Creation of spectral bands for a periodic domain with small windows

TL;DR

This paper investigates how small windows in a periodic domain influence the spectral properties of a Schroedinger operator, revealing that virtual levels become spectral bands as the windows close, with detailed analysis of their structure and asymptotics.

Contribution

It introduces a novel analysis of spectral band formation from virtual levels in periodic domains with small windows, extending understanding of spectral behavior in such systems.

Findings

Spectral bands emerge from virtual levels as windows close.

Asymptotic descriptions of spectral band structure.

Analysis of the transition from virtual levels to spectral bands.

Abstract

We consider a Schroediner operator in a periodic system of strip-like domains coupled by small windows. As the windows close, the domain decouples into an infinite series of identical domains. The operator similar to the original one but on one copy of these identical domains has an essential spectrum. We show that once there is a virtual level at the threshold of this essential spectrum, the windows turns this virtual level into the spectral bands for the original operator. We study the structure and the asymptotic behavior of these bands.