The quantum spectral statistical properties, i.e. the nearest neighbor
level spacing distribution (NNSD), the mode fluctuation distribution (MFD),
the spectral rigidity etc. of a spin-1/2 particle in two- and
three-dimensional coupled quartic oscillator potentials are
numerically studied. Changing coupling parameters the system is
continuously transformed from an integrable one to chaotic ones. In
the chaotic regime, selecting non-zero parameters, various kinds of
the Gaussian ensembles(GEs): GOE, GUE and GSE can be achieved
[1]. Especially, it is a rare example for the GSE. We have been
interested in the Gaussian ensembles in the aspect of the intermediate
system between the integrable one and the GEs. In order to have
reliable statistics of quantum levels, it is necessary to evaluate
thousands of energy levels from the ground state without missing any. We
compute the quantum energy levels by numerical diagonalization of the
truncated matrix of the Hamiltonian in the basis of harmonic
oscillators. Most of our calculations start with the system of the
superposition of two or three truncated harmonic oscillators
[2,3]. The method with the spherical Bessel functions and the surface
harmonics has been also developed. Several types of the interpolation
formula of the Poissonian and the Wigner-like distribution of each GEs
for the NNSD are examined. It is found that the MFD is more sensitive
to the integrability of the system than the chaoticity as in the case
of a spin-less particle. The distribution of 
nearest neighbors
is also examined [4].
References
[1] E. Caurier and B. Grammaticos, Phys. Lett. A136 387(1989).
[2] M. Tomiya and N. Yoshinaga, Phys. Rev. E58 8017(1998).
[3] M. Tomiya and N. Yoshinaga, J. Phys. Soc. Jpn. 69 2786(2000).
[4] M. L. Mehta, Random Matrices, 2nd ed., Academic Press, San Diego, 1991.