Laguerre polynomials and singular differential operators
The second order differential equations having as solutions the classical orthogonal polynomials are some of the best known examples to illustrate the well-developed theory of second order, singular, formally symmetric differential expressions. In this paper, the author develops the appropriate right- and left-definite boundary value problems associated with the second order Laguerre operator: \[ M[y]\quad =-(py')'+qy = \lambda wy, \] where \(p=x^{\alpha +1}e^{-x}\), \(q=(\alpha +1/2)w(x)\) and \(w=x^{\alpha}e^{-x}\), \(\alpha >-1\). The Laguerre polynomials \(\{L^{\alpha}_ n(x)\}\) are eigenfunctions of M[\(\cdot]\); they form a complete orthogonal set in the Hilbert space \(L\) \(2_ w(0,\infty)\), consisting of all Lebesgue measure functions f: (0,\(\infty)\to {\mathbb{C}}\) satisfying \(\int^{\infty}_{0}| f|\) 2 w dx\(<\infty\). In the classical ways of Titchmarsh, the right-definite problem is studied in the space \(L\) \(2_ w(0,\infty)\) and the author establishes the self- adjointness of an unbounded operator \(T_{\alpha}\), generated by M[\(\cdot]\), on a suitably restricted domain of \(L\) \(2_ w(0,\infty)\) having the Laguerre polynomials as eigenfunctions. The left-definite problem concerns the study of M[\(\cdot]\) in the Hilbert function space H, consisting of complex-valued functions f on (0,\(\infty)\) that are locally absolutely continuous and satisfy \(p^{1/2}f',q^{1/2}f\in L\) 2(0,\(\infty)\), with inner product \[ (f,g)_ H=\int^{\infty}_{0}\{pf'\bar g'+qf\bar g\}dx. \] Using operator- theoretic techniques, the author proves the existence of a bounded self- adjoint operator on H, whose inverse is an unbounded, self-adjoint operator \(S_{\alpha}\) having the Laguerre polynomials as eigenfunctions. The interesting feature of this left-definite study is that there does not appear to be a way of relating the operator \(S_{\alpha}\) in H with \(T_{\alpha}\) in \(L\) \(2_ w(0,\infty)\). This work continues earlier work of \textit{W. N. Everitt} [Ordinary and partial differential equations, Proc. 5th Conf., Dundee 1978, Lect. Notes Math. 827, 83-106 (1980; Zbl 0453.34021)] who made a similar analysis of the Legendre operator.
scientific article; zbMATH DE number 4050010