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Mathematics in Science and Technology

VERTEX COUPLINGS IN QUANTUM GRAPHS: APPROXIMATIONS BY SCALED SCHRÖDINGER OPERATORS

Pavel Exner

Published Jun 1, 2011• 7 citations
Publisher

Abstract

We review recent progress in understanding the physical meaning of quantum graph models through analysis of their vertex coupling approximations.
DOI
10.1142/9789814338820_0004

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We present an integral formulation of the vacuum energy of Schrödinger operators on finite metric graphs. Local vertex matching conditions on the graph are classified according to the general scheme of Kostrykin and Schrader. While the vacuum energy of the graph can contain finite ambiguities the Casimir force on a bond with compactly supported potential is well defined. The vacuum energy is determined from the zeta function of the graph Schrödinger operator which is derived from an appropriate secular equation via the argument principle. A quantum graph has an associated probabilistic classical dynamics which is generically both ergodic and mixing. The results therefore present an analytic formulation of the vacuum energy of this quasi-one-dimensional quantum system which is classically chaotic.
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