On essential spectra of singular linear Hamiltonian systems

Abstract

The paper is concerned with essential spectra of singular linear Hamiltonian systems of arbitrary order with arbitrary equal defect indices. Several sufficient conditions for the essential spectral points are given in terms of the number of linearly independent square integrable solutions of the corresponding Hamiltonian system, and a sufficient and necessary condition for the essential spectral points is obtained for Hamiltonian systems of even-order with one singular endpoint. An advantage of these results is that one can determine the essential spectral points of Hamiltonian systems by the information of solutions obtained by numerous tools available in the fundamental theory of differential equations. In addition, two illustrative examples are provided to show how to get some information about the essential spectrum by our results.

Stability of essential spectra of singular Sturm‐Liouville differential operators under perturbations small at infinity

“…For every a >0, consider the restriction τ a of τ to [ a , ∞ ). We have σ e ( τ )= σ e ( τ a ) by Sun and Shi, Theorem 4.1 since equation τ y = λ y is a special case of the Hamiltonian systems discussed in Sun and Shi and the numbers of linearly independent square integrable solutions of τ y = λ y and τ a y = λ y are equal. Let T a ,0 be the preminimal operator corresponding to τ a defined similarly as T 0 .…”

“…Then , and hence is transformed into From , we get that y ∈ L 2 [0, ∞ ) if and only if , and hence the number of linearly independent solutions of equation in L 2 [0, ∞ ) is equal to that of linearly independent solutions of equation in . Then, by Sun and Shi, Theorem 4.1. Using …”

“…The spectral theory for singular differential operators, especially for singular Sturm‐Liouville differential operators, has been investigated extensively by using many methods, and many good results have been established (cf, eg, previous studies() and their references). Among them, many sufficient and necessary conditions for relatively compact perturbations of singular symmetric Sturm‐Liouville differential operators, higher order differential operators, and Dirac operators were established under certain assumptions (see, eg, Anderson and Hinton, Balslev and Gamelin, and Melescue).…”

Regular approximations of spectra of singular discrete linear Hamiltonian systems with one singular endpoint

“…The study of fundamental theory of regular difference equations has a long history and their spectral theory has formed a relatively complete theoretical system such as eigenvalue problems, orthogonality of eigenfunctions and expansion theory (cf., [2,17,27,36,39,41]). Spectral problems for singular difference equations were firstly studied by Atkinson [2] in 1964, and some significant progresses have been made since then (cf., e.g., [5,6,8,16,21,22,24,25,28,32,33,37,38]). Especially, research on spectral theory of singular discrete Hamiltonian systems has attracted a great deal of interest and some good results have been obtained (cf., [22,24,25,28,37,38], and references cited therein).…”

Essential spectrum of singular discrete linear Hamiltonian systems

“…Spectra of singular symmetric differential operators have been studied intensively by researchers using various methods such as the asymptotic analysis, the oscillation theory, and the singular sequence approaches, see , , , , , . Especially, spectra of singular symmetric differential operators can be considered using the numbers of linearly independent square‐integrable solutions (cf., e.g., , , , , , , , ). An advantage of this method is the availability of numerous tools in the fundamental theory of differential equations for the study of spectra of differential operators.…”