Semiclassical Density of States for the Quantum Asymmetric Top

Details Semiclassical Density of States for the Quantum Asymmetric Top Semiclassical Density of States for the Quantum Asymmetric Top

2007-06-28

arxiv.org/abs/0706.4127v1

Physics

Mathematical Physics

Scientific paper

In the quantization of a rotating rigid body, a {\it top,} one is concerned with the Hamiltonian operator $L_\alpha=\alpha_0^2 L_x^2 + \alpha_1^2 L_y^2 + \alpha_2^2 L_z^2,$ where $\alpha_0 < \alpha_1 <\alpha_2.$ An explicit formula is known for the eigenvalues of $L_\alpha$ in the case of the spherical top ($\alpha_1 = \alpha_2 = \alpha_3$) and symmetrical top ($\alpha_1 = \alpha_2 \neq \alpha_3$) \cite{LL}. However, for the asymmetrical top, no such explicit expression exists, and the study of the spectrum is much more complex. In this paper, we compute the semiclassical density of states for the eigenvalues of the family of operators $L_\alpha=\alpha_0^2 L_x^2 + \alpha_1^2 L_y^2 + \alpha_2^2 L_z^2$ for any $\alpha_0 < \alpha_1 <\alpha_2$.

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