Finitely many sphere interactions in quantum mechanics: Nonseparated boundary conditions

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Abstract
Let Ḣ be the closure of the restriction of the three‐dimensional Laplacian −Δ on the domain C∞0(R3\Σ), where Σ=∪Nj=1∂K(0,Rj) and ∼(K(0,Rj)) is a closed ball of radius Rj centered at the origin in R3. It is well known that Ḣ is a closed symmetric operator with deficiency indices (∞,∞). In this paper all self‐adjoint (s.a.) extensions of Ḣ are constructed; these extensions contain as particular cases the quantum Hamiltonian describing concentric δ‐ and δ’‐sphere interactions. It is also shown that the s.a. extensions of Ḣ may be obtained as norm‐resolvent limits of momentum cutoff and scaled separable potentials.