A new treatment for some periodic Schr\"{o}dinger operators I: the eigenvalue

TL;DR

This paper investigates the Floquet properties of periodic Schrödinger operators with elliptic potentials, revealing new relations between Floquet exponents and monodromy that extend classical theory and produce consistent asymptotic solutions.

Contribution

It introduces novel relations between Floquet exponents and monodromy for elliptic potentials, expanding understanding of spectral solutions in periodic Schrödinger operators.

Findings

Identified relations between Floquet exponent and monodromy for elliptic potentials.

Discovered two new relations not explained by classical Floquet theory.

Produced both old and new asymptotic solutions consistent with existing results.

Abstract

We study the problem of how the Floquet property manifests for periodic Schr\"{o}dinger operators which are known to have multiple of asymptotic spectral solutions. The main conclusions are made for elliptic potentials, we demonstrate that for each period of the elliptic function there is a relation about the Floquet exponent and the monodromy of wave function. Among them there are two relations not explained by the classical Floquet theory. These relations produce both old and new asymptotic solutions consistent with results already known.