TY - JOUR

T1 - Scattering on periodic metric graphs

AU - Korotyaev, Evgeny

AU - Сабурова, Наталья

N1 - Funding Information: Evgeny Korotyaev was supported by the RSF grant No. 18-11-00032. Natalia Saburova was supported by the RFBR grant No. 19-01-00094. Publisher Copyright: © 2020 World Scientific Publishing Company. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/9/1

Y1 - 2020/9/1

N2 - We consider the Laplacian on a periodic metric graph and obtain its decomposition into a direct fiber integral in terms of the corresponding discrete Laplacian. Eigenfunctions and eigenvalues of the fiber metric Laplacian are expressed explicitly in terms of eigenfunctions and eigenvalues of the corresponding fiber discrete Laplacian and eigenfunctions of the Dirichlet problem on the unit interval. We show that all these eigenfunctions are uniformly bounded. We apply these results to the periodic metric Laplacian perturbed by real integrable potentials. We prove the following: (a) the wave operators exist and are complete, (b) the standard Fredholm determinant is well-defined and is analytic in the upper half-plane without any modification for any dimension, (c) the determinant and the corresponding S-matrix satisfy the Birman-Krein identity.

AB - We consider the Laplacian on a periodic metric graph and obtain its decomposition into a direct fiber integral in terms of the corresponding discrete Laplacian. Eigenfunctions and eigenvalues of the fiber metric Laplacian are expressed explicitly in terms of eigenfunctions and eigenvalues of the corresponding fiber discrete Laplacian and eigenfunctions of the Dirichlet problem on the unit interval. We show that all these eigenfunctions are uniformly bounded. We apply these results to the periodic metric Laplacian perturbed by real integrable potentials. We prove the following: (a) the wave operators exist and are complete, (b) the standard Fredholm determinant is well-defined and is analytic in the upper half-plane without any modification for any dimension, (c) the determinant and the corresponding S-matrix satisfy the Birman-Krein identity.

KW - Direct integral

KW - Fredholm determinant

KW - metric Laplacian

KW - periodic metric graph

KW - scattering

KW - Schrödinger operators

KW - DISCRETE SCHRODINGER-OPERATORS

KW - THEOREM

KW - POTENTIALS

KW - SPECTRAL PROPERTIES

KW - SELF-ADJOINT EXTENSIONS

KW - LAPLACIAN

KW - Schrodinger operators

UR - http://www.scopus.com/inward/record.url?scp=85091424798&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/e60f6b9a-4b6d-3ef7-a4a9-ab6c286461d8/

U2 - 10.1142/S0129055X20500245

DO - 10.1142/S0129055X20500245

M3 - Review article

AN - SCOPUS:85091424798

VL - 32

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

SN - 0129-055X

IS - 8

M1 - 2050024

ER -