Extensions of the Quadratic Form of the Transverse Laplace Operator
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.
KeywordsQuadratic Form Laplace Operator Boundary Term Symmetric Operator Deficiency Index
Unable to display preview. Download preview PDF.
- 1.K. Friedrichs, “Spektraltheorie halbbeschränkter Operatoren,” Math. Ann., 109, 465–487 (1934); M. Stone, in Linear Transformations in Hilbert spaces and Their Applications in Analysis, Amer. Math. Soc. Colloquium Publication, 15, Providence, R.I. (1932); or see Theorem X.23 of .Google Scholar
- 2.M. Reed and B. Simon, Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-adjointness, Academic Press (1975).Google Scholar
- 3.M. G. Krein, “The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications,” Mat. Sbornik, N. S., 20 (62), 431–495 (1947); Mat. Sbornik, N. S., 21 (64), 365–404 (1947).Google Scholar
- 5.S. Albeverio and P. Kurasov, Singular Perturbation of Differential Operators. Solvable Schrödinger Type Operators, Cambridge Univ. Press (2000).Google Scholar