Abstract
Large classes of nonnegative Schrödinger operators on \(\Bbb R^2\) and \(\Bbb R^3\) with the following properties are described:
1. The restriction of each of these operators to an appropriate unbounded set of measure zero in \(\Bbb R^2\) (in \(\Bbb R^3\)) is a nonnegative symmetric operator (the operator of a Dirichlet problem) with compact preresolvent;
2. Under certain additional assumptions on the potential, the Friedrichs extension of such a restriction has continuous (sometimes absolutely continuous) spectrum filling the positive semiaxis.
The obtained results give a solution of a problem by M. S. Birman.
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Funding
This work was supported by the Ministry of Science and Higher Education of Russian Federation, agreement 075-15-2021-602.
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2023, Vol. 57, pp. 111–116 https://doi.org/10.4213/faa4085.
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Translated by M. M. Malamud
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Malamud, M.M. On the Birman Problem in the Theory of Nonnegative Symmetric Operators with Compact Inverse. Funct Anal Its Appl 57, 173–177 (2023). https://doi.org/10.1134/S0016266323020090
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DOI: https://doi.org/10.1134/S0016266323020090