WKB Methods for Finite Difference Schrodinger Equations

TL;DR

This thesis develops WKB methods for finite difference Schrödinger equations, creating an all-order algorithm, analyzing solutions like Bessel functions, and deriving connection formulas to study spectral properties.

Contribution

It introduces a comprehensive WKB framework for finite difference Schrödinger equations, including quantum momentum, all-order corrections, and connection formulas for spectral analysis.

Findings

All-order WKB algorithm for finite difference Schrödinger equations.

Discovery of additional periodic factors affecting solutions.

Construction of connection formulas for spectral problems.

Abstract

In this thesis, we develop WKB techniques for the finite difference Schrodinger equation, following the construction of the WKB approach for the standard differential Schrodinger equation. In particular, we will develop an all-order WKB algorithm to get arbitrary hbar-corrections and construct a general quantum momentum, underlining the various properties of its coefficients and the quantities that will be used when constructing the quantization condition. In doing so, we discover the existence of additional periodic factors that need to be considered in order to obtain the most general solution to the problem at hand. We will then proceed to study the simplest non trivial example, the linear potential case and the Bessel functions, that provide a solution to the linear problem. After studying the resurgence properties of the Bessel functions from an analytical and numerical point of view, we will then proceed to use those results in order to build general connection formulae, allowing us to connect the local solutions defined on two sides of a turning point into a smooth solution on the whole real line. With those connection formulae, we will analyse a selection of problems, constructing the discrete spectrum of various finite difference Schrodinger problems and comparing our results with existing literature.