Anisotropic Kepler Problem and Critical Level Statistics

The subject of this chapter is quantum chaos (QC), in particular, the QC that occurs in the Anisotropic Kepler Problem (AKP). In QC, one studies a quantum system whose classical counter part is chaotic, and one investigates how the chaotic property in the classical theory shows up its imprints in the quantum theory. To elucidate this quantum-classical correspondence is the first mission of the quantum chaos study. With the advent of nanophysics techniques, QC has become very important concern also at the experimental side; for pioneering works, we refer to conductance fluctuations in quantum dot (Marcus et. al., 1992), magnetoresistance on a superlattice of antidots (Weiss et al., 1991), a cold-atom realization of the kicked top (Chaudhury et al., 2009). Also for an interesting experimental observation of quantum scars of classical orbits (Heller, 1984; 1989), we refer to (Stein & Stockmann, 1992). Furthermore, quantum chaos study has been developed under far-reachingmutual influences with related areas. In order to explain where our study in AKP stands, let us briefly review a few aspects of quantum chaos study. Let us first consider the randommatrix theory (RMT) (Mehta, 2004). It is proposed by (Wigner, 1951) to predict the universal spectral property of complex nucleus, and the mathematical basement is set by (Dyson, 1962), (Dyson & Mehta, 1963) and (Mehta & Dyson, 1963). In RMT, the hamiltonian of the physical system is described by a random matrix in the three basic ensembles. This implies that the intrinsic quantum property of the physical system is determined by the time reversal symmetry and internal symmetry only, and does not depend on the details of the system hamiltonian. If the time-reversal by the operator T is broken in a system, the relevant ensemble is gaussian random ensemble of hermite matrices for the hamiltonian, admitting the invariance of the hamiltonian under the unitary transformation (GUE), with Dyson parameter β = 2. If the T invariance holds with T2 = 1, it is gaussian orthogonal ensemble (GOE) of real symmetric matrices with β = 1 , while with T2 = −1, it is gaussian symplectic ensemble (GSE) of quarternion selfdual matrices with β = 4. It is conjectured by (Bohigas et al., 1984) that, irrespective to the details of the system hamiltonian, the stochastic spectral property of energy levels of an physical system (including AKP) is uniquely described by the relevant RMT ensemble chosen by the above symmetry property Anisotropic Kepler Problem and Critical Level Statistics