Abstract
Let Y be a countable set of points in ℝ d, d = 2, 3, such that d *(Y) := inf{|y − y′| : y, y′ ∈ Y, y ≠ y′} > 0. Using the connection between the Sobolev Space W 22 (ℝ d) and the Hilbert space ℓ2, it is proved that the system of Dirac’s delta functions {δ(x − y), y ∈ Y, x ∈ Rd, d = 2, 3} forms the Riesz basis in its linear hull in W − 22 (ℝ d). The properties of the Friedrichs and Krein extensions for a nonnegative symmetric operator A Y , d := −Δ:
are studied. Boundary triplets for the operators A * Y,2 and A * Y,3 are constructed in a formally unified way.
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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 11, No. 2, pp. 203–226, April–May, 2014.
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Kovalev, Y. To the theory of nonnegative point Hamiltonians on a plane and in the space. J Math Sci 204, 315–332 (2015). https://doi.org/10.1007/s10958-014-2204-5
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DOI: https://doi.org/10.1007/s10958-014-2204-5