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arXiv · 1012.2588

Symmetry preserving self-adjoint extensions of Schr\"odinger operators with singular potentials

D. M. Gitman·A. G. Smirnov·I. V. Tyutin·B. L. Voronov
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Abstract

We develop a general technique for finding self-adjoint extensions of a symmetric operator that respect a given set of its symmetries. Problems of this type naturally arise when considering two- and three-dimensional Schr\"odinger operators with singular potentials. The approach is based on constructing a unitary transformation diagonalizing the symmetries and reducing the initial operator to the direct integral of a suitable family of partial operators. We prove that symmetry preserving self-adjoint extensions of the initial operator are in a one-to-one correspondence with measurable families of self-adjoint extensions of partial operators obtained by reduction. The general construction is applied to the three-dimensional Aharonov-Bohm Hamiltonian describing the electron in the magnetic field of an infinitely thin solenoid.

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D. M. Gitman, A. G. Smirnov, I. V. Tyutin, B. L. Voronov. 2010-12-12. Symmetry preserving self-adjoint extensions of Schr\"odinger operators with singular potentials. https://arxiv.org/abs/1012.2588

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