On the Essential Spectrum of Quantum Graphs

Abstract

Let $\Gamma \subset \mathbb {R}^{n}$ be a graph periodic with respect to the action of a group $\mathbb {G}$ isomorphic to $\mathbb {Z}^{m},1\le m\le n.$ We consider a one-dimensional Schrödinger operator

$$\begin{aligned} S_{q}u(x)=\left( -\frac{d^{2}}{dx^{2}}+q(x)\right) u(x),u\in C_{0}^{\infty }(\Gamma \backslash \mathcal {V)},q\in L^{\infty }(\Gamma ) \end{aligned}$$

defined on the edges of the graph $\Gamma$, where $\mathcal {V}$ is the set of the vertices of $\Gamma$. The operator $S_{q}$ is extended to a closed unbounded operator $\mathcal {H}_{q}$ in $L^{2}(\Gamma )$ with domain $\tilde{H} ^{2}(\Gamma )$ consisting of functions u belonging to the Sobolev space $H^{2}(e)$ on the edges e of the graph $\Gamma$ and satisfying the Kirchhoff–Neumann conditions at the vertices of $\Gamma .$ For the unbounded operator $\mathcal {H}_{q}$ we introduce a family $Lim (\mathcal {H}_{q})$ of limit operators $\mathcal {H}_{q}^{g}$ defined by the sequences $\mathbb {G\ni }g_{m}\rightarrow \infty$ and prove that

$$\begin{aligned} sp_{ess}\mathcal {H}_{q}= {\displaystyle \bigcup \limits _{\mathcal {H}_{q}^{g}\in Lim(\mathcal {H}_{q})}} sp\mathcal {H}_{q}^{g}. \end{aligned}$$

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.