A Mathematical Aspect of Bloch's Theorem
Yan Li, Bin Yang, Aihui Zhou
TL;DR
This paper investigates the solutions of periodic Schrödinger equations in multiple dimensions, revealing that not all solutions are Bloch solutions, and explores properties of solutions and quasimomenta.
Contribution
It classifies solutions of multidimensional periodic Schrödinger equations and clarifies the relationship between bounded solutions and Bloch solutions.
Findings
Not all solutions are Bloch solutions in multidimensional cases
Properties of solutions and quasimomenta are characterized
Relationship between bounded solutions and Bloch solutions is elucidated
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.
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Taxonomy
TopicsHistory and Theory of Mathematics · Mathematics and Applications