Eigenvalue bounds for non-self-adjoint Schrödinger operators with the inverse-square potential

Abstract

The purpose of this paper is to study spectral properties of non-self-adjoint Schrödinger operators $-\Delta -\frac{(n-2{)}^2}{4|x{|}^2}+V$ on ${\mathbb{R}}^n$ with complex-valued potentials $V\in L^{p,\infty }$, $p>n/2$. We prove Keller type inequalities which measure the radius of a disc containing the discrete spectrum, in terms of the $L^{p,\infty }$ norm of $V$. Similar inequalities also hold if the inverse-square potential is replaced by a large class of subcritical potentials with critical singularities at the origin. The main new ingredient in the proof is the uniform Sobolev inequality of Kenig-Ruiz-Sogge type for Schrödinger operators with strongly singular potentials, which is of independent interest.

22 pages, 1 figure; references added

References in corpus (1)

• Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials

Cited by in corpus (1)

• Remarks on $L^p$-limiting absorption principle of Schrödinger operators and applications to spectral multiplier theorems