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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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On a mathematical model of irreversible quantum graphs
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by M. Z. Solomyak
Translated by: R. Shterenberg
St. Petersburg Math. J. 17 (2006), 835-864
DOI: https://doi.org/10.1090/S1061-0022-06-00932-0
Published electronically: July 27, 2006

Abstract:

The “irreversible quantum graph” model, suggested by U. Smilansky, is considered. Mathematically, the problem is in the investigation of the spectrum of the operator $\mathbf A_\alpha$ determined by an infinite system of ordinary differential equations on a graph and by a system of boundary conditions, such as conditions on the jumps of derivatives. The operator depends on a parameter $\alpha \ge 0$ involved in the boundary conditions only.

In the paper, the point spectrum and the absolute continuous spectrum of the operator $\mathbf A_\alpha$ are studied in detail in their dependence on $\alpha$. Some special effects appear, the main one being a “phase transition” for some value $\alpha =\alpha _0$ that depends on the geometry of the graph: the spectral properties of the operator for $\alpha <\alpha _0$ and $\alpha >\alpha _0$ differ greatly.

References
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Bibliographic Information
  • M. Z. Solomyak
  • Affiliation: Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
  • Email: michail.solomyak@weizmann.ac.il
  • Received by editor(s): December 21, 2004
  • Published electronically: July 27, 2006

  • Dedicated: In fond memory of Ol$’$ga Aleksandrovna Ladyzhenskaya
  • © Copyright 2006 American Mathematical Society
  • Journal: St. Petersburg Math. J. 17 (2006), 835-864
  • MSC (2000): Primary 35Q40, 34L40
  • DOI: https://doi.org/10.1090/S1061-0022-06-00932-0
  • MathSciNet review: 2241428