The energy level spacing for two harmonic oscillators with golden mean ratio of frequencies

Abstract

Berry and Tabor discussed, among other things, a beautiful problem about the energy level spacing distribution for a system of two harmonic oscillators. They gave some interesting theoretical arguments which show that there is no level clustering for generic harmonic oscillators, and various numerical experiments were exposed and discussed. But the main question they posed about the existence of the limit distribution of the level spacing remained open. The present paper discusses this question in the case when the ratio of the frequencies is the golden mean σ=(√5−1)/2. The approach enables one to study the generic case of the frequency ratio as well, which is done elsewhere.