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Bargmann type estimates of the counting function for general Schrödinger operators

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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 ⩽ 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. Bibliography: 32 titles.

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Authors and Affiliations

  1. University of North Carolina at Charlotte, Charlotte, NC, 28223, USA

    S. Molchanov & B. Vainberg

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  1. S. Molchanov
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  2. B. Vainberg
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Correspondence to B. Vainberg.

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Translated from Problems in Mathematical Analysis 65, May, 2012, pp. 77-118.

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Molchanov, S., Vainberg, B. Bargmann type estimates of the counting function for general Schrödinger operators. J Math Sci 184, 457–508 (2012). https://doi.org/10.1007/s10958-012-0877-1

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