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Spectral Analysis of the Schrödinger Operator for the Incommensurate System

Yan Li, Yujian Song, Aihui Zhou

TL;DR

The paper addresses spectral analysis for incommensurate (quasi-periodic) Schrödinger operators by embedding the system into a higher-dimensional periodic setting and introducing a regularization to obtain an elliptic, periodic model. It proves spectral equivalence between the original operator ${\mathscr{H}}$ and a self-adjoint extended operator ${\tilde{\mathscr{H}}}$, and shows that the spectrum can be approximated by the spectra of the regularized operators ${\tilde{\mathscr{H}}^{\delta}}$ as $\delta\to0^{+}$, with Bloch-type solutions that are likewise approximable by regularized Bloch solutions. The framework enables the use of Bloch-Floquet theory and existing periodic-system numerics to study incommensurate layered materials, offering rigorous convergence guarantees and a path to scalable computation in higher dimensions and multiple layers. This provides a solid theoretical foundation for understanding and computing observable quantities in incommensurate systems and suggests broad applicability beyond the current two-layer scenario.

Abstract

Many novel and unique physical phenomena of incommensurate systems can be illustrated and predicted using the spectra of the associated Schrödinger 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 Schrödinger operator for the incommensurate system can be approximated by the spectra of a family of regularized Schrödinger operators, which are elliptic, retain periodicity, and enjoy favorable analytic and spectral properties. We also show the existence of Bloch-type solutions to the Schrödinger 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.

Spectral Analysis of the Schrödinger Operator for the Incommensurate System

TL;DR

The paper addresses spectral analysis for incommensurate (quasi-periodic) Schrödinger operators by embedding the system into a higher-dimensional periodic setting and introducing a regularization to obtain an elliptic, periodic model. It proves spectral equivalence between the original operator and a self-adjoint extended operator , and shows that the spectrum can be approximated by the spectra of the regularized operators as , with Bloch-type solutions that are likewise approximable by regularized Bloch solutions. The framework enables the use of Bloch-Floquet theory and existing periodic-system numerics to study incommensurate layered materials, offering rigorous convergence guarantees and a path to scalable computation in higher dimensions and multiple layers. This provides a solid theoretical foundation for understanding and computing observable quantities in incommensurate systems and suggests broad applicability beyond the current two-layer scenario.

Abstract

Many novel and unique physical phenomena of incommensurate systems can be illustrated and predicted using the spectra of the associated Schrödinger 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 Schrödinger operator for the incommensurate system can be approximated by the spectra of a family of regularized Schrödinger operators, which are elliptic, retain periodicity, and enjoy favorable analytic and spectral properties. We also show the existence of Bloch-type solutions to the Schrödinger 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.
Paper Structure (14 sections, 17 theorems, 143 equations)

This paper contains 14 sections, 17 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Incommensurate system
  4. Bloch-Floquet transform
  5. The extended Schrödinger operator $\tilde{\mathscr{H}}$
  6. Self-adjoint extension
  7. Bloch-Floquet transform on $\mathcal{G}$
  8. A regularized Schrödinger operator $\tilde{\mathscr{H}}^{\delta}$
  9. Regularization
  10. Relationship between $\sigma_{\mathcal{G}}(\tilde{\mathscr{H}})$ and $\sigma(\tilde{\mathscr{H}}^{\delta})$
  11. Spectrum of the Schrödinger operator $\mathscr{H}$
  12. Relationship between $\sigma(\mathscr{H})$ and $\sigma(\tilde{\mathscr{H}}^{\delta})$
  13. Bloch-type solution
  14. Concluding remarks

Key Result

Lemma 2.3

\newlabellemma:isome0 The Bloch-Floquet transform $\mathcal{U}_{\tilde{\mathcal{R}}}$ acting on $L^{2}(\mathbb{R}^{2d})$ has the following properties:

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 3.1
  • Proof 1
  • Lemma 3.2
  • Proposition 3.3
  • Proof 2
  • Theorem 3.4
  • ...and 21 more