A new treatment for some periodic Schr\"odinger operators II: the wave function

TL;DR

This paper develops asymptotic wave functions for periodic Schrödinger operators, linking them to supersymmetric gauge theory and supporting monodromy relations, with implications for mathematical physics and number theory.

Contribution

It introduces a novel method to derive asymptotic wave functions using eigenvalues, extending previous work and connecting to instanton partition functions.

Findings

Asymptotic wave functions are explicitly derived from eigenvalues.

Supports monodromy relations for Floquet exponents.

Links wave functions to N=2 supersymmetric gauge theory.

Abstract

Following the approach of our previous paper we continue to study the asymptotic solution of periodic Schr\"{o}dinger operators. Using the eigenvalues obtained earlier the corresponding asymptotic wave functions are derived. This gives further evidence in favor of the monodromy relations for the Floquet exponent proposed in the previous paper. In particular, the large energy asymptotic wave functions are related to the instanton partition function of N=2 supersymmetric gauge theory with surface operator. A relevant number theoretic dessert is appended.