2004/06/08 by Bernhelm Booß–Bavnbek, Bernhelm Booss-Bavnbek, Booss-Bavnbek, Bernhelm +2
Mathematics · #Advanced Mathematical Physics Problems #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Primary 58J30 #Secondary 53D12 #Spectral Theory in Mathematical Physics #Symplectic Geometry (math.SG) #math.DG #math.SG #msc:53D12 #msc:58J30
paper · pdf · doi:10.48550/arxiv.math/0406139
arxiv created 2004/06/08 · openalex publication_date 2004/06/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a continuous curve of self-adjoint Fredholm extensions of a curve of closed symmetric operators with fixed minimal domain Dm and fixed \it intermediate domain DW. Our main example is a family of symmetric generalized operators of Dirac type on a compact manifold with boundary with varying well-posed boundary conditions. Here DW is the first Sobolev space and Dm the subspace of sections with support in the interior. We express the spectral flow of the operator curve by the Maslov index of a corresponding curve of Fredholm pairs of Lagrangian subspaces of the quotient Hilbert space DW/Dm which is equipped with continuously varying \it weak symplectic structures induced by the Green form. In this paper, we specify the continuity conditions; define the Maslov index in weak symplectic analysis; discuss the required weak inner Unique Continuation Property; derive a General Spectral Flow Formula; and check that the assumptions are natural and all are satisfied in geometric and pseudo-differential context. Applications are given to L2 spectral flow formulae; to the splitting of the spectral flow on partitioned manifolds; and to linear Hamiltonian systems.