The asymptotic behavior of the spectral function for elliptic operators in an~unbounded region

G. I. Bass

Matematičeskie zametki, Tome 5 (1969) no. 2, pp. 245-251

Résumé

We consider elliptic self-adjoint differential operators $L$ of order $2m$ in a bounded region $D\subset R_n$. An asymptotic formula for the function $N(\lambda)=\sum\limits_{\lambda_n\lambda}1$ the number of eigenvalues of the operator $L$ less than $\lambda$ is proved: $$ N(\lambda)=M_0\lambda{n/2m}+o(\lambda^{n/2m}) $$ where $\lambda\to+\infty$ and $M_0$ is the following constant: $$ M_0=\frac{V_D}{(2\pi)^n\Gamma(1+n/2m)}\int_{R_n}e^{-L(s)}\,ds. $$