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.
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To Petr Petrovich Kulish on the occasion of his 70th anniversary
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 433, 2015, pp. 78–110.
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Bolokhov, T.A. Extensions of the Quadratic Form of the Transverse Laplace Operator. J Math Sci 213, 671–693 (2016). https://doi.org/10.1007/s10958-016-2731-3
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DOI: https://doi.org/10.1007/s10958-016-2731-3