TY - JOUR

T1 - Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides

AU - Nazarov, S. A.

N1 - Publisher Copyright: © 2020 RAS(DoM) and LMS.

PY - 2020/12

Y1 - 2020/12

N2 - We describe and classify the thresholds of the continuous spectrum and the resulting resonances for general formally self-adjoint elliptic systems of second-order differential equations with Dirichlet or Neumann boundary conditions in domains with cylindrical and periodic outlets to infinity (in waveguides). These resonances arise because there are "almost standing"waves, that is, non-trivial solutions of the homogeneous problem which do not transmit energy. We consider quantum, acoustic, and elastic waveguides as examples. Our main focus is on degenerate thresholds which are characterized by the presence of standing waves with polynomial growth at infinity and produce effects lacking for ordinary thresholds. In particular, we describe the effect of lifting an eigenvalue from the degenerate zero threshold of the spectrum. This effect occurs for elastic waveguides of a vector nature and is absent from the scalar problems for cylindrical acoustic and quantum waveguides. Using the technique of self-adjoint extensions of differential operators in weighted spaces, we interpret the almost standing waves as eigenvectors of certain operators and the threshold as the corresponding eigenvalue. Here the threshold eigenvalues and the corresponding vector-valued functions not decaying at infinity can be obtained by approaching the threshold (the virtual level) either from below or from above. Hence their properties differ essentially from the customary ones. We state some open problems.

AB - We describe and classify the thresholds of the continuous spectrum and the resulting resonances for general formally self-adjoint elliptic systems of second-order differential equations with Dirichlet or Neumann boundary conditions in domains with cylindrical and periodic outlets to infinity (in waveguides). These resonances arise because there are "almost standing"waves, that is, non-trivial solutions of the homogeneous problem which do not transmit energy. We consider quantum, acoustic, and elastic waveguides as examples. Our main focus is on degenerate thresholds which are characterized by the presence of standing waves with polynomial growth at infinity and produce effects lacking for ordinary thresholds. In particular, we describe the effect of lifting an eigenvalue from the degenerate zero threshold of the spectrum. This effect occurs for elastic waveguides of a vector nature and is absent from the scalar problems for cylindrical acoustic and quantum waveguides. Using the technique of self-adjoint extensions of differential operators in weighted spaces, we interpret the almost standing waves as eigenvectors of certain operators and the threshold as the corresponding eigenvalue. Here the threshold eigenvalues and the corresponding vector-valued functions not decaying at infinity can be obtained by approaching the threshold (the virtual level) either from below or from above. Hence their properties differ essentially from the customary ones. We state some open problems.

KW - almost standing waves

KW - Dirichlet or Neumann boundary conditions

KW - elliptic systems

KW - selfadjoint extensions of differential operators

KW - spaces with separated asymptotic conditions

KW - threshold resonances

KW - thresholds of continuous spectrum

KW - virtual levels

UR - http://www.scopus.com/inward/record.url?scp=85099026138&partnerID=8YFLogxK

U2 - 10.1070/IM8928

DO - 10.1070/IM8928

M3 - Article

AN - SCOPUS:85099026138

VL - 84

SP - 1105

EP - 1160

JO - Izvestiya Mathematics

JF - Izvestiya Mathematics

SN - 1064-5632

IS - 6

ER -