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St. Petersburg Mathematical Journal

This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.

ISSN 1547-7371 (online) ISSN 1061-0022 (print)

The 2020 MCQ for St. Petersburg Mathematical Journal is 0.68.

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On the spectrum of the Laplace operator on the infinite Dirichlet ladder
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by S. A. Nazarov
Translated by: A. I. Plotkin
St. Petersburg Math. J. 23 (2012), 1023-1045
DOI: https://doi.org/10.1090/S1061-0022-2012-01228-8
Published electronically: September 17, 2012

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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Bibliographic Information
  • S. A. Nazarov
  • Affiliation: Institute of Mechanical Engineering Problems, Bol’shoĭ pr. V.O. 61, 199178 St.Petersburg, Russia
  • MR Author ID: 196508
  • Email: srgnazarov@yahoo.co.uk
  • Received by editor(s): January 25, 2010
  • Published electronically: September 17, 2012
  • Additional Notes: Supported by RFBR (grant no. 09-01-00759)
  • © Copyright 2012 American Mathematical Society
  • Journal: St. Petersburg Math. J. 23 (2012), 1023-1045
  • MSC (2010): Primary 35J05
  • DOI: https://doi.org/10.1090/S1061-0022-2012-01228-8
  • MathSciNet review: 2962184