Some asymptotic spectral properties of singular operators

A class of uniformly elliptic, positive operators in $R^n$ with discrete spectrum is considered for which the coefficients of the derivatives of even order and the free term increase at the same rate, while the other coefficients play a subordinate role. The first term of the asymptotic expansion of the spectral function and $N(\lambda)$ is found for such operators; here $N(\lambda)=\sum_{\lambda_n\leqslant\lambda}1$, where the $\lambda_n$ are the eigenvalues of the operator.