On a mathematical model of irreversible quantum graphs
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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
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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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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
- © 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
Dedicated: In fond memory of Ol$’$ga Aleksandrovna Ladyzhenskaya