Integr. Equ. Oper. Theory 88 (2017), 339–362
Published online July 6, 2017
Springer International Publishing AG 2017
and Operator Theory
On the Essential Spectrum of Quantum
Abstract. Let Γ ⊂ R
be a graph periodic with respect to the action of a
group G isomorphic to Z
, 1 ≤ m ≤ n. We consider a one-dimensional
(Γ\V),q ∈ L
deﬁned on the edges of the graph Γ, where V is the set of the vertices of Γ.
The operator S
is extended to a closed unbounded operator H
(Γ) consisting of functions u belonging to the Sobolev
(e) on the edges e of the graph Γ and satisfying the Kirchhoﬀ–
Neumann conditions at the vertices of Γ. For the unbounded operator
we introduce a family Lim(H
) of limit operators H
deﬁned by the
sequences G g
→∞and prove that
We apply this result to the calculation of the essential spectra of self-
adjoint Schr¨odinger operators with periodic potentials perturbed by
terms slowly oscillating at inﬁnity. We show that such perturbations
signiﬁcantly change the structure of the spectrum of Schr¨odinger oper-
ators with periodic potentials.
Mathematics Subject Classiﬁcation. Primary: 34B45, 35R02, 81Q35.
Keywords. Quantum graphs, Fredholm theory, Essential spectrum,
In mathematics and physics, a quantum graph is a linear, network-shaped
structure of vertices connected by edges with a diﬀerential or pseudo-
diﬀerential operator acting on functions deﬁned on its edges. Works devoted
This work is partially supported by the National System of Investigators of Mexico (SNI),
and the Conacyt Project SB-179872.