A pragmatic approach to the problem of the self-adjoint extension of Hamilton operators with the Aharonov-Bohm potential

Audretsch, Juergen; Jasper, Ulf; Skarzhinsky, Vladimir D.

doi:10.1088/0305-4470/28/8/026

Cited by 28 publications

(43 citation statements)

References 17 publications

(27 reference statements)

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“…Self-adjoint extensions of the Dirac Hamiltonian in 3 + 1 dimensions were found in [11]. The works [12,13] present an alternative method of treating the Hamiltonian extension problem in 2 + 1 and in 3 + 1 dimensions. It was shown in [14] that in 2 + 1 dimensions only two values of the extension parameter correspond to the presence of the point-like magnetic field at the origin.…”

Section: Introductionmentioning

confidence: 99%

Dirac equation in magnetic-solenoid field

Gavrilov

Гитман²,

Smirnov³

et al. 2003

Eur Phys J C

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We consider the Dirac equation in the magnetic-solenoid field (the field of a solenoid and a collinear uniform magnetic field). For the case of Aharonov-Bohm solenoid, we construct self-adjoint extensions of the Dirac Hamiltonian using von Neumann's theory of deficiency indices. We find self-adjoint extensions of the Dirac Hamiltonian and boundary conditions at the AB solenoid. Besides, for the first time, solutions of the Dirac equation in the magnetic-solenoid field with a finite radius solenoid were found. We study the structure of these solutions and their dependence on the behavior of the magnetic field inside the solenoid. Then we exploit the latter solutions to specify boundary conditions for the magnetic-solenoid field with Aharonov-Bohm solenoid.

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Section: Introductionmentioning

confidence: 99%

Dirac equation in magnetic-solenoid field

Gavrilov

Гитман²,

Smirnov³

et al. 2003

Eur Phys J C

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“…In the literature it has been already shown that the standard procedure for the self-adjoint extension of the Dirac Hamiltonian gives rise to a one-parameter family of boundary conditions in the background of an Aharonov-Bohm gauge field [55][56][57]. The specific value of the parameter is related to the physical details of the magnetic field distribution inside a more realistic finite radius flux tube (for a more detailed discussion and for specific models with finite radius magnetic flux see [58][59][60][61][62][63][64][65][66][67][68][69]). The idealized model under consideration is a limiting case of the latter.…”

Section: Fermion Condensatementioning

confidence: 99%

Finite temperature fermion condensate, charge and current densities in a ( $$2+1$$ 2 + 1 )-dimensional conical space

et al. 2016

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We evaluate the fermion condensate and the expectation values of the charge and current densities for a massive fermionic field in (2 + 1)-dimensional conical spacetime with a magnetic flux located at the cone apex. The consideration is done for both irreducible representations of the Clifford algebra. The expectation values are decomposed into the vacuum expectation values and contributions coming from particles and antiparticles. All these contributions are periodic functions of the magnetic flux with the period equal to the flux quantum. Related to the non-invariance of the model under the parity and time-reversal transformations, the fermion condensate and the charge density have indefinite parity with respect to the change of the signs of the magnetic flux and chemical potential. The expectation value of the radial current density vanishes. The azimuthal current density is the same for both the irreducible representations of the Clifford algebra. It is an odd function of the magnetic flux and an even function of the chemical potential. The behavior of the expectation values in various asymptotic regions of the parameters are discussed in detail. In particular, we show that for points near the cone apex the vacuum parts dominate. For a massless field with zero chemical potential the fermion condensate and charge density vanish. Simple expressions are derived for the part in the total charge induced by the planar angle deficit and magnetic flux. Combining the results for separate irreducible representations, we also consider the fermion condensate, charge and current densities in parity and time-reversal symmetric models. Possible applications to graphitic nanocones are discussed. a

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“…Our first order of business is to characterize the self-adjoint realizations of the operator in (1.1); for general references on self-adjoint realizations and their applications to physics see, e.g., [2,6,10,16,17,18,32,33,34,35,42,47]. To do so, we first need to determine the maximal domain of ∆:…”

Section: The Maximal Domainmentioning

confidence: 99%

“…Taking logarithms, we see that as |x| → ∞ for x ∈ Υ, we have log F (ix) ∼ c + log(log x − κ) − 1 2 log x + xR + log 1 + 1 8xR + 9 2(8xR) 2 …”

Section: The ζ-Function With Dirichlet Conditions At R = Rmentioning

confidence: 99%

The very unusual properties of the resolvent, heat kernel, and zeta function for the operator −d2∕dr2−1∕(4r2)

Kirsten

Loya

Park

2006

Journal of Mathematical Physics

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Abstract.In this article we analyze the resolvent, the heat kernel and the spectral zeta function of the operator −d 2 /dr 2 −1/(4r 2 ) over the finite interval. The structural properties of these spectral functions depend strongly on the chosen self-adjoint realization of the operator, a choice being made necessary because of the singular potential present. Only for the Friedrichs realization standard properties are reproduced, for all other realizations highly nonstandard properties are observed. In particular, for k ∈ N we find terms like (log t) −k in the small-t asymptotic expansion of the heat kernel. Furthermore, the zeta function has s = 0 as a logarithmic branch point.

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A pragmatic approach to the problem of the self-adjoint extension of Hamilton operators with the Aharonov-Bohm potential

Cited by 28 publications

References 17 publications

Dirac equation in magnetic-solenoid field

Dirac equation in magnetic-solenoid field

Finite temperature fermion condensate, charge and current densities in a ( $$2+1$$ 2 + 1 )-dimensional conical space

The very unusual properties of the resolvent, heat kernel, and zeta function for the operator −d2∕dr2−1∕(4r2)

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