Thursday, August 16, 2012

1208.3072 (Ralf Rueckriemen et al.)

Trace Formulae for quantum graphs with edge potentials    [PDF]

Ralf Rueckriemen, Uzy Smilansky
This work explores the spectra of quantum graphs where the Schr\"odinger operator on the edges is equipped with a potential. The scattering approach, which was originally introduced for the potential free case, is extended to this case and used to derive a secular function whose zeros coincide with the eigenvalue spectrum. Exact trace formulas for both smooth and $\delta$-potentials are derived, and an asymptotic semiclassical trace formula (for smooth potentials) is presented and discussed.
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