Spectral asymptotics of non-self-adjoint elliptic systems of differential operators in bounded domains (Q803817)

Let \(A=A(x,D)\) be a differential operator whose principal symbol \[ P(x,\xi)=\sum_{| \alpha | =2m}a_{\alpha}(x)\xi^{\alpha}\quad (x\in {\bar \Omega},\quad \xi \in {\mathbb{R}}^ n\setminus \{0\}) \] has only eigenvalues on the semi-axis \({\mathbb{R}}_+\), and denote by N(t) and N(t;x,\(\xi\)) the number of eigenvalues \(\lambda\) of A(x,D) with Re \(\lambda\) \(<t\) and eigenvalues \(\mu\) of P(x,\(\xi\)) with \(0<\mu <t\), respectively. The authors prove the asymptotic formula \[ N(t)\sim (1/(2\pi)^ n)\int_{\Omega}\int_{{\mathbb{R}}^ n}N(t,x,\xi)d\xi dx. \]