# On Leighton's Comparison Theorem

## TL;DR

This paper presents a simple, flexible comparison theorem for a class of differential equations, including Schrödinger and Jacobi difference equations, with broad applicability due to adaptable auxiliary functions.

## Contribution

It introduces a new, simplified proof of a comparison theorem that allows for flexible auxiliary functions, extending its applicability to various differential and difference equations.

## Key findings

- The theorem applies to equations with integrable coefficients.
- It encompasses Schrödinger equations with distributional potentials.
- Includes Jacobi difference equations as special cases.

## Abstract

We give a simple proof of a fairly flexible comparison theorem for equations of the type $-(p(u'+su))'+rp(u'+su)+qu=0$ on a finite interval where $1/p$, $r$, $s$, and $q$ are real and integrable. Flexibility is provided by two functions which may be chosen freely (within limits) according to the situation at hand. We illustrate this by presenting some examples and special cases which include Schr\"odinger equations with distributional potentials as well as Jacobi difference equations.