THE LEFT-DEFINITE LEGENDRE TYPE BOUNDARY-PROBLEM

The left-definite Legendre type boundary problem concerns the study of a fourth-order singular differential expression M(k)[.] in a weighted Sobolev space H generated by a Dirichlet inner product. The fourth-order differential equation M(k)[y] = lambda-y has orthogonal polynomial eigenfunctions, called the Legendre type polynomials, associated with the eigenvalues lambda-n = n(n + 1)(n2 + n + 4-alpha - 2) + k. In this paper, we show that the space C2[-1, 1] is dense in H, from which it follows that the spectrum of the self-adjoint left-definite operator S(k)[.] associated with M(k)[.] is a purely point spectrum and consists only of the eigenvalues lambda-n. Comparisons between S(k)[.] and the associated right-definite operator T(k)[.] are made. This work extends earlier work of Everitt, Krall, Littlejohn, and Williams.