On an extension of the Ikehara Tauberian theorem
In order to study the asymptotic behavior of eigenvalues for a class of hypoelliptic pseudodifferential operators P, we need to extend the Ikehara Tauberian theorem. Let H be a Hilbert space and P a densely defined positive self-adjoint operator with domain K on H. We regard K equipped with the graph norm as a Hilbert space. We assume that the canonical injection from K to H is compact. Then it is well known that the spectrum \(\sigma\) (P) is discrete. Thus we denote the eigenvalues of P by: \(0<\lambda _ 1\leq \lambda _ 2\leq..\). \((\lambda _ k\to \infty)\) with repetition according to multiplicity and the number of \(\lambda _ k\) so that \(\lambda _ k\leq \lambda\) by \(N_ P(\lambda)\). Then we have: Theorem. Let P be an operator as above. Assume that (i) \(P^{-s}\) is of trace class for large Re s\(>0\) and Tr \(P^{-s}\) has a meromorphic extension \(Z_ p(s)\) in the complex plane \({\mathbb{C}}\) whose poles are distributed on the real line. (ii) \(Z_ P(s)\) has the first singularity at \(s=a\) \((>0)\) and \[ Z_ P(s)-\sum ^{p}_{j=1}\frac{A_ j}{(j- 1)!}(-\frac{d}{ds})^{j-1}\frac{1}{s-a} \] is holomorphic in \(\{s\in {\mathbb{C}}\); Re\(s>a-\delta\}\) for some \(\delta >0\). (iii) \(Z_ P(s)\) is of polynomial order with respect to Im s in all vertical strips, excluding neighborhoods of the poles. Then we have for some \(\delta _ 0>0\), \[ N_ P(\lambda)=\sum ^{p}_{j=1}\frac{A_ j}{(j- 1)!}(\frac{d}{ds})^{j-1}(\frac{\lambda ^ s}{s})| _{s=a}+O(\lambda ^{a-\delta _ 0})\text{ as } \lambda \to +\infty. \]
hypoelliptic pseudodifferential operators
scientific article; zbMATH DE number 4002712