Spectral and Scattering Theory for the Aharonov–bohm Operators

Pankrashkin, Konstantin; Richard, Serge

doi:10.1142/s0129055x11004205

Cited by 38 publications

(57 citation statements)

References 31 publications

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“…We also stress that the operators that we study are extremely natural and appear in many situations. For example, they describe oscillations of a conical membrane, the Aharonov-Bohm effect [7,28], and sticky diffusion [16]. They also play an important role in the study of the wave and Klein-Gordon equations on anti-de Sitter spacetime, see for example [5,15,19] and references therein.…”

Section: Introductionmentioning

confidence: 99%

On Schrödinger Operators with Inverse Square Potentials on the Half-Line

Dereziński

Richard

2016

Ann. Henri Poincaré

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135

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Abstract. The paper is devoted to operators given formally by the expressionThis expression is homogeneous of degree minus 2. However, when we try to realize it as a self-adjoint operator for real α, or closed operator for complex α, we find that this homogeneity can be broken. This leads to a definition of two holomorphic families of closed operators on L 2 (R+), which we denote Hm,κ and H ν 0 , with m 2 = α, −1 < Re(m) < 1, and where κ, ν ∈ C ∪{∞} specify the boundary condition at 0. We study these operators using their explicit solvability in terms of Bessel-type functions and the Gamma function. In particular, we show that their point spectrum has a curious shape: a string of eigenvalues on a piece of a spiral. Their continuous spectrum is always [0, ∞[. Restricted to their continuous spectrum, we diagonalize these operators using a generalization of the Hankel transformation. We also study their scattering theory. These operators are usually non-self-adjoint. Nevertheless, it is possible to use concepts typical for the self-adjoint case to study them. Let us also stress that −1 < Re(m) < 1 is the maximal region of parameters for which the operators Hm,κ can be defined within the framework of the Hilbert space L 2 (R+).

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Section: Introductionmentioning

confidence: 99%

On Schrödinger Operators with Inverse Square Potentials on the Half-Line

Dereziński

Richard

2016

Ann. Henri Poincaré

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135

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“…(Result obtained independently by Bruneau, Georgescu, and myself in [4], and by Richard and Pankrashkin in [10]).…”

Section: Homogeneous Schrödinger Operatorsmentioning

confidence: 83%

“…However, the non-selfadjoint case seems to heve been first described in [6]. A number of new exact formulas about these operators is contained in [4,10,6] and [2].…”

Section: Introductionmentioning

confidence: 99%

Homogeneous Rank One Perturbations and Inverse Square Potentials

Dereziński

2019

Trends in Mathematics

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Following [2,4,6], I describe several exactly solvable families of closed operators on L 2 [0, ∞[. Some of these families are defined by the theory of singular rank one perturbations. The remaining families are Schrödinger operators with inverse square potentials and various boundary conditions. I describe a close relationship between these families. In all of them one can observe interesting "renormalization group flows" (action of the group of dilations).Mathematics Subject Classification (2010). 34L40, 33C10.

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“…(e) It is worth mentioning that an important technical point in the proofs is the quadratic form L κ associated with H κ , that is, L κ (ψ) := Ωa |(i∇ + A κ )ψ| 2 + V |ψ| 2 dxdy, with domain H 1 0 (Ω a ) ∩ domV 1/2 (H 1 0 is the usual Sobolev space). (f) The case a = 0 with no additional potential V and Dirichlet condition is known [11] to have absolutely continuous spectrum [0, ∞).…”

Section: Introduction and Resultsmentioning

confidence: 99%

Aharonov-Bohm effect without contact with the solenoid

Oliveira

Romano

2017

Journal of Mathematical Physics

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We add a confining potential to the Aharonov-Bohm model resulting in no contact of the particle with the solenoid (border); this is characterized by a unique self-adjoint extension of the initial Hamiltonian operator. It is shown that the spectrum of such extension is discrete and the first eigenvalue is found to be a nonconstant 1-periodic function of the magnetic flux circulation with a minimum at integers and maximum at half-integer circulations. This is a rigorous verification of the effect.

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Spectral and Scattering Theory for the Aharonov–bohm Operators

Abstract: We review the spectral and the scattering theory for the Aharonov-Bohm model on R 2 . New formulae for the wave operators and for the scattering operator are presented. The asymptotics at high and at low energy of the scattering operator are computed.

Cited by 38 publications

References 31 publications

On Schrödinger Operators with Inverse Square Potentials on the Half-Line

On Schrödinger Operators with Inverse Square Potentials on the Half-Line

Homogeneous Rank One Perturbations and Inverse Square Potentials

Aharonov-Bohm effect without contact with the solenoid

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