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The paper concerns upper and lower estimates for the number of negativeeigenvalues of one- and two-dimensional Schr\"{o}dinger operators and moregeneral operators with the spectral dimensions $d\leq 2$. The classicalCwikel-Lieb-Rosenblum (CLR) upper estimates require the corresponding Markovprocess to be transient, and therefore the dimension to be greater than two. Weobtain CLR estimates in low dimensions by transforming the underlying recurrentprocess into a transient one using partial annihilation. As a result, theestimates for the number of negative eigenvalues are not translation invariantand contain Bargmann type terms. The general theorems are illustrated byanalysis of several classes of the Schr\"{o}dinger type operators (on theRiemannian manifolds, lattices, fractals, etc.). We provide estimates frombelow which prove that the results obtained are sharp. Lieb-Thirring estimatesfor the low-dimensional Schr\"{o}dinger operators are also studied.

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