 Page 20 of 24 J. Boman et al. IEOT Appendix A: Elementary spectral properties In this appendix we provide proofs for the elementary spectral properties of quantum graphs already mentioned in Sect. 2. Most of these results are well-known, but we need them for the sake of completeness. Proofofformula (2.2). For a bounded from below self-adjoint operator A with discrete spectrum, deﬁne the eigenvalue counting function E : R → N, by E (λ)=#{λ ∈ σ(A) | λ ≤ λ}. A j j The standard Laplacian is positive (hence the lower estimate in (2.2)isin- teresting only if n>M ) therefore when calculating the eigenvalue counting function we assume that λ ≥ 0. Consider ﬁrst the Laplace operator L on a single interval I of length πn with Dirichlet conditions at the end-points. The eigenvalues are λ = , n =1, 2,... So the eigenvalue counting function for λ ≥ 0 is in fact given by E D (λ)= , L (I) where square-brackets denote the integer-part of the argument. Returning to Γ we note that if we impose Dirichlet conditions on the vertices of Γ—denote the operator by L (Γ)—then we really just have a decoupled set of intervals and therefore the set of eigenvalues is just the union of the eigenvalues for each interval (counting multiplicities). Therefore the corresponding counting function E D is given by L (Γ) E D (λ)= E D (λ) L (Γ) L (E ) 0 0 n n=1 √ √ √ √ λ λ λ λ = + + ··· + ≤ L , (A.1) 1 2 N π π π π since taking integer parts may only decrease the value. First adding and then taking integer parts may—compared to adding the integer parts—at most raise the value by the number of terms −1, i.e. the number of edges −1, so conversely we also have L − N +1 ≤ E D (λ). (A.2) L (Γ) Formulas (A.1) and (A.2) give eﬀective two-sided bounds for the eigenvalues of the Dirichlet Laplacian on Γ. D −1 st −1 We now show that (L (Γ) − λ) − (L (Γ) − λ) is of ﬁnite rank. Take 0 0 λ = 0, and suppose that D st (L − λ)u = f, (L − λ)u = f. D st 0 0 2 2 Then for the diﬀerential operator −d /dx we have that − − λ (u − u )=0. D st dx IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 21 of 24 40 D st Since functions in the domain of L and L are continuous, to determine the 0 0 rank of the resolvent diﬀerence, we need to determine dim ker − − λ on dx continuous functions. Prescribing values u at each vertex V in Γ, we see j j that a unique solution to this boundary value problem is given as follows: on an edge E =[x ,x ] between V and V ,set n 2n−1 2n i j sin k(x − x ) sin k(x − x ) 2n 2n−1 u(x):= u + u i j sin k(x − x ) sin k(x − x ) 2n−1 2n 2n 2n−1 2 2 2 for k = λ. Then u is continous on Γ and solves (−d /dx − λ)u =0, so 2 2 dim ker(−d /dx − λ) ≤ M . Therefore we have E st (λ) ≤ E D (λ)+ M ≤ L + M. (A.3) L (Γ) L (Γ) The lower estimate estimate (A.2) can be modiﬁed in a similar way, but st D instead we shall take into account that L (Γ) ≤ L (Γ). Really Dirichlet con- 0 0 ditions in particular imply the continuity of functions in the domain of the quadratic form. Passing to standard conditions means weakening the con- ditions on functions in the domain of the quadratic form, since now only continuity is required at the vertices. Therefore the domain of the quadratic form Q st is larger than that of Q D , so by the Min-Max principle (see Propo- 0 0 sition 4.1) eigenvalues can only go up when imposing Dirichlet conditions. In particular, the lower bound (A.2) on the eigenvalue counting function E D L (Γ) is also valid for E st . L (Γ) Putting the lower and upper estimates together we have √ √ λ λ st L − N +1 ≤ E (λ) ≤ L + M (A.4) L (Γ) π π Setting λ = n we obtain n − N +1 ≤ E st n ≤ n + M, L (Γ) 0 2 so λ ≤ n ≤ λ . n−N +1 n+M π 2 Setting n = n + M we get λ ≥ (n − M ) and similarly we ﬁnd λ ≤ n 2 n π 2 (n + N − 1) , which proves the theorem. Proofofformula (2.3). The Weyl asymptotics follow from the relation 2 2 π π 2 st 2 (n − M ) ≤ λ (L ) ≤ (n + N − 1) . L L Proofofformula (2.4). For ψ corresponding to λ = k we have 1 k 2π 2 2 ψ ≥ |ψ(x)| dx ≥ max |ψ(x)| , L (Γ) x∈E n 2 2π k where [·] denotes the integer part of the argument. [ k/2π] may be equal to zero for only ﬁnitely many k since the eigenvalues satisfy Weyl asymptotics. 40 Page 22 of 24 J. Boman et al. IEOT Since [ k/2π]/k is bounded for k ∈ R this implies the existence of a k- independent bound c(Γ). Proofofformula (3.1). Let x denote a global minimum for ψ. Then in min 2 2 particular |ψ(x )| ≤ψ /. We then have min 2 2 |ψ(x)| = |ψ(x )| +2 ψ(y)ψ (y)dy min min ≤|ψ(x )| +2 |ψ(y)ψ (y)|dy min 2 2 2 ≤|ψ(x )| + |ψ (y)| dy + |ψ(y)| dy min 0 0 2 2 2 ≤ψ / + ψ + ψ 2 2 2 1 1 2 2 = ψ + + ψ . 2 2 For < we have 1/ > 1/ and the claim follows. References [1] Ambartsumian, V.: Uber eine Frage der Eigenwerttheorie. Z. Phys. 53, 690–695 (1929) [2] Avdonin, S., Kurasov, P.: Inverse problems for quantum trees. Inverse Probl. Imaging 2, 1–21 (2008) [3] Avdonin, S., Kurasov, P., Nowaczyk, M.: Inverse problems for quantum trees II: recovering matching conditions for star graphs. Inverse Probl. Imaging 4, 579–598 (2010) [4] Band, R., Parzanchevski, O., Ben-Shach, G.: The isospectral fruits of repre- sentation theory: quantum graphs and drums. J. Phys. A 42 (2009) [5] von Below, J.: Can one hear the shape of a network. In: Partial Diﬀerential Equations on Multistructures (Lecture Notes in Pure and Appliced Mathemat- ics vol. 219) (New York: Dekker) pp. 19–36 (2000) [6] Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs. Mathematical Surveys and Monographs, Vol. 186. AMS (2013) [7] Borg, G.: ‘Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bes- timmung der Diﬀerentialgleichung durch die Eigenwerte (German). Acta Math. 78, 1–96 (1946) [8] Brown, B.M., Weikard, R.: A Borg-Levinson theorem for trees. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461, 3231–3243 (2005) [9] Carlsson, R.: Inverse eigenvalue problems on directed graphs. Trans. Am. Math. Soc. 351, 4069–88 (1999) [10] Carlson, R., Pivovarchik, V.: Ambartsumian’s theorem for trees. Electron. J. Diﬀer. Equ. 142, 1–9 (2007) [11] Carlson, R., Pivovarchik, V.: Spectral asymptotics for quantum graphs with equal edge lengths. J. Phys. A 41 (2008), no. 14, 145202, 16 pp IEOT Schr¨ odinger Operators on Graphs and Geometry II Page 23 of 24 40 [12] Davies, E.B.: An inverse spectral theorem. J. Oper. Theory 69, 195–208 (2013) [13] Friedlander, L.: Extremal properties of eigenvalues for a metric graph. Ann. Inst. Fourier 55, 199–211 (2005) [14] Gerasimenko, N.I., Pavlov, B.S.: A scattering problem on noncompact graphs, (Russian) Teoret. Mat. Fiz.74 (1988) 345–359; translation in Theoret. and Math. Phys74 (1988) 230–240 [15] Gutkin, B., Smilansky, U.: Can one hear the shape of a graph? J. Phys. A: Math. Gen. 34, 6061–8 (2001) [16] Hardy, G.M., Wright, E.M.: revised by Heath-Brown, D.R., Silverman, J.H.: An Introduction to the Theory of Numbers, 6th edn., Oxford University Press, Oxford (2008) [17] Kac, I., Pivovarchik, V.: On multiplicity of a quantum graph spectrum. J. Phys. A44 (2011) [18] Kurasov, P., Nowaczyk, M.: Inverse spectral problem for quantum graphs. J. Phys. A: Math. Gener. 38, 4901–4915 (2005) [19] Kurasov, P.: Graph Laplacians and topology. Ark. Mat. 46, 95–111 (2008) [20] Kurasov, P.: Schr¨ odinger operators on graphs and geometry I: essentially bounded potentials. J. Func. Anal. 254, 934–953 (2008) [21] Kurasov, P.: Inverse problems for Aharonov-Bohm rings. Math. Proc. Camb. Philos. Soc. 148, 331–362 (2010) [22] Kurasov, P.: Can one distinguish quantum trees from the boundary? Proc. Amer. Math. Soc. 140, 2347–2356 (2012) [23] Kurasov, P.: Inverse scattering for lasso graph. J. Math. Phys. 54 (2013) [24] Kurasov, P., Enerb¨ ack, M.: Aharonov-Bohm ring touching a quantum wire: how to model it and to solve the inverse problem. Rep. Math. Phys. 68, 271– 287 (2011) [25] Kurasov, P., Naboko, S.: Rayleigh estimates for diﬀerential operators on graphs. J. Spectral Theory 4 (2014) [26] Kurasov, P.: Quantum Graphs (to be published) [27] Law, C.K., Yanagida, E.: A Solution to an Ambarsumyan Problem on Trees. Kodai Math. J. 35, 358–373 (2012) [28] Nicaise, S.: Spectre des reseaux topologiques ﬁnis. Bull. Sci. Math. 11, 401–413 (1987) [29] Pivovarchik, V.N.: Ambartsumian’s Theorem for Sturm-Liouville boundary value problem on a star-shaped graph. Funct. Anal. Appl. 39, 148–151 (2005) [30] Pivovarchik, V.: Inverse problem for the Sturm-Liouville equation on a star- shaped graph. Math. Nachr. 280, 1595–1619 (2007) [31] Pivovarchik, V., Rozhenko, N.: Inverse Sturm-Liouville problem on equilateral regular tree. Appl. Anal. 92, 784–798 (2013) [32] Reed, M., Simon, B.: Methods of Modern Mathematical Physics I–IV. Aca- demic, New York (1972–1979) [33] Rundell, W., Sacks, P.: Inverse eigenvalue problem for a simple star graph. J. Spectr. Theory 5, 363–380 (2015) [34] Visco-Comandini, F., Mirrahimi, M., Sorine, M.: Some inverse scattering prob- lems on star-shaped graphs. J. Math. Anal. Appl. 378, 343–358 (2011) [35] Yang, C.F.: Inverse spectral problems for the Sturm-Liouville operator on a d-star graph. J. Math. Anal. Appl. 365, 742–749 (2010) 40 Page 24 of 24 J. Boman et al. IEOT [36] Yang, C.F., Huang, Z.Y., Yang, X.P.: Ambarzumyan-type theorems for the Sturm-Liouville equation on a graph. Rocky Mountain J. Math. 39, 1353–1372 (2009) [37] Yurko, V.: Inverse spectral problems for Sturm-Liouville operators on graphs. Inverse Probl. 21, 1075–1086 (2005) [38] Yurko, V.: Inverse problems for Sturm-Liouville operators on graphs with a cycle’. Oper. Matrices 2, 543–553 (2008) [39] Yurko, V.: Reconstruction of Sturm-Liouville operators from the spectra on a graph with a cycle’ (Russian), Mat. Sb. 200 (2009) 147–160; translation in Sb. Math. 200(2009), 1403–1415 [40] Yurko, V.: Uniqueness of recovering Sturm-Liouville operators on a-graphs from spectra. Results Math. 55, 199–207 (2009) [41] Yurko, V.: An inverse problem for Sturm-Liouville diﬀerential operators on A-graphs. Appl. Math. Lett. 23, 875–879 (2010) Jan Boman, Pavel Kurasov ( ) and Rune Suhr Department of Mathematics Stockholm University 106 91 Stockholm Sweden e-mail: kurasov@math.su.se Jan Boman e-mail: jabo@math.su.se Rune Suhr e-mail: suhr@math.su.se Received: January 19, 2018. Revised: April 23, 2018.

Journal

Integral Equations and Operator Theory
– Springer Journals

Published: May 30, 2018

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References

Über eine Frage der Eigenwerttheorie

Ambartsumian, V

Inverse problems for quantum trees

Avdonin, S; Kurasov, P

Inverse problems for quantum trees II: recovering matching conditions for star graphs

Avdonin, S; Kurasov, P; Nowaczyk, M

‘Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte (German)

Borg, G

A Borg-Levinson theorem for trees

Brown, BM; Weikard, R

Inverse eigenvalue problems on directed graphs

Carlsson, R

Ambartsumian’s theorem for trees

Carlson, R; Pivovarchik, V

An inverse spectral theorem

Davies, EB

Extremal properties of eigenvalues for a metric graph

Friedlander, L

Can one hear the shape of a graph?

Gutkin, B; Smilansky, U

Inverse spectral problem for quantum graphs

Kurasov, P; Nowaczyk, M

Graph Laplacians and topology

Kurasov, P

Schrödinger operators on graphs and geometry I: essentially bounded potentials

Kurasov, P

Inverse problems for Aharonov-Bohm rings

Kurasov, P

Can one distinguish quantum trees from the boundary?

Kurasov, P

Aharonov-Bohm ring touching a quantum wire: how to model it and to solve the inverse problem

Kurasov, P; Enerbäck, M

A Solution to an Ambarsumyan Problem on Trees

Law, CK; Yanagida, E

Spectre des reseaux topologiques finis

Nicaise, S

Ambartsumian’s Theorem for Sturm-Liouville boundary value problem on a star-shaped graph

Pivovarchik, VN

Inverse problem for the Sturm-Liouville equation on a star-shaped graph

Pivovarchik, V

Inverse Sturm-Liouville problem on equilateral regular tree

Pivovarchik, V; Rozhenko, N

Inverse eigenvalue problem for a simple star graph

Rundell, W; Sacks, P

Some inverse scattering problems on star-shaped graphs

Visco-Comandini, F; Mirrahimi, M; Sorine, M

Inverse spectral problems for the Sturm-Liouville operator on a d-star graph

Yang, CF

Ambarzumyan-type theorems for the Sturm-Liouville equation on a graph

Yang, CF; Huang, ZY; Yang, XP

Inverse spectral problems for Sturm-Liouville operators on graphs

Yurko, V

Inverse problems for Sturm-Liouville operators on graphs with a cycle’

Yurko, V

Uniqueness of recovering Sturm-Liouville operators on a-graphs from spectra

Yurko, V

An inverse problem for Sturm-Liouville differential operators on A-graphs

Yurko, V

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