We study implications of unitarity for pseudo-orbit expansions of thespectral determinants of quantum maps and quantum graphs. In particular, weadvocate to group pseudo-orbits into sub-determinants. We show explicitly thatthe cancellation of long orbits is elegantly described on this level and thatunitarity can be built in using a simple sub-determinant identity which has anon-trivial interpretation in terms of pseudo-orbits. This identity yields muchmore detailed relations between pseudo orbits of different length than knownpreviously. We reformulate Newton identities and the spectral density in termsof sub-determinant expansions and point out the implications of thesub-determinant identity for these expressions. We analyse furthermore theeffect of the identity on spectral correlation functions such as theauto-correlation and parametric cross correlation functions of the spectraldeterminant and the spectral form factor.
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