00-171 Minami N.

On Level Clustering in Regular Spectra (44K, LATeX 2e) Apr 7, 00

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Abstract. This expository paper, which is an elaboration of a part of the author's previous note ([10] and archived as mp_arc 00-163), aims at giving a mathematical formulation of level statistics based on the idea of Berry and Tabor, and applying it to regular spectra, obtained by quantizing a classically integrable system. We define strict and wide sense level clustering, and prove some preliminary results for later references. These results are formulated in analogy with corresponding propositions in the theory of stationary point processes. Then we shall apply the level statistics thus formulated to regular spectra, and discuss the closely related theorems by Sinai and Major. We argue that although it is probably very difficult to apply theorems of Sinai and Major to prove the strict sense level clustering in generic regular spectra, there is some hope in proving the wide sense level clustering for some concrete Hamiltonian such as rectangular billiard. In conclusion, we argue that "level clustering" should not always mean strict Poissonian property of the spectrum.

Files: 00-171.tex