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A star-graph model via operator extension

Published online by Cambridge University Press:  10 April 2007

B. PAVLOV
B. PAVLOV*
Affiliation:
Department of Mathematics, the University of Auckland, Auckland, New Zealand, Private Bag 92019. e-mail: pavlov@math.auckland.ac.nz
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Abstract

One usually assumes that the Schrödinger operator on a thin quantum network (“fattened graph”) can be simulated by the ordinary Schrödinger operator on the corresponding one-dimensional quantum graph. On the other hand, each quantum graph can be constructed of standard elements – star graphs. We prove that a thin star-shaped quantum network which consists of a compact domain and a few straight semi-infinite quantum wires, with a two-dimensional Schrödinger operator on it, can be simulated by the corresponding solvable model in form of a one-dimensional star graph: an outer space, with an ordinary Schrödinger operator on the leads, a resonance vertex supplied with an inner space and a finite matrix in it and an appropriate boundary condition connecting the inner and outer components of elements from the domain of the model. The model is (locally) quantitatively consistent: the scattering matrix of the model on a certain spectral interval serves an approximation of the scattering matrix of the network. The role of the constructed star-graph model as a “jump-start” in analytic perturbation procedure on continuous spectrum is discussed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 2 , March 2007 , pp. 365 - 384
Copyright
Copyright © Cambridge Philosophical Society 2007

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