Quasi-Weyl Asymptotics of the Spectrum of the Vector Dirichlet Problem

In a space of vector functions, we consider the spectral problem $\mu Au=\mathcal{P}u$, $u=u(x)$, where $A=(A_{jk})$, $j,k=1,\dots,n$, $A_{jk}=\sum_\alpha a_{\alpha jk}D^{2\alpha}$, $\mathcal{P}=(p_{jk})$, $A\ge c_0>0$, $\mathcal{P}=\mathcal{P}^*$, the $a_{\alpha jk}$ and $p_{jk}$ are constants, $x\in\Omega$, and $\Omega$ is a bounded open set. The boundary conditions correspond to the Dirichlet problem. Let $N_\pm(\mu)$ be the positive and negative spectral counting functions. We establish the asymptotics $N_\pm(\mu)\sim(\operatorname{mes}_m\Omega)\varphi_\pm(\mu)$ as $\mu\to+0$. The functions $\varphi_\pm(\mu)$ are independent of $\Omega$. In the nonelliptic case, these asymptotics are in general different from the classical (Weyl) asymptotics.