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My understanding of the so-called correspondence principle in quantum chaos, is that it is a connection between the behaviour of a classical Hamiltonian system (chaotic/completely integrable) and the behaviour of the quantized system in the semi-classical limit $\hbar \to 0$.

For the spectrum of the quantized system, we seem to have two heuristics on the statistics of the spectrum.

  • When the classical system is completely integrable, one expects the spectrum to behave like a Poisson process. For example, Berry-Tabor conjectures that the nearest spacing statistics of the spectrum distributes like the waiting time between consecutive events of a Poisson process.

  • When the classical system is chaotic, one expects the spectrum to distribute like a suitable Gaussian orthogonal ensemble. For example, Bohigas-Giannoni-Schmit conjectures that the nearest spacing statistics to distribute like the consecutive level spacing distribution of some Gaussian orthogonal ensemble.

Question: These heuristics only seem to hold for compact Riemannian manifolds though. What heuristics are there in more general cases, for example in finite-volume manifolds, as long as the spectrum is expected to have a discrete part and is large enough so that understanding its distribution makes sense? Are there any reference on this issue?

The rough heuristic above certainly seems false in that case. For example, For $\Gamma \backslash \mathbb{H}$ where $\Gamma \leq SL_2(\mathbb{R})$ is some lattice, the geodesic flow is certainly chaotic (ergodic, Anosov,..). My impression is that if $\Gamma$ is a non-arithmetic lattice (whatever that means), one expects the statistics of the spectrum of the Laplacian to be modeled by GOE as before; but for arithmetic lattice, the statistics of the spectrum for Laplacian seems to be Poissonian again.

My vague speculative impression is that the heuristics above seems to apply even for finite-volume manifolds as long as it is "non-arithmetic", but then I still have no idea on what to expect for the spectrum of arithmetic manifolds. In the case of surfaces, Sarnak (Spectra of hyperbolic surface) seems to suggest that spectrum of Laplacian would be Poissonian/GOE-like depending on whether the lattice is arithmetic/non-arithmetic. But I have not been able to find references/speculations for higher dimensions.

Edit: The answer below is helpful, but I would like a more definitive reference/source on the current expectations/state of the art. The best reference on these issues so far remain to be Sarnak's Schur lecture notes "Arithmetic Quantum Chaos" back in 1993. There has to be a more updated survey after 20 years. Would anyone point me into more reference?

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A recent study of the spectral statistics of arithmetic billiards in the semiclassical limit is Arithmetic and pseudo-arithmetic billiards (2015):

The arithmetic triangular billiards are classically chaotic but have Poissonian energy level statistics, in ostensible violation of the Bohigas-Giannoni-Schmit conjecture. From the semiclassical point of view, the peculiar properties of arithmetic systems result from constructive interference of contributions of an abnormally large number of periodic orbits with exactly the same length and action. We establish the boundary conditions under which the quantum billiard is “genuinely arithmetic”, i.e., has Poissonian level statistics; otherwise the billiard is ”pseudoarithmetic” and belongs to the GOE universality class.

For a higher-dimensional study, see Arithmetic quantum chaos of Maass waveforms (2004):

We compute numerically the eigenvalues and eigenfunctions of the Laplacian that describes the quantum mechanics of a point particle moving freely in the non-integrable three-dimensional hyperbolic space of constant negative curvature generated by the Picard group. The Picard group is arithmetic and we find that our results are in accordance with the conjecture that arithmetic quantum chaos produces a Poisson distribution of the eigenvalue distribution.

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