Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some Studies on Arithmetical Chaos in Classical and Quantum Mechanics   Several aspects of classical and quantum mechanics applied to a class of
strongly chaotic systems are studied. These consist of single particles moving
without external forces on surfaces of constant negative Gaussian curvature
whose corresponding fundamental groups are supplied with an arithmetic
structure. It is shown that the arithmetic features of the considered systems
lead to exceptional properties of the corresponding spectra of lengths of
periodic orbits. The most significant one is an exponential growth of
degeneracies in these length spectra. Furthermore, the arithmetical systems are
distinguished by a structure that appears as a generalization of geometric
symmetries. These pseudosymmetries occur in the quantization of the classical
arithmetic systems as Hecke operators, which form an infinite algebra of
self-adjoint operators commuting with the Hamiltonian. The statistical
properties of quantum energies in the arithmetical have previously been
identified as exceptional. They do not fit into the general scheme of random
matrix theory. It is shown with the help of a simplified model for the spectral
form factor how the spectral statistics in arithmetic quantum chaos can be
understood by the properties of the corresponding classical length spectra. A
decisive is played by the exponentially increasing multiplicities of lengths.
The model developed for the level spacings distribution and for the number
variance is compared to the corresponding quantities obtained from quantum
energies for a specific arithmetical system.

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The categories of this paper are: adap-org
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The categories of this paper are: econ
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The categories of this paper are: mtrl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-th
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: math-ph
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The categories of this paper are: physics
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-ex