# Inequalities between Neumann and Dirichlet eigenvalues of Schr\"odinger operators

## TL;DR

This paper establishes inequalities between Neumann and Dirichlet eigenvalues of Schrödinger operators on bounded domains, highlighting differences between one-dimensional and higher-dimensional cases based on potential properties.

## Contribution

It introduces new inequalities relating Neumann and Dirichlet eigenvalues for Schrödinger operators, considering potential monotonicity and convexity, extending classical Laplacian results.

## Key findings

- Inequalities depend on potential's monotonicity and convexity.
- Results differ between one-dimensional and higher-dimensional cases.
- Classical inequalities are extended to Schrödinger operators.

## Abstract

Given a Schr\"odinger operator with a real-valued potential on a bounded, convex domain or a bounded interval we prove inequalities between the eigenvalues corresponding to Neumann and Dirichlet boundary conditions, respectively. The obtained inequalities depend partially on monotonicity and convexity properties of the potential. The results are counterparts of classical inequalities for the Laplacian but display some distinction between the one-dimensional case and higher dimensions.