A sub-determinant approach for pseudo-orbit expansions of spectral determinants

We study implications of unitarity for pseudo-orbit expansions of the spectral determinants of quantum maps and quantum graphs. In particular, we advocate to group pseudo-orbits into sub-determinants. We show explicitly that the cancellation of long orbits is elegantly described on this level and that unitarity can be built in using a simple sub-determinant identity which has a non-trivial interpretation in terms of pseudo-orbits. We reformulate Newton identities and the spectral density in terms of sub-determinant expansions and point out the implications of the subdeterminant identity for these expressions. We analyse furthermore the effect of the identity on spectral correlation functions such as the auto-correlation and parametric cross correlation functions of the spectral determinant and the spectral form factor. ar X iv :1 20 9. 31 31 v1 [ nl in .C D ] 1 4 Se p 20 12 A sub-determinant approach for pseudo-orbit expansions of spectral determinants 2