Extensions of the quadratic form of the transverse Laplace operator

We review the quadratic form of the Laplace operator in spehrical coordinates which acts on the transverse components of vector functions on the $3$-dimensional space. 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. Eigenfuctions of the extensions in question represent a stable soliton-like solutions of the physical system with the quadratic form being a potential energy.