Demuth, Michael (ed.) et al., Spectral theory, microlocal analysis, singular manifolds. Berlin: Akademie Verlag. Math. Top. 14, 138-199 (1997).
Let be a bounded domain in with -dimensional boundary , and be a elliptic operator of order on . Consider the boundary value problem
where are partial differential operators of order with coefficients defined only on .
The paper treats the case that the coefficients of the operators are not always smooth, and moreover the operators are not always formally selfadjoint. Let be a sector in the complex plane with vertex at the origin. We assume that the operator is elliptic with parameter in . Let us consider the closed densely defined operator acting in , with domain
We see that in this case the resolvent set is nonvoid (contains all with large ) and the resolvent is compact. Thus the spectrum of as an operator in is discrete. If we consider the operators , in spaces, then they are spectrally equivalent (Section 3). The concern of the paper is to investigate the spectral property of the operator and to get a result on the asymptotic behavior in the absolute value of the eigenvalues by using the asymptotic behavior of the trace of the resolvent in parameter . The main results are essentially Theorem 4.1 and Theorem 5.1. In these theorems the asymptotic behavior of the trace in as is given when .