00-163 Minami N.

Level statistics for quantum Hamiltonians--Some preliminary ideas toward mathematical justification of the theory of Berry and Tabor (16K, LATeX 2e) Apr 5, 00

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Abstract. According to Berry and Tabor, the spectrum of a classically integrable quantum Hamiltonian (the regular spectrum) should generically have the same statistical property of the Poisson point process. In particular, the spacings between energy levels obey the exponential distribution, which is the phenomena called "level clustering". Strict justification of this assertion seems to be a very difficult mathematical problem. In this note, the author shall introduce a weakened notion of level clustering and give a sufficient condition for this in terms of factorial moments of the number of levels in intervals of fixed length. A preliminary result is also stated, which is closely related to the justification of this sufficient condition in the case of regular spectra. All proofs are omitted, and will be published elsewhere.

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