Left-Definite Regular Hamiltonian Systems

Linear Hamiltonian systems allow us to generalize, as well as consider, self-adjoint problems of any even order. Such left-definite problems are interesting, not only because of the generalization, but also because of the new intricacies they expose, some of which have made it possible to go beyond fourth order scale problems. We explore the left definite Sobolev settings for such problems, which are in general subspaces determined by boundary conditions. We show that the Hamiltonian operator remains self-adjoint, and inherits the same resolvent and spectral resolution from its original L² space when set in the left-definite Sobolev space.