Spectral analysis of the sixth-order Krall differential expression. (English) Zbl 1481.47036
Authors’ abstract: In this paper, we construct a self-adjoint operator \(\widehat{T}_{A,B}\) generated by the sixth-order Krall differential expression in the extended Hilbert space \(L_{2}(-1,1)\oplus \mathbb{C}_{2}\). To obtain \(\widehat{T}_{A,B}\), we apply a new general theory, the so-called GKN-EM theory, developed recently by L. L. Littlejohn and R. Wellman [Oper. Matrices 13, No. 3, 667–704 (2019; Zbl 1426.47003)], that extends the classical Glazman-Krein-Naimark theory using a complex symplectic geometric approach developed by W. N. Everitt and L. Markus [Trans. Am. Math. Soc. 351, No. 12, 4905–4945 (1999; Zbl 0936.34005)]. This work extends earlier studies of the Krall expression by both L. L. Littlejohn [Nonclassical orthogonal polynomials and differential equations. The Pennsylvania State University, State College (PhD Thesis) (1981); Quaest. Math. 5, 255–265 (1982; Zbl 0514.33008)] and S. M. Loveland [Spectral analysis of the Legendre equations. Logan, UT: Utah State University (PhD Thesis) (1990)].
Reviewer: Bilender P. Allahverdiev (Isparta)
MSC:
| 47B25 | Linear symmetric and selfadjoint operators (unbounded) |
| 33C65 | Appell, Horn and Lauricella functions |
| 34B30 | Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) |
| 34B20 | Weyl theory and its generalizations for ordinary differential equations |
| 47B65 | Positive linear operators and order-bounded operators |
Keywords:
Krall polynomials; orthogonal polynomials; Glazman-Krein-Naimark theory; symplectic geometry; Lagrangian symmetry; self-adjoint operator; boundary conditionsReferences:
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