To the theory of nonnegative point Hamiltonians on a plane and in the space

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: $$ \mathrm{d}\mathrm{o}\mathrm{m}\left({A}_{Y,d}\right)=\left\{f\in {W}_2^2\left({\mathbb{R}}^d\right):f(y)=0,\;y\in Y\right\} $$ are studied. Boundary triplets for the operators A * Y,2 and A * Y,3 are constructed in a formally unified way.