J -self-adjoint extensions for second-order linear difference equations with complex coefficients
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RESEARCH
Open Access
J-self-adjoint extensions for second-order linear difference equations with complex coefficients Huaqing Sun1* and Guojing Ren2 *
Correspondence: [email protected] 1 Department of Mathematics, Shandong University at Weihai, Weihai, Shandong 264209, P.R. China Full list of author information is available at the end of the article
Abstract This paper is concerned with second-order linear difference equations with complex coefficients which are formally J-symmetric. Both J-self-adjoint subspace extensions and J-self-adjoint operator extensions of the corresponding minimal subspace are completely characterized in terms of boundary conditions. MSC: Primary 39A70; secondary 47A06 Keywords: difference equation; complex coefficient; J-Hermitian subspace; J-self-adjoint subspace extension; defect index
1 Introduction In this paper, we consider the following second-order linear difference equation with complex coefficients: τ (x)(t) := –∇ p(t)x(t) + q(t)x(t) = λw(t)x(t),
t ∈ I,
(.)
where I is the integer set {t}bt=a , a is a finite integer or –∞, and b is a finite integer or +∞ with b – a ≥ ; and ∇ are the forward and backward difference operators, respectively, i.e., x(t) = x(t + ) – x(t) and ∇x(t) = x(t) – x(t – ); p(t) and q(t) are complex with p(t) = for t ∈ I, p(a – ) = if a is finite and p(b + ) = if b is finite; w(t) > for t ∈ I; and λ is a spectral parameter. Equation (.) is formally symmetric if and only if both p(t) and q(t) are real numbers. Therefore, if p(t) or q(t) are complex, then Eq. (.) is formally nonsymmetric. To study nonsymmetric operators, Glazman introduced the concept of J-symmetric operators in [] where J is a conjugation operator (see Definition .). The minimal operators generated by Sturm-Liouville and some higher-order differential and difference expressions with complex coefficients are J-symmetric operators in the related Hilbert spaces (e.g., [–]). Here, we remark that a bounded J-symmetric operator is also called a complex symmetric operator (cf. [, ]). The operators generated by singular differential and difference expressions are not bounded in general. It is well known that the study of spectra of symmetric (J-symmetric) differential expressions is to consider the spectra of self-adjoint (J-self-adjoint) operators generated by such expressions. In general, under a certain definiteness condition, a formally differential expression can generate a minimal operator in a related Hilbert space and its adjoint © 2013 Sun and Ren; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Sun and Ren Advances in Difference Equations 2013, 2013:3 http://www.advancesindifferenceequations.com/content/2013/1/3
is the corresponding maximal operator (see, e.g., [, ]). Generally, the self-adjoint (Jself-adjoint) operators are gen
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