Extensions of the quadratic form of the transverse Laplace operator

TL;DR

This paper reviews and extends the quadratic form of the transverse Laplace operator in 3D spherical coordinates, focusing on self-adjoint extensions for operators with angular momentum 1 and 2, crucial for mathematical physics applications.

Contribution

It develops self-adjoint extensions for fourth-order symmetric differential operators acting on transverse vector components with specific angular momentum, advancing the mathematical framework of the Laplace operator.

Findings

Operators are fourth order with deficiency indices (1,1).

Self-adjoint extensions are constructed for these operators.

Modified scalar product for angular momentum 2 remains local in radial variable.

Abstract

We review the quadratic form of the Laplace operator in 3 dimensions in spehrical coordinates which acts on the transverse components of vector functions. Operators, acting on the parametrizing functions of one of the transverse components with angular momentum 1 and 2, appear to be fourth order symmetric differential operators with deficiency indices (1,1). We develop self-adjoint extensions of these operators and propose correspondent extensions for the initial quadratic form. The relevant scalar product for the angular momentum 2 differs from the original product in the space of vector functions, but nevertheless it is still local in radial variable.