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Journal of Mathematical Physics

Two-dimensional quantum scattering by non-isotropic interactions localized on a circle, applications to open billiards

E. de Prunelé

Published Sep 18, 2018• 2 citations

Abstract

Two-dimensional quantum scattering by isotropic and non-isotropic interactions localized on a circle is considered. The expansion of the interaction on the circle in a Fourier series allows us to express basic objects of scattering theory (resolvent, T operator, differential cross length, cross length, and cross length averaged over all orientations of the incident particle), in terms of operations on matrices. For numerical applications, these matrices are truncated to a given order. If the interaction is isotropic, this general formulation reduces to the usual one, and the resonances in the isotropic cases are studied because they allow us to interpret resonances in some non-isotropic cases. Applications to open circular billiards are given. A first approach to the open quantum circular billiard is an interaction equal to zero on some parts of the circle and to a great value λ on the other parts. Then, it turns out that the limit λ → +∞ does not lead to an explicit result in this approach. A second approach to the open circular billiard is given by replacing apertures (where the interaction is zero) by pseudo-apertures (where the interaction is small but non-zero), and the limit λ → +∞ then leads to an explicit result. In the high energy regime, the two approaches give similar results and allow interpretation of numerous results, in particular, for differential cross lengths, in terms of classical mechanics.

DOI: 10.1063/1.5004638

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We present quasi-analytical and numerical calculations of Gaussian wave packet solutions of the Schr\\"odinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and

N \\to \\infty

regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wavefunctions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wavepackets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasi-classical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.

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Quantum systems which interact with their environment are often modeled by maximal dissipative operators or so-called Pseudo-Hamiltonians. In this paper the scattering theory for such open systems is considered. First it is assumed that a single maximal dissipative operator

A_D

in a Hilbert space $\sH$ is used to describe an open quantum system. In this case the minimal self-adjoint dilation

\widetilde K

of

A_D

can be regarded as the Hamiltonian of a closed system which contains the open system $\{A_D,\sH\}$, but since

\widetilde K

is necessarily not semibounded from below, this model is difficult to interpret from a physical point of view. In the second part of the paper an open quantum system is modeled with a family

\{A(μ)\}

of maximal dissipative operators depending on energy

μ

, and it is shown that the open system can be embedded into a closed system where the Hamiltonian is semibounded. Surprisingly it turns out that the corresponding scattering matrix can be completely recovered from scattering matrices of single Pseudo-Hamiltonians as in the first part of the paper. The general results are applied to a class of Sturm-Liouville operators arising in dissipative and quantum transmitting Schrödinger-Poisson systems.