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In the quantization of a rotating rigid body, a {\it top,} one is concernedwith 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 formulais 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 explicitexpression exists, and the study of the spectrum is much more complex. In thispaper, we compute the semiclassical density of states for the eigenvalues ofthe 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 $\alpha_0 < \alpha_1 <\alpha_2$.

Semiclassical Density of States for the Quantum Asymmetric Top