**Overview**

The principal goal in the field of quantum chaos is often described as *finding fingerprints of classical chaos in quantum mechanics*.The Schroedinger equation, governing the time evolution of quantum systems, is linear and hence no chaotic behaviour emerges in quantum dynamics. On the other hand, classical Hamiltonian dynamics can be either regular or completely irregular, hence chaotic, or of any intermediate type in between these extremes.

In many examples of quantum systems it was found that certain of their characteristic features are implied by the fact that their corresponding classical counterparts are either integrable (i.e., regular) or chaotic. Most obviously, this is seen in the correlation of energy eigenvalues. Berry and Tabor conjectured that energy eigenvalues of quantum systems with integrable classical limit are uncorrelated, alike outcomes of a Poissonian random process. Subsequently, Bohigs, Giannoni and Schmit conjectured that the eigenvalue correlations in spectra of quantum systems with chaotic classical limit arethe same as the expected spectral correlations in random hermitian matrices of the same symmetry type.

In the classically chaotic case the corresponding eigenfunctions (stationary quantum states) satisfy two properties: Berry conjectured that their correlations are the same as in random waves, whereas following a theorem of Shnirelman, Zelditch, Colin de Verdiere and others classical ergodicity implies an equidistribution of almost all eigenfunctions on the energy shell in phase space (quantum ergodicity).

Many of the aforementioned properties have been studied in great detail in a number of model systems, such as:

- Laplacians on manifolds of negative curvature
- Quantum billiards
- Quantum maps
- Quantum graphs

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