Instability, Index Theorem, and Exponential Trichotomy for Linear Hamiltonian PDEs

Instability, Index Theorem, and Exponential Trichotomy for Linear Hamiltonian PDEs Synopsis

"Consider a general linear Hamiltonian system tu JLu in a Hilbert space . We assume that L induces a bounded and symmetric bi-linear form L, on , which has only finitely many negative dimensions n (L). There is no restriction on the anti-self-dual operator J D(J) . We first obtain a structural decomposition of into the direct sum of several closed subspaces so that L is blockwise diagonalized and JL is of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy of etJL. In particular, etJL has at most algebraic growth in the finite co-dimensional center subspace. Next we prove an instability index theorem to relate n (L) and the dimensions of generalized eigenspaces of eigenvalues of JL, some of which may be embedded in the continuous spectrum. This generalizes and refines previous results, where mostly J was assumed to have a bounded inverse. More explicit i