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arXiv · 1411.5351

Reduction by symmetries in singular quantum-mechanical problems: general scheme and application to Aharonov-Bohm model

A. G. Smirnov
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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 scheme is applied to the three-dimensional Aharonov-Bohm Hamiltonian describing the electron in the magnetic field of an infinitely thin solenoid. We construct all self-adjoint extensions of this Hamiltonian, invariant under translations along the solenoid and rotations around it, and explicitly find their eigenfunction expansions.

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A. G. Smirnov. 2014-11-19. Reduction by symmetries in singular quantum-mechanical problems: general scheme and application to Aharonov-Bohm model. https://doi.org/10.1063/1.4936305

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