GKN theory for linear Hamiltonian systems

Abstract

In this paper, we give the definition of maximal and minimal operators for linear Hamiltonian systems and investigate the relationship between the conjugate scalar product in a weighted Hilbert space and the skew-symmetric boundary form of the associated singular Hamiltonian operator, namely, the one-to-one correspondence between the set of self-adjoint extensions of the minimal operator and the set of Lagrangian symplectic subspaces. These results extend and improve the classical Glazman–Krein–Naimark (GKN) theory for quasi-differential operators.