On negative eigenvalues of low-dimensional Schr\"{o}dinger operators

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. We show that a classical form of CLR estimates can not be valid for operators with recurrent underlying Markov processes. We provide estimates from below which prove that the obtained results are sharp. Lieb-Thirring estimates for the low-dimensional Schr\"{o}dinger operators are also studied.