Bargmann type estimates of the counting function for general Schrödinger operators

Abstract

The paper concerns upper and lower estimates for the number of negative eigenvalues of one- and two-dimensional Schrödinger operators and more general operators with the spectral dimensions $d\le 2$. 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 Schrödinger 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 Schrödinger operators are also studied.

Some changes in section 5 on 2-D operators. arXiv admin note: substantial text overlap with arXiv:1105.0937

References in corpus (2)

• On the negative spectrum of two-dimensional Schrödinger operators with radial potentials
• On negative eigenvalues of low-dimensional Schrödinger operators

Cited by in corpus (1)

• On spectral estimates for the Schrödinger operators in global dimension 2