Periodic quantum graphs with predefined spectral gaps

Andrii Khrabustovskyi

doi:10.1088/1751-8121/aba98b

Abstract

Let Γ be an arbitrary -periodic metric graph, which does not coincide with a line. We consider the Hamiltonian on Γ with the action −ɛ −1d2/dx 2 on its edges; here ɛ > 0 is a small parameter. Let . We show that under a proper choice of vertex conditions the spectrum of has at least m gaps as ɛ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of as ɛ → 0 can be completely controlled through a suitable choice of coupling constants standing in those vertex conditions. We also show how to ensure for fixed (small enough) ɛ the precise coincidence of the left endpoints of the first m spectral gaps with predefined numbers.

Highlights

The name quantum graph refers to a pair (Γ, H ), where Γ is a network-shaped structure of vertices connected by edges of certain positive lengths and H is a second order self-adjoint differential operator on Γ (Hamiltonian)
Quantum graphs arise naturally in mathematics, physics, chemistry and engineering as simplified models of wave propagation in quasi-one-dimensional systems looking like narrow neighborhoods of graphs
Our goal is to improve this result: we show that under a proper choice of α j one can ensure the precise coincidence the left endpoints of the spectral gaps of Hε [α, β, γ] with prescribed numbers

Summary

INTRODUCTION

The name quantum graph refers to a pair (Γ, H ), where Γ is a network-shaped structure of vertices connected by edges of certain positive lengths (metric graph) and H is a second order self-adjoint differential operator on Γ (Hamiltonian). Given a fixed graph we “decorate” it changing its geometrical structure at each vertex: either one attaches to each vertex a copy of certain fixed compact graph [28] (see [39] where similar idea was used for discrete graphs) or in each vertex one disconnects the edges emerging from it and connects their loose endpoints by a certain additional graph (“spider”) [8, 37] Another way to open spectral gaps is to use “advanced” vertex conditions. When designing materials with prescribed properties it is desirable to open up spectral gaps, and be able to control their location and length – via a suitable choice of operator coefficients or/and geometry of the medium We addressed this problem for various classes of periodic operators in a series of papers [4, 11, 17,18,19].

SETTING OF THE PROBLEM AND MAIN RESULT

CONTROL OVER THE ENDPOINTS OF SPECTRAL GAPS