Maximizing Riesz means of anisotropic harmonic oscillators

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 $(\mathbb{N}-\frac{1}{2}) \times (\mathbb{N}-\frac{1}{2})$ inside triangles with vertices $(0, 0), (0, \lambda \sqrt{\beta})$ and $(\lambda / \sqrt{\beta}, 0)$ with respect to $\beta \gt 0$, 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 $\sigma, \tau \gt -1$. 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, $\sigma = \tau = -\frac{1}{2}$, is a critical case.