Spectral properties of generalized Toeplitz matrices

The asymptotic behavior of $N_n(\lambda)$, the number of eigenvalues less than $\lambda$, as $n\to\infty$ is found for a sequence of generalized Toeplitz operators $A_n$, along with the asymptotic behavior of $\operatorname{det}A_n$. It is shown that both asymptotic formulas are quasiclassical and connected with the quantization of classical mechanics whose phase spaces are products of two-dimensional spheres.
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