On the Essential Spectrum of Quantum Graphs

On the Essential Spectrum of Quantum Graphs Let $$\Gamma \subset \mathbb {R}^{n}$$ Γ ⊂ R n be a graph periodic with respect to the action of a group $$\mathbb {G}$$ G isomorphic to $$\mathbb {Z}^{m},1\le m\le n. $$ Z m , 1 ≤ m ≤ n . We consider a one-dimensional Schrödinger operator $$\begin{aligned} S_{q}u(x)=\left( -\frac{d^{2}}{dx^{2}}+q(x)\right) u(x),u\in C_{0}^{\infty }(\Gamma \backslash \mathcal {V)},q\in L^{\infty }(\Gamma ) \end{aligned}$$ S q u ( x ) = - d 2 d x 2 + q ( x ) u ( x ) , u ∈ C 0 ∞ ( Γ \ V ) , q ∈ L ∞ ( Γ ) defined on the edges of the graph $$\Gamma $$ Γ , where $$\mathcal {V}$$ V is the set of the vertices of $$\Gamma $$ Γ . The operator $$S_{q}$$ S q is extended to a closed unbounded operator $$\mathcal {H}_{q}$$ H q in $$L^{2}(\Gamma )$$ L 2 ( Γ ) with domain $$\tilde{H} ^{2}(\Gamma )$$ H ~ 2 ( Γ ) consisting of functions u belonging to the Sobolev space $$H^{2}(e)$$ H 2 ( e ) on the edges e of the graph $$\Gamma $$ Γ and satisfying the Kirchhoff–Neumann conditions at the vertices of $$\Gamma .$$ Γ . For the unbounded operator $$\mathcal {H}_{q}$$ H q we introduce a family $$Lim (\mathcal {H}_{q})$$ L i m ( H q ) of limit operators $$\mathcal {H}_{q}^{g}$$ H q g defined by the sequences $$\mathbb {G\ni }g_{m}\rightarrow \infty $$ G ∋ g m → ∞ and prove that $$\begin{aligned} sp_{ess}\mathcal {H}_{q}= {\displaystyle \bigcup \limits _{\mathcal {H}_{q}^{g}\in Lim(\mathcal {H}_{q})}} sp\mathcal {H}_{q}^{g}. \end{aligned}$$ s p e s s H q = ⋃ H q g ∈ L i m ( H q ) s p H q g . We apply this result to the calculation of the essential spectra of self-adjoint Schrödinger operators with periodic potentials perturbed by terms slowly oscillating at infinity. We show that such perturbations significantly change the structure of the spectrum of Schrödinger operators with periodic potentials. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Integral Equations and Operator Theory Springer Journals

On the Essential Spectrum of Quantum Graphs

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer International Publishing AG
Subject
Mathematics; Analysis
ISSN
0378-620X
eISSN
1420-8989
D.O.I.
10.1007/s00020-017-2386-6
Publisher site
See Article on Publisher Site

Abstract

Let $$\Gamma \subset \mathbb {R}^{n}$$ Γ ⊂ R n be a graph periodic with respect to the action of a group $$\mathbb {G}$$ G isomorphic to $$\mathbb {Z}^{m},1\le m\le n. $$ Z m , 1 ≤ m ≤ n . We consider a one-dimensional Schrödinger operator $$\begin{aligned} S_{q}u(x)=\left( -\frac{d^{2}}{dx^{2}}+q(x)\right) u(x),u\in C_{0}^{\infty }(\Gamma \backslash \mathcal {V)},q\in L^{\infty }(\Gamma ) \end{aligned}$$ S q u ( x ) = - d 2 d x 2 + q ( x ) u ( x ) , u ∈ C 0 ∞ ( Γ \ V ) , q ∈ L ∞ ( Γ ) defined on the edges of the graph $$\Gamma $$ Γ , where $$\mathcal {V}$$ V is the set of the vertices of $$\Gamma $$ Γ . The operator $$S_{q}$$ S q is extended to a closed unbounded operator $$\mathcal {H}_{q}$$ H q in $$L^{2}(\Gamma )$$ L 2 ( Γ ) with domain $$\tilde{H} ^{2}(\Gamma )$$ H ~ 2 ( Γ ) consisting of functions u belonging to the Sobolev space $$H^{2}(e)$$ H 2 ( e ) on the edges e of the graph $$\Gamma $$ Γ and satisfying the Kirchhoff–Neumann conditions at the vertices of $$\Gamma .$$ Γ . For the unbounded operator $$\mathcal {H}_{q}$$ H q we introduce a family $$Lim (\mathcal {H}_{q})$$ L i m ( H q ) of limit operators $$\mathcal {H}_{q}^{g}$$ H q g defined by the sequences $$\mathbb {G\ni }g_{m}\rightarrow \infty $$ G ∋ g m → ∞ and prove that $$\begin{aligned} sp_{ess}\mathcal {H}_{q}= {\displaystyle \bigcup \limits _{\mathcal {H}_{q}^{g}\in Lim(\mathcal {H}_{q})}} sp\mathcal {H}_{q}^{g}. \end{aligned}$$ s p e s s H q = ⋃ H q g ∈ L i m ( H q ) s p H q g . We apply this result to the calculation of the essential spectra of self-adjoint Schrödinger operators with periodic potentials perturbed by terms slowly oscillating at infinity. We show that such perturbations significantly change the structure of the spectrum of Schrödinger operators with periodic potentials.

Journal

Integral Equations and Operator TheorySpringer Journals

Published: Jul 6, 2017

References

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