Vol. 4, No. 5, 2011

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Non-Weyl resonance asymptotics for quantum graphs

E. Brian Davies and Alexander Pushnitski

Vol. 4 (2011), No. 5, 729–756
Abstract

We consider the resonances of a quantum graph G that consists of a compact part with one or more infinite leads attached to it. We discuss the leading term of the asymptotics of the number of resonances of G in a disc of a large radius. We call G a Weyl graph if the coefficient in front of this leading term coincides with the volume of the compact part of G. We give an explicit topological criterion for a graph to be Weyl. In the final section we analyze a particular example in some detail to explain how the transition from the Weyl to the non-Weyl case occurs.

Keywords
quantum graph, resonance, Weyl asymptotics
Mathematical Subject Classification 2000
Primary: 34B45
Secondary: 35B34, 47E05
Milestones
Received: 22 March 2010
Revised: 2 August 2010
Accepted: 14 September 2010
Published: 16 February 2012

Proposed: Terence Tao
Authors
E. Brian Davies
Department of Mathematics
King’s College London
Strand
London WC2R 2LS
United Kingdom
http://www.kcl.ac.uk/nms/depts/mathematics/people/atoz/daviesb.aspx
Alexander Pushnitski
Department of Mathematics
King’s College London
Strand
London WC2R 2LS
United Kingdom
http://www.mth.kcl.ac.uk/staff/a_pushnitski.html