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 (N−12)×(N−12) inside triangles with vertices (0,0),(0,λβ−−√) and (λ/β−−√,0) with respect to β>0, for fixed λ≥0. This lattice point formulation of the problem naturally leads to a family of generalized problems where one instead considers the shifted lattice (N+σ)×(N+τ), for σ,τ>−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, σ=τ=−12, is a critical case.

We provide some characterizations for SOC-monotone and SOC-convex functions by using differential analysis. From these characterizations, we particularly obtain that a continuously differentiable function defined in an open interval is SOC-monotone (SOC-convex) of order <i>n</i> 3 if and only if it is 2-matrix monotone (matrix convex), and furthermore, such a function is also SOC-monotone (SOC-convex) of order <i>n</i> 2 if it is 2-matrix monotone (matrix convex). In addition, we also prove that Conjecture 4.2 proposed in Chen (Optimization 55:363385, 2006) does not hold in general. Some examples are included to illustrate that these characterizations open convenient ways to verify the SOC-monotonicity and the SOC-convexity of a continuously differentiable function defined on an open interval, which are often involved in the solution methods of the convex second-order cone optimization.

We discuss operators of the type H=-\Delta+ V (x)-lpha\delta (x-\Sigma) with an attractive interaction, H=-\Delta+ V (x)-lpha\delta (x-\Sigma) , in H=-\Delta+ V (x)-lpha\delta (x-\Sigma) , where H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is an infinite surface, asymptotically planar and smooth outside a compact, dividing the space into two regions, of which one is supposed to be convex, and H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is a potential bias being a positive constant H=-\Delta+ V (x)-lpha\delta (x-\Sigma) in one of the regions and zero in the other. We find the essential spectrum and ask about the existence of the discrete one with a particular attention to the critical case, H=-\Delta+ V (x)-lpha\delta (x-\Sigma) . We show that H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is then empty if the bias is supported in theexterior'region, while in the opposite case isolated eigenvalues may exist.

We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian \mathsf {H} on an unbounded, radially symmetric (generalized) parabolic layer \mathsf {H} . It was known before that \mathsf {H} has an infinite number of eigenvalues below the threshold of its essential spectrum. In the present paper, we find the discrete spectrum asymptotics for \mathsf {H} by means of a consecutive reduction to the analogous asymptotic problem for an effective one-dimensional Schrdinger operator on the half-line with the potential the behaviour of which far away from the origin is determined by the geometry of the layer \mathsf {H} at infinity.

We analyze a family of singular Schrdinger operators describing a Neumann waveguide with a periodic array of singular traps of a delta\prime type. We show that in the limit when perpendicular size of the guide tends to zero and the delta\prime interactions are appropriately scaled, the first spectral gap is determined exclusively by geometric properties of the traps.