Aharonov-Bohm effect on the Poincaré disk

Lisovyy, O.

doi:10.1063/1.2738751

Cited by 16 publications

(39 citation statements)

References 35 publications

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“…Then the spectrum consists of a finite number of infinitely degenerated eigenvalues in the lower part of the spectrum (also called the 'Landau levels'), and continuous spectrum in the higher part. Comtet [6] gives a physical interpretation of the spectral structure; the low-energy classical particles in H subjected to a homogeneous magnetic field are trapped by the magnetic field, but the high-energy particles can escape to infinity due to the negative curvature of H. Moreover, Bulaev, Geyler and Margulis [5] and Lisovyy [22] show that the Landau levels are still infinitely degenerated even if we add one pointlike magnetic field (or, the Aharonov-Bohm magnetic field) to the homogeneous field. We can easily show that the same is true if we add a finite number of pointlike magnetic fields.…”

Section: Motivationmentioning

confidence: 98%

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Landau levels on the hyperbolic plane in the presence of Aharonov–Bohm fields

Mine¹,

Nomura²

2012

Journal of Functional Analysis

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We consider the magnetic Schrödinger operators on the Poincaré upper half plane with constant Gaussian curvature −1. We assume the magnetic field is given by the sum of a constant field and the Dirac δ measures placed on some lattice. We give a sufficient condition for each Landau level to be an infinitely degenerated eigenvalue. We also prove the lowest Landau level is not an eigenvalue if the above condition fails. In particular, the infinite degeneracy of the lowest Landau level is equivalent to the infiniteness of the zeromodes of the two-dimensional Pauli operator.

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Section: Motivationmentioning

confidence: 98%

“…We assume (28), and we need some upper bound of the function |u|, where u is the function given in (22).…”

Section: Infiniteness Of the Lowest Landau Modesmentioning

confidence: 99%

Landau levels on the hyperbolic plane in the presence of Aharonov–Bohm fields

Mine¹,

Nomura²

2012

Journal of Functional Analysis

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“…The outline of the calculation is similar to [21] and the reader is referred to this paper for more details.…”

Section: Full One-vortex Green Functionmentioning

confidence: 99%

“…Using the technique described in the Appendix A of [21], one may compute the integral for G (0) (z, z ′ ) in terms of hypergeometric functions. The result reads…”

Section: Full One-vortex Green Functionmentioning

confidence: 99%

“…Partial Green functions are then calculated relatively easily in each channel with fixed angular momentum; the difficult part is the summation of these partial contributions to a closed-form expression. We have solved this problem by writing radial solutions of the Dirac equation as Sommerfeldtype superpositions of horocyclic waves, similarly to a simpler scalar case [21]. This allows to perform the summation and to obtain a simple integral representation for the one-vortex Green function.…”

Section: Introductionmentioning

confidence: 99%

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On Painlevé VI transcendents related to the Dirac operator on the hyperbolic disk

Lisovyy

2008

Journal of Mathematical Physics

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Dirac hamiltonian on the Poincaré disk in the presence of an Aharonov-Bohm flux and a uniform magnetic field admits a one-parameter family of self-adjoint extensions. We determine the spectrum and calculate the resolvent for each element of this family. Explicit expressions for Green functions are then used to find Fredholm determinant representations for the tau function of the Dirac operator with two branch points on the Poincaré disk. Isomonodromic deformation theory for the Dirac equation relates this tau function to a one-parameter class of solutions of the Painlevé VI equation with γ = 0. We analyze long distance behaviour of the tau function, as well as the asymptotics of the corresponding Painlevé VI transcendents as s → 1. Considering the limit of flat space, we also obtain a class of solutions of the Painlevé V equation with β = 0.

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Schrödinger and Dirac Operators with Aharonov–Bohm and Magnetic-Solenoid Fields

Гитман¹,

Tyutin²,

Voronov³

2012

Self-Adjoint Extensions in Quantum Mechanics

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We construct all self-adjoint Schrödinger and Dirac operators (Hamiltonians) with both the pure Aharonov-Bohm (AB) field and the so-called magnetic-solenoid field (a collinear superposition of the AB field and a constant magnetic field). We perform a spectral analysis for these operators, which includes finding spectra and spectral decompositions, or inversion formulae. In constructing the Hamiltonians and performing their spectral analysis, we follow, respectively, the von Neumann theory of self-adjoint extensions of symmetric operators and the Krein method of guiding functionals.

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Aharonov-Bohm effect on the Poincaré disk

Cited by 16 publications

References 35 publications

Landau levels on the hyperbolic plane in the presence of Aharonov–Bohm fields

Landau levels on the hyperbolic plane in the presence of Aharonov–Bohm fields

On Painlevé VI transcendents related to the Dirac operator on the hyperbolic disk

Schrödinger and Dirac Operators with Aharonov–Bohm and Magnetic-Solenoid Fields

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