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On the spectrum of the Laplace operator on the infinite Dirichlet ladder

Author: S. A. Nazarov
Translated by: A. I. Plotkin
Original publication: Algebra i Analiz, tom 23 (2011), nomer 6.
Journal: St. Petersburg Math. J. 23 (2012), 1023-1045
MSC (2010): Primary 35J05
Published electronically: September 17, 2012
MathSciNet review: 2962184
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Abstract: The spectrum of the Dirichlet problem is studied in the case of a periodic infinite planar domain having the form of an accommodation ladder: two parallel strips-uprights of thickness $ h>0$ are linked by treads of the same thickness. It is shown that, for $ h$ small, a gap is always opened between the second and third segments of the essential spectrum of the problem operator. The gap between the first and second segments is also discussed: its presence and characteristics depend on the distance between the uprights. It is shown that variation of the thickness of finitely many treads leads to the arising of any prescribed number of discrete spectrum points below the essential spectrum as well as inside the open gap.

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Additional Information

S. A. Nazarov
Affiliation: Institute of Mechanical Engineering Problems, Bol’shoĭ pr. V.O. 61, 199178 St.Petersburg, Russia

Keywords: Periodic junction of thin domains, essential spectrum, Dirichlet problem, gaps, discrete spectrum
Received by editor(s): January 25, 2010
Published electronically: September 17, 2012
Additional Notes: Supported by RFBR (grant no. 09-01-00759)
Article copyright: © Copyright 2012 American Mathematical Society