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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

On a mathematical model of irreversible quantum graphs


Author: M. Z. Solomyak
Translated by: R. Shterenberg
Original publication: Algebra i Analiz, tom 17 (2005), nomer 5.
Journal: St. Petersburg Math. J. 17 (2006), 835-864
MSC (2000): Primary 35Q40, 34L40
Published electronically: July 27, 2006
MathSciNet review: 2241428
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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.


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Additional Information

M. Z. Solomyak
Affiliation: Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
Email: michail.solomyak@weizmann.ac.il

DOI: http://dx.doi.org/10.1090/S1061-0022-06-00932-0
PII: S 1061-0022(06)00932-0
Keywords: Quantum graphs, spectrum, Jacobi matrices
Received by editor(s): December 21, 2004
Published electronically: July 27, 2006
Dedicated: In fond memory of Ol$’$ga Aleksandrovna Ladyzhenskaya
Article copyright: © Copyright 2006 American Mathematical Society