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On the relation between the S-matrix and the spectrum of the interior Laplacian

100%

Ramm A. G.

Bulletin of the Polish Academy of Sciences. Mathematics

2009

tom Vol. 57, no 2

181-188

The main results of this paper are: 1) a proof that a necessary condition for 1 to be an eigenvalue of the S-matrix is real analyticity of the boundary of the obstacle, 2) a short proof that if 1 is an eigenvalue of the S-matrix, then k² is an eigenvalue of the Laplacian of the interior problem, and that in this case there exists a solution to the interior Dirichlet problem for the Laplacian, which admits an analytic continuation to the whole space R³ as an entire function.

Proposal to improve the behaviour of self-energy contributions to the S-matrix

100%

Tóth G.

Open Physics

2010

tom 8

nr 4

527-541

A simple modification of the definition of the S-matrix is proposed. It is expected that the divergences related to nonzero self-energies are considerably milder with the modified definition than with the usual one. This conjecture is verified in a few examples using perturbation theory. The proposed formula is written in terms of the total Hamiltonian operator and a free Hamiltonian operator and is therefore applicable in any case when these Hamiltonian operators are known.

On the analytic properties of the S-matrix for the unknown interactions surrounded by centrifugal and rapidly decreasing potentials

88%

Olkhovsky V.

Open Physics

2011

tom 9

nr 1

13-44

The analytic structure of the non-relativistic unitary and non-unitary S-matrix is investigated for the cases of the unknown interactions with the unknown motion equations inside a sphere of radius a, surrounded by the centrifugal and rapidly decreasing (exponentially or by the Yukawian law or by the more rapidly decreasing) potentials. The one-channel case and special examples of many-channel cases are considered. Some kinds of symmetry conditions are imposed. The Schroedinger equation for r > a for the particle motion and the condition of the completeness of the correspondent wave functions are assumed. The connection of the obtained results with the usual (temporal) causality is examined. Finally a scientific program is presented as a clear continuation and extension of the obtained results.

On the S-matrix of Schrodinger operator with nonlocal 6-interaction

88%

Główczyk A. , Kużel S.

Opuscula Mathematica

2021

tom Vol. 41, no. 3

413--435

Schrodinger operators with nonlocal δ-interaction are studied with the use of the Lax-Phillips scattering theory methods. The condition of applicability of the Lax-Phillips approach in terms of non-cyclic functions is established. Two formulas for the S-matrix are obtained. The first one deals with the Krein-Naimark resolvent formula and the Weyl-Titchmarsh function, whereas the second one is based on modified reflection and transmission coefficients. The S-matrix S(z) is analytical in the lower half-plane C- when the Schrodinger operator with nonlocal δ-interaction is positive self-adjoint. Otherwise, S(z) is a meromorphic matrix-valued function in C- and its properties are closely related to the properties of the corresponding Schrodinger operator. Examples of S-matrices are given.