03-189 J.L. Borg, J.V. Pule

Pauli Approximations to the Self-Adjoint Extensions of the Aharonov-Bohm Hamiltonian (272K, PDF) Apr 23, 03

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Abstract. It is well known that the formal Aharonov-Bohm Hamiltonian operator, describing the interaction of a charged particle with a magnetic vortex, has a four-parameter family of self-adjoint extensions, which reduces to a two-parameter family if one requires that the Hamiltonian commutes with the angular momentum operator. The question we study here is which of these self-adjoint extensions can considered as limits of regularised Aharonov-Bohm Hamiltonians, that is Pauli Hamiltonians in which the magnetic field corresponds to a flux tube of non-zero diameter. We show that not all the self-adjoint extensions in this two-parameter family can be obtained by these approximations, but only two one-parameter subfamilies. In these two cases we can choose the gyromagnetic ratio in the approximating Pauli Hamiltonian in such a way that we get convergence in the norm resolvent sense to the corresponding self-adjoint extension.

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