Approximation of one-dimensional relativistic point interactions by regular potentials revised
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Approximation of one-dimensional relativistic point interactions by regular potentials revised Matej ˇ Tušek1 Received: 1 April 2019 / Revised: 1 April 2019 / Accepted: 3 August 2020 / Published online: 12 August 2020 © Springer Nature B.V. 2020
Abstract We show that the one-dimensional Dirac operator with quite general point interaction may be approximated in the norm resolvent sense by the Dirac operator with a scaled regular potential of the form 1/ε h(x/ε) ⊗ B, where B is a suitable 2 × 2 matrix. Moreover, we prove that the limit does not depend on the particular choice of h as long as it integrates to a constant value. Keywords Dirac operator · Point interaction · Approximations Mathematics Subject Classification 46N50 · 81Q10 · 81Q80
1 Introduction The one-dimensional Dirac operator perturbed at one point is an important exactly solvable model of relativistic quantum mechanics. Mathematically, the perturbation is described by a boundary condition at the interaction point. Following [1], we will write it as ψ(0+) = ψ(0−),
(1)
where ψ is a two-component spinor and is from a four-parametric family of admissible matrices that lead to self-adjoint realizations of the Dirac operator and will be explicitly characterized later. Note that for convenience and without loss of generality, the interaction point coincides with the origin. The question how to approximate the point interactions by regular potentials is important for two major reasons. First, an approximation sequence may tell us much
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Matˇej Tušek [email protected] Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 120 00 Prague, Czechia
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M. Tušek
more about the nature of the point interaction rather than an abstract boundary condition. Second, various short-range interactions may be well approximated by the point interactions, and the latter are described by analytically solvable models. In the relativistic case, this question was addressed rigorously for the first time by Šeba [2]. He focused exclusively on the so-called electrostatic and Lorentz scalar point interactions. Taking the former, for example, he started with the Dirac operator with the potential 1/ε h(x/ε)⊗ I for some h ∈ L 1 (R; R). Then, he proved that as ε → 0+, this operator converges in the norm resolvent sense to the Dirac operator with the point interaction described by the boundary condition η (ψ1 (0+) + ψ1 (0−)) = i(ψ2 (0+) − ψ2 (0−)) 2 η (ψ2 (0+) + ψ2 (0−)) = i(ψ1 (0+) − ψ1 (0−)) 2
(2)
η = sgn (h)|h|1/2 , (1 − K 2 )−1 |h|1/2 L 2 (R) .
(3)
with
Here, K is the integral operator on L 2 (R) with the kernel K (x, y) =
i |h(x)|1/2 sgn(x − y) sgn(h(y))|h(y)|1/2 . 2
Next results are due to Hughes who found smooth local approximations to all types of the point interactions but only in the strong resolvent topology [3,4]. She showed that if h integrates to one, then the Dirac operator with the potential i/ε h(x/ε) ⊗ σ1 A converges to the Dirac operator with the point interac
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