Extensions of the Quadratic Form of the Transverse Laplace Operator

Abstract

We study the quadratic form of the Laplace operator in 3 dimensions written in spherical coordinates and acting on transverse components of vector-functions. Operators which act on parametrizing functions of one of the transverse components with angular momentum 1 and 2 appear to be fourth-order symmetric operators with deficiency indices (1, 1). We consider self-adjoint extensions of these operators and propose the corresponding extensions for the initial quadratic form. The relevant scalar product for angular momentum 2 differs from the original product in the space of vector-functions, but, nevertheless, it is still local in radial variable. Eigenfunctions of the operator extensions in question can be treated as stable soliton-like solutions of the corresponding dynamical system whose quadratic form is a functional of the potential energy.