Semiclassical Density of States for the Quantum Asymmetric Top

TL;DR

This paper derives the semiclassical density of states for the eigenvalues of the quantum asymmetric top, a complex problem where explicit solutions are unknown, providing new insights into its spectral properties.

Contribution

It computes the semiclassical density of states for the quantum asymmetric top, extending spectral analysis beyond known explicit eigenvalue formulas.

Findings

Derived explicit semiclassical density of states for the asymmetric top

Extended spectral analysis to cases without explicit eigenvalue formulas

Provided a framework for understanding eigenvalue distribution in asymmetric tops

Abstract

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 $\alpha_0 < \alpha_1 <\alpha_2$.