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Quadratic Forms for Aharonov-Bohm Hamiltonians
No full textWe consider a charged quantum particle immersed in an axial magnetic field, comprising a local Aharonov-Bohm singularity and a regular perturbation. Quadratic form techniques are used to characterize different self-adjoint realizations of the reduced two-dimensional Schrödinger operator, including the Friedrichs Hamiltonian and a family of singular perturbations indexed by 2×2 Hermitian matrices. The limit of the Friedrichs Hamiltonian when the Aharonov-Bohm flux parameter goes to zero is discussed in terms of Γ - convergence
Magnetic perturbations of anyonic and Aharonov-Bohm Schrödinger operators
No full textWe study the Hamiltonian describing two anyons moving in a plane in the presence of an external magnetic field and identify a one-parameter family of self-adjoint realizations of the corresponding Schrödinger operator. We also discuss the associated model describing a quantum particle immersed in a magnetic field with a local Aharonov–Bohm singularity. For a special class of magnetic potentials, we provide a complete classification of all possible self-adjoint extensions
The Casimir energy anomaly for a point interaction
No full textThe Casimir energy for a massless, neutral scalar field in presence of a point interaction is analyzed using a general zeta-regularization approach developed in earlier works. In addition to a regular bulk contribution, there arises an anomalous boundary term which is infinite despite renormalization. The intrinsic nature of this anomaly is briefly discussed
Vacuum polarization with zero-range potentials on a hyperplane
Full text linkThe quantum vacuum fluctuations of a neutral scalar field induced by background zero-range potentials concentrated on a flat hyperplane of co-dimension 1 in (d+1)-dimensional Minkowski spacetime are investigated. Perfectly reflecting and semitransparent surfaces are both taken into account, making reference to the most general local, homogeneous and isotropic boundary conditions compatible with the unitarity of the quantum field theory. The renormalized vacuum polarization is computed for both zero and non-zero mass of the field, implementing a local version of the zeta regularization technique. The asymptotic behaviors of the vacuum polarization for small and large distances from the hyperplane are determined to leading order. It is shown that boundary divergences are softened in the specific case of a pure Dirac delta potential
The semiclassical limit on a star-graph with Kirchhoff conditions
No full textWe consider the dynamics of a quantum particle of mass m on a n-edges star-graph with Hamiltonian HK=−(2m)−1ħ2Δ and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting classical dynamics through a Liouville operator on the graph, obtained by means of Kreĭn’s theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study the semiclassical limit of the wave and scattering operators for the couple (HK,H⊕D), where H⊕D is the Hamiltonian with Dirichlet conditions in the vertex
On inverses of Kreın’s Q-functions
Full text linkLet AQ be the self-adjoint operator defined by the Q-function Q: z 7→ Qz through the Kreın-like resolvent formula (−AQ + z)−1 = (−A0 + z)−1 + GzWQ−z1V G∗z ̄ , z ∈ ZQ , where V and W are bounded operators and ZQ := {z ∈ ρ(A0): Qz and Qz ̄ have a bounded inverse}. We show that ZQ 6= ∅ = ZQ = ρ(A0) ∩ ρ(AQ) . We do not suppose that Q is represented in terms of a uniformly strict, operator-valued Nevanlinna function (equivalently, we do not assume that Q is associated to an ordinary boundary triplet), thus our result extends previously known ones. The proof relies on simple algebraic computations stemming from the first resolvent identity
Scattering Theory for Delta-Potentials Supported by Locally Deformed Planes
No full textWe give a self-contained description of the main results from the paper (Cacciapuoti et al., J Math Anal Appl 473(1):215–257, 2019). We focus on the fundamental concepts and on the chief achievements, omitting some auxiliary results and a number of technical details given in the original paper. We discuss the scattering problem for a quantum particle in dimension three in the presence of a semitransparent unbounded obstacle, modelled by a δ-interaction supported on a surface obtained through a local, Lipschitz continuous deformation of a flat plane. We discuss existence and asymptotic completeness of the wave operators with respect to a suitable reference dynamics. Additionally, we provide an explicit expression for the related scattering matrix and show that it converges to the identity as the deformation goes to zero (giving a quantitative estimates on the rate of convergence)
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