  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.
