Another look at qualitative properties of eigenvalues using effective Hamiltonians

TL;DR

This paper reviews qualitative properties of eigenvalues in various problems using effective Hamiltonians and the Hopf-Cole transform, highlighting differences between quadratic and non-quadratic cases.

Contribution

It introduces a novel perspective on eigenvalue properties via effective Hamiltonians and the Hopf-Cole transform, providing new insights into monotonicity and scaling limits.

Findings

Revisits monotonicity results for eigenvalues.

Derives scaling limits using Donsker-Varadhan formula.

Identifies differences between quadratic and non-quadratic Hamiltonians.

Abstract

The goal of this paper is to review several qualitative properties of well-known eigenvalue problems using a different perspective based on the theory of effective Hamiltonians, working exclusively on the Hopf-Cole transform of the equation. We revisit some monotonicity results as well as the derivation of several scaling limits by means of the Donsker-Varadhan formula, and we point out several differences between the case of quadratic Hamiltonians and non-quadratic ones.