Hilbert space, boundary value problems and orthogonal polynomials
The book has two main themes. The first concerns the extension of the Titchmarsh-Weyl theory associated with second-order linear symmetric differential equations to linear Hamiltonian systems of the form \[ JY'=(\lambda A+ B)Y\tag{1} \] on some interval \((a,b)\subseteq \mathbb{R}\), where \(J\), \(A\), \(B\) are square matrices satisfying \(J^2=-I\), \(A= A^*\geq 0\), \(B= B^*\), and \(\lambda\) is the complex spectral parameter. A detailed spectral analysis of (1) is given and this includes a comprehensive treatment of the \(M(\lambda)\)-matrices, which are the analogues for (1) of the Titchmarsh-Weyl \(m(\lambda)\)-function for second-order scalar problems, and related topics like the existence of \(L^2_A(a,b)\) solutions of (1) and boundary conditions which generate selfadjoint operators. The account is influenced by the work of F. V. Atkinson, H. D. Niessen and especially that of D. B. Hinton and J. K. Shaw. The author has also made significant contributions in this area. The second theme is orthogonal polynomials. The emphasis is on the determination of orthogonal polynomials which satisfy differential equations of even order. Examples discussed, other than the classical orthogonal polynomials satisfying second-order equations, include the H. L. Krall, Littlejohn, Koekoek and generalized Laguerre-type polynomials which satisfy differential equations of order greater than 2. This problem goes back to the work of the author's father, \textit{H. L. Krall} [Duke Math. J. 4, 705--718 (1938; Zbl 0020.02002)], and has been a topic of intensive study during the last 20 years.
scientific article; zbMATH DE number 1818055