Spectral Analysis of the Schr\"odinger Operator for the Incommensurate System

TL;DR

This paper develops a spectral analysis framework for incommensurate systems by embedding them into higher dimensions and using regularization, enabling approximation of their spectra and solutions with periodic, elliptic operators.

Contribution

Introduces a method to approximate spectra of incommensurate Schr"odinger operators via regularized periodic operators in higher dimensions.

Findings

Spectra of incommensurate systems can be approximated by regularized operators.

Existence of Bloch-type solutions for incommensurate Schr"odinger equations.

Regularized operators retain favorable spectral properties.

Abstract

Many novel and unique physical phenomena of incommensurate systems can be illustrated and predicted using the spectra of the associated Schr\"odinger operators. However, the absence of periodicity in these systems poses significant challenges for obtaining the spectral information. In this paper, by embedding the system into higher dimensions together with introducing a regularization technique, we prove that the spectrum of the Schr\"odinger operator for the incommensurate system can be approximated by the spectra of a family of regularized Schr\"odinger operators, which are elliptic, retain periodicity, and enjoy favorable analytic and spectral properties. We also show the existence of Bloch-type solutions to the Schr\"odinger equation for the incommensurate system, which can be well approximated by the Bloch solutions to the equations associated with the regularized operators. Our analysis provides a theoretical support for understanding and computing the incommensurate systems.