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 the non-self-adjoint Schrödinger operator $-\Delta -\frac{(n-2{)}^2}{4|x{|}^2}+V$ on ${\mathbb{R}}^n$ with complex-valued potential $V\in L^{p,\infty }$, $p>n/2$. 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 may be of independent interest.