Minami N.
On Level Clustering in Regular Spectra
(44K, LATeX 2e)
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.