4 M ay 2 00 5 General Spectral Flow Formula for Fixed Maximal Domain

We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by the Cauchy data spaces. We provide rigorous definitions of the underlying concepts of spectral theory and symplectic analysis and give a full (and surprisingly short) proof of our General Spectral Flow Formula for the case of fixed maximal domain. As a side result, we establish local stability of weak inner unique continuation property (UCP) and explain its role for parameter dependent spectral theory. 1. Statement of the problem and main result 1.1. Statement of the problem. Roughly speaking, the spectral flow counts the net number of eigenvalues changing from the negative real half axis to the non-negative one. The definition goes back to a famous paper by M. Atiyah, V. Patodi, and I. Singer [3], and was made rigorous by J. Phillips [23] for continuous paths of bounded selfadjoint Fredholm operators, by K.P. Wojciechowski [29] and C. Zhu 2000 Mathematics Subject Classification. Primary 58J30; Secondary 53D12.