Essential spectra of singular Hamiltonian differential operators of arbitrary order under a class of perturbations

The main object of this paper is to study the essential spectrum of a Hamiltonian system of arbitrary order with one singular endpoint under a class of perturbations. We first present a characterization of the essential spectrum in terms of singular sequences and then give the concept of perturbations small at singular endpoints of Hamiltonian systems. Based on the above characterization, the invariance of essential spectra of Hamiltonian systems under these perturbations is shown. It is noted that these perturbations are given by using the associated pre-minimal operator H00(L), which provides great convenience in the study of essential spectra of Hamiltonian systems since each element of the domain D(H00(L)) of H00(L) has compact support. As applications, some sufficient conditions for the invariance of essential spectra of some systems are obtained in terms of coefficients of systems and perturbations terms. Further, essential spectra of Hamiltonian systems with different weight functions are discussed. Here, Hamiltonian systems may be non-symmetric.