Bargmann type estimates of the counting function for general Schr\"{o}dinger operators

TL;DR

This paper develops new upper and lower bounds for the number of negative eigenvalues of low-dimensional Schrödinger operators, extending classical estimates to cases where the underlying processes are recurrent, using partial annihilation techniques.

Contribution

It introduces Bargmann type estimates for low-dimensional Schrödinger operators by transforming recurrent processes into transient ones, broadening the applicability of CLR estimates.

Findings

Established CLR estimates in low dimensions via process transformation.

Provided sharp lower bounds matching the upper estimates.

Analyzed Schrödinger operators on various structures like manifolds and fractals.

Abstract

The paper concerns upper and lower estimates for the number of negative eigenvalues of one- and two-dimensional Schr\"{o}dinger operators and more general operators with the spectral dimensions $d\leq 2$. The classical Cwikel-Lieb-Rosenblum (CLR) upper estimates require the corresponding Markov process to be transient, and therefore the dimension to be greater than two. We obtain CLR estimates in low dimensions by transforming the underlying recurrent process into a transient one using partial annihilation. As a result, the estimates for the number of negative eigenvalues are not translation invariant and contain Bargmann type terms. The general theorems are illustrated by analysis of several classes of the Schr\"{o}dinger type operators (on the Riemannian manifolds, lattices, fractals, etc.). We provide estimates from below which prove that the results obtained are sharp. Lieb-Thirring estimates for the low-dimensional Schr\"{o}dinger operators are also studied.