Maximizing Riesz means of anisotropic harmonic oscillators

TL;DR

This paper investigates the asymptotic behavior of eigenvalues of anisotropic harmonic oscillators, focusing on Riesz means and heat kernel traces, by translating the problem into lattice point counting within specific geometric regions.

Contribution

It introduces a lattice point reformulation of eigenvalue minimization for anisotropic oscillators and analyzes the impact of shifts in the lattice, identifying the harmonic oscillator case as critical.

Findings

The problem reduces to counting lattice points in triangles for fixed parameters.

Different shift parameters lead to qualitatively different problem behaviors.

The harmonic oscillator case with specific shifts is identified as a critical scenario.

Abstract

We consider problems related to the asymptotic minimization of eigenvalues of anisotropic harmonic oscillators in the plane. In particular we study Riesz means of the eigenvalues and the trace of the corresponding heat kernels. The eigenvalue minimization problem can be reformulated as a lattice point problem where one wishes to maximize the number of points of $(\mathbb{N}-\tfrac12)\times(\mathbb{N}-\tfrac12)$ inside triangles with vertices $(0, 0), (0, \lambda \sqrt{\beta})$ and $(\lambda/{\sqrt{\beta}}, 0)$ with respect to $\beta>0$, for fixed $\lambda\geq 0$. This lattice point formulation of the problem naturally leads to a family of generalized problems where one instead considers the shifted lattice $(\mathbb{N}+\sigma)\times(\mathbb{N}+\tau)$, for $\sigma, \tau >-1$. We show that the nature of these problems are rather different depending on the shift parameters, and in particular that the problem corresponding to harmonic oscillators, $\sigma=\tau=-\tfrac12$, is a critical case.