Integr. Equ. Oper. Theory 88 (2017), 373–386
Published online May 29, 2017
Springer International Publishing 2017
and Operator Theory
Eﬀects of Rayleigh Waves on the Essential
Spectrum in Perturbed Doubly Periodic
F. L. Bakharev, G. Cardone
, S. A. Nazarov and J. Taskinen
Abstract. We give an example of a scalar second order diﬀerential oper-
ator in the plane with double periodic coeﬃcients and describe its mod-
iﬁcation, which causes an additional spectral band in the essential spec-
trum. The modiﬁed operator is obtained by applying to the coeﬃcients a
mirror reﬂection with respect to a vertical or horizontal line. This change
gives rise to Rayleigh type waves localized near the line. The results are
proven using asymptotic analysis, and they are based on high contrast
of the coeﬃcient functions.
Mathematics Subject Classiﬁcation. Primary 35P05, Secondary 47A75.
Keywords. Periodic media, Open waveguides, High contrast of coeﬃ-
cients, Asymptotics, Spectral bands.
A satisfactory theory for spectral elliptic boundary-value problems in dou-
ble periodic media containing open waveguides does not exist yet, and the
topic contains a lot of unanswered questions. An open waveguide consists of
a semi-inﬁnite foreign inclusion, cf. Fig. 1, and being a non-compact domain
perturbation, it can in general change the essential spectrum of the problem,
when compared to the corresponding problem on an intact domain without
perturbation. This topic was studied for example in the recent paper ,
which contains a complete description of the essential spectrum σ
a large class of elliptic second order systems with Neumann boundary condi-
tions, satisfying a Korn inequality. The following question
course of the investigation: is the formula
The question has been asked by a referee of the paper , among others.