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arXiv:1008.2906 · Journal of Physics A Mathematical and Theoretical

Scattering and self-adjoint extensions of the Aharonov–Bohm Hamiltonian

César R de Oliveira, Marciano Pereira

Published Aug 12, 2010• 22 citations
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Abstract

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.

DOI
10.1088/1751-8113/43/35/354011

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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.
In the space R3\mathbb R^3R3, for Robin parameter L>0L>0L>0, it is shown that there is a family of Schrödinger operators with penetrable toroidal solenoid and variable conductivity that approximates the magnetic Aharonov-Bohm operator with a Robin boundary condition at the solenoid (border). It is also shown that approximations via smooth potentials and then a barrier in the solenoid interior give Dirichlet boundary conditions. The approximations are in the strong resolvent sense and obtained through the ΓΓΓ-convergence technique, and they hold for the more general setting of smooth, closed and compact surfaces and continuous and bounded magnetic potentials.
We study all the s.a. Schrodinger and Dirac operators (Hamiltonians) both with pure AB field and with magnetic-solenoid field. Then, we perform a complete spectral analysis for these operators, which includes finding spectra and spectral decompositions, or inversion formulas. In constructing the Hamiltonians and performing their spectral analysis, we respectively follow the von Neumann theory of s.a. extensions of symmetric differential operators and the Krein method of guiding functionals. The examples of similar consideration are given by us in arXiv:0903.5277, where a nonrelativistic particle in the Calogero potential field is considered and in Theor. Math. Phys. 150 (1) (2007) 34, where a Dirac particle in the Coulomb field of arbitrary charge is considered. However, due to peculiarities of the three-dimensional problems under consideration, we elaborated a generalization of the approach used in the study of the Dirac particle.
In this work, we review two methods used to approach singular Hamiltonians in (2+1) dimensions. Both methods are based on the self-adjoint extension approach. It is very common to find singular Hamiltonians in quantum mechanics, especially in quantum systems in the presence of topological defects, which are usually modelled by point interactions. In general, it is possible to apply some kind of regularization procedure, as the vanishing of the wave function at the location of the singularity, ensuring that the wave function is square integrable and then can be associated with a physical state. However, an analysis based on the self-adjoint extension approach can lead to more general boundary conditions that still gives acceptable physical states. We exemplify the methods by exploring the bound and scattering scenarios of a spin 1/2 charged particle with an anomalous magnetic moment in the Aharonov-Bohm potential in the conical space.
The quantum-mechanical problems of electron scattering by an infinitely thin solenoid and by a half of an infinitely thin solenoid are examined from the viewpoint of constructing a self-adjoint Hamiltonian. It is demonstrated that in both problems there exist unique self-adjoint operators with a ``non-singular'' domain, that, due to physical reasons, are identified with the corresponding Hamiltonians. In the case of quantized values of magnetic flow along the solenoid, the electron does not experience any scattering by the string. It is shown that the scattering amplitude and wave function of an electron in the problem of scattering by an infinitely long solenoid of radius a in the limit a->0 turn into the corresponding expressions for the problem of an infinitely thin solenoid. In particular, at a quantized value of magnetic flow along the solenoid, scattering disappears at a->0.