The Scalar Products of the Regular Analytic Vectors of the Laplace Operator in the Solenoidal Subspace

Abstract

The Laplace operator on the subspace of solenoidal vector functions of three variables vanishing together with their first derivatives at selected points x_n, n = 1, . . .,N, is a symmetric operator with deficiency indices (3N, 3N). The calculation of the scalar products of its regular analytic vectors is the key step in the construction of the resolvents of its self-adjoint extensions via Krein’s formula.