On the singularities of the zeta and eta functions of an elliptic operator (Q2888818)

Let \(P : C^{\infty} (M, E) \rightarrow C^{\infty} (M, E)\) be a self-adjoint elliptic pseudodifferential operator of order \(m\), \(m\in \mathbb{N}\), acting on the sections of a Hermitian vector bundle \(E\) over a compact Riemannian manifold of dimensional \(n\).NEWLINENEWLINEThe zeta and eta functions of \(P\) are two of the most important spectral functions that can be associated to \(P\). Not only do they study the spectral properties of \(P\), but when \(P\) is a Laplace-type or Dirac-type operator they may carry a lot of a geometric information on the manifold \(M\). In particular, the residues of the zeta and eta functions are of special interest. For instance, the residues of the eta function at integer points naturally come into the index formula for Dirac operators on manifolds with singularities (see [\textit{J. Brüning} and \textit{R. Seeley}, Am. J. Math. 110, No. 4, 659--714 (1988; Zbl 0664.58035)]). In addition, the residues at integer points of the zeta functions of the square of the Dirac operators are important in the context of noncommutative geometry (see [\textit{R. Ponge}, Lett. Math. Phys. 83, No. 1, 19--32 (2008; Zbl 1136.58017)]).NEWLINENEWLINEGeneral arguments show that the zeta and eta functions may have poles only at points of the form \(\frac{k}{m}\), where \(k\) ranges over all nonzero integers less than \(n+1\). The authors construct elementary and explicit examples of perturbations of \(P\) which make the zeta and eta functions become singular at all points at which they are allowed to have singularities. The authors proceed within three classes of operators: Dirac-type operators, self-adjoint first-order differential operators and self-adjoint elliptic pseudodifferential operators. As consequences, genericity results for the singularities of the zeta and eta functions in those settings are obtained. In particular, in the setting of Dirac-type operators, the authors obtain a purely analytical proof of a well-known result of \textit{T. P. Branson} and \textit{P. B. Gilkey} [J. Funct. Anal. 108, No. 1, 47--87 (1992; Zbl 0756.58048)], which was obtained by invoking Riemannian invariant theory.