Poles of zeta and eta functions for perturbations of the Atiyah-Patodi-Singer problem

The zeta and eta functions of a differential operator of Dirac-type on a co mpact n-dimensional manifold, provided with a well-posed pseudodifferential boundary condition, have been shown in [G99] to be meromorphic on C with s imple or double poles on the real axis. Extending results from [G99] we sho w how perturbations of the boundary condition of order - J affect the poles ; in particular they preserve a possible regularity of zeta at 0 and a poss ible simple pole of eta at 0 when J greater than or equal to n. This applie s to perturbations of spectral boundary conditions, also when the structure is non-product and the problem is non-selfadjoint.