  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.
