Spectra of a class of non-symmetric operators in Hilbert spaces with applications to singular differential operators

Abstract

This paper is concerned with a class of non-symmetric operators, that is, 𝒥-symmetric operators, in Hilbert spaces. A sufficient condition for λ ∈ C being an element of the essential spectrum of a 𝒥-symmetric operator is given in terms of the number of linearly independent solutions of a certain homogeneous equation, and a characterization for points of the essential spectrum plus the set of all eigenvalues of a 𝒥-symmetric operator is obtained in terms of the numbers of linearly independent solutions of certain inhomogeneous equations. As direct applications, the corresponding results are obtained for singular 𝒥-symmetric Hamiltonian systems and their special forms of singular Sturm-Liouville equations with complex-valued coefficients, which enable us to study the spectra of singular 𝒥-symmetric differential expressions using numerous tools available in the fundamental theory of differential equations.

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

“…[Refs. [37,Theorem 4.1] and [39,Theorem 4.2]] Assume that (A) holds for system (1) with = 2 , ∈ ℕ. Then the following results hold:…”