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Quantum mechanics says that the state function phi for a quantum mechanical system at energy level E satisfies Schrödinger's equation L phi = E phi. Here L is the Hamiltonian operator for the system (a differential operator). Part of quantum chaos is concerned with the statistics of the energy levels E for quantum systems whose classical motion is chaotic. In practice this means that one looks at the histograms of the energy levels (or differences of energy levels). The operator L is an operator on a subset of an infinite-dimensional Hilbert space, so this physical problem is a problem in infinite-dimensional linear algebra or functional analysis. Of course, in practice physicists can find approximations to only a finite number of energy levels for a given physical system. You might think that this would throw a monkey wrench into the works, but amazingly the physicists have found some beautiful results that have significance in number theory and other situations. We only sketch a bit of their work here. The interested reader should look at [1], [2], [4], and [13].
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