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Russian Journal of Mathematical Physics

Norm Convergence of Self-Adjoint Extensions of the Aharonov–Bohm Hamiltonian as Solenoid Length Goes to Infinity

Marciano Pereira

Published Jul 1, 2018• 0 citations
Publisher

Abstract

With respect to the Aharonov-Bohm effect, we consider only solenoids of zero radius and their action on its plane of symmetry. We find all self-adjoint extensions in case of finite length solenoids and compare them with the case of infinitely long one. It is then shown a convergence in the uniform resolvent sense for each angular momentum sector of such self-adjoint extensions as the solenoid length goes to infinity.

DOI
10.1134/s1061920818030068

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With respect to the Aharonov-Bohm effect, we consider only solenoids of zero radius and their action on its plane of symmetry. We find all self-adjoint extensions in case of finite length solenoids and compare them with the case of infinitely long one. It is then shown a convergence in the uniform resolvent sense for each angular momentum sector of such self-adjoint extensions as the solenoid length goes to infinity.
We consider the Hamiltonian operator associated with planar sections of infinitely long cylindrical solenoids and with a homogeneous magnetic field in their interior.First, in the Sobolev space H 2 , we characterize all generalized boundary conditions on the solenoid border compatible with quantum mechanics, i.e. the boundary conditions, so that the corresponding Hamiltonian operators are self-adjoint.Then we study and compare the scattering of the most usual boundary conditions, that is, Dirichlet, Neumann and Robin.
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 regularized Aharonov–Bohm Hamiltonians, that is Pauli Hamiltonians in which the magnetic field corresponds to a flux tube of nonzero 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.
We add a scalar potential to the 2D Aharonov-Bohm (AB) model which properly diverges both at the solenoid border and at infinity so that the resulting operator is essentially self-adjoint and has a discrete spectrum; the former property is interpreted as no contact of the particle with the solenoid border since there is no need of boundary conditions. We study gauge transformations to get the usual periodic behavior of the AB properties as a function of the magnetic flux. The presence of the AB effect is proven through the ground state energy, which is shown to be smooth in case it is simple and with a nonzero derivative if the ground state is real valued; such properties are verified in the case of circular solenoids, for which it is shown to be a nonconstant periodic function with a minimum at integer and a maximum at half-integer circulations (at half-integer circulations, it is doubly degenerated).
The scattering amplitude of a charged particle from a long hard cylinderical solenoid is derived by solving the time independent Schrödinger equation on a double connected plane. It is a summation over the angular momentum quantum number (partial wave summation). It is shown that only negative mechanical angular momenta contribute to the amplitude when the radius of the solenoid goes to zero limit without varying the magnetic induction flux (Flux line). Original Aharonov-Bohm result is obtained with this limit.