Seikei University, Japan

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.