Negative eigenvalue estimates for polyharmonic Schrödinger operators with measure-potentials: the subcritical case

2026/07/22 by Medet Nursultanov, Grigori Rozenblum
#math.SP

paper · pdf

Abstract

We study spectral estimates for polyharmonic Schrödinger operators -Δ^l-μ in the subcritical regime 2l<N. The measure potential μ is assumed to satisfy a capacitary smallness condition which guarantees that the corresponding operator is semibounded and self-adjoint. With such a measure μ we associate an Otelbaev function, which reflects both the local concentration and the spatial distribution of the potential. In terms of this function, we obtain two-sided estimates for the distribution function of the negative eigenvalues, and derive a sufficient condition and a necessary condition for the discreteness of the negative spectrum. As an application, we establish two-sided estimates of Lieb-Thirring-type, improving the classical ones.

Related

• Negative eigenvalue estimates for the 1D Schrödinger operator with measure-potential
• Negative eigenvalues of non-local Schrödinger operators with sign-changing potentials
• Eigenvalues of Schroedinger operators with potential asymptotically homogeneous of degree -2
• Estimates for the number of eigenvalues of two dimensional Schroedinger operators lying below the essential spectrum
• On negative eigenvalues of low-dimensional Schrödinger operators