Geodesic
On the Birman problem in the theory of nonnegative symmetric operators with compact inverse
Funkcionalʹnyj analiz i ego priloženiâ, Tome 57 (2023) no. 2, pp. 111-116

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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.
Keywords: Schrödinger operator, symmetric nonnegative operator, compact preresolvent, Friedrichs extension, continuous spectrum. 
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     author = {M. M. Malamud},
     title = {On the {Birman} problem in the theory of nonnegative symmetric operators with compact inverse},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {111--116},
     publisher = {mathdoc},
     volume = {57},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2023_57_2_a8/}
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M. M. Malamud. On the Birman problem in the theory of nonnegative symmetric operators with compact inverse. Funkcionalʹnyj analiz i ego priloženiâ, Tome 57 (2023) no. 2, pp. 111-116. http://geodesic.mathdoc.fr/item/FAA_2023_57_2_a8/