We develop a general technique for finding self-adjoint extensions of asymmetric operator that respect a given set of its symmetries. Problems of thistype naturally arise when considering two- and three-dimensional Schr\"odingeroperators with singular potentials. The approach is based on constructing aunitary transformation diagonalizing the symmetries and reducing the initialoperator to the direct integral of a suitable family of partial operators. Weprove that symmetry preserving self-adjoint extensions of the initial operatorare in a one-to-one correspondence with measurable families of self-adjointextensions of partial operators obtained by reduction. The general scheme isapplied to the three-dimensional Aharonov-Bohm Hamiltonian describing theelectron in the magnetic field of an infinitely thin solenoid. We construct allself-adjoint extensions of this Hamiltonian, invariant under translations alongthe solenoid and rotations around it, and explicitly find their eigenfunctionexpansions.
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