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A Mathematical Aspect of Bloch's Theorem

Yan Li, Bin Yang, Aihui Zhou

Abstract

In this paper, by studying a class of 1-D Sturm-Liouville problems with periodic coefficients, we show and classify the solutions of periodic Schrodinger equations in a multidimensional case, which tells that not all the solutions are Bloch solutions. In addition, we also provide several properties of the solutions and quasimomenta and illustrate the relationship between bounded solutions and Bloch solutions.

A Mathematical Aspect of Bloch's Theorem

Abstract

In this paper, by studying a class of 1-D Sturm-Liouville problems with periodic coefficients, we show and classify the solutions of periodic Schrodinger equations in a multidimensional case, which tells that not all the solutions are Bloch solutions. In addition, we also provide several properties of the solutions and quasimomenta and illustrate the relationship between bounded solutions and Bloch solutions.

Paper Structure

This paper contains 7 sections, 19 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Bloch-Floquet transform
  3. Classfication of solutions
  4. 1-D Strum-Liouville problem
  5. Multidimensions
  6. Concluding remarks
  7. Details of one dimensional cases

Key Result

Theorem \oldthetheorem

Let $\bar{\mathbb{T}}$ be the closure of $\mathbb{T}$, then $\mathcal{U}_{\mathcal{R}}$ has the following properties:

Theorems & Definitions (39)

  • Definition \oldthetheorem: kuchment2016overviewgontier2016convergence
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Proof 1
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • ...and 29 more