Vacuum energy and closed orbits in quantum graphs

Abstract

The vacuum (Casimir) energy of a quantized scalar field in a given geometrical situation is a certain moment of the eigenvalue density of an asso- ciated self-adjoint differential operator. For various classes of quantum graphs it has been calculated by several methods: (1) Direct calculation from the explicitly known spectrum is feasible only in simple cases. (2) Analysis of the secular equation determining the spectrum, as in the Kottos-Smilansky derivation of the trace formula, yields a sum over periodic orbits in the graph. (3) Construction of an associated integral kernel by the method of images yields a sum over closed (not necessarily periodic) orbits. We show that for the Kirchhoff and other scale-invariant boundary conditions the sum over non- periodic orbits in fact makes no contribution to the total energy, whereas for more general (frequency-dependent) vertex scattering matrices it can make a nonvanishing contribution, which, however, is localized near vertices and hence can be indexed a posteriori by truly periodic orbits. For the scale-invariant cases complete calculations have been done by both methods (2) and (3), with identical results. Indeed, applying the image method to the resolvent kernel provides an alternative derivation of the trace formula.