The left-definite Legendre type boundary problem (Q1176327)

The authors reconsider the fourth-order Legendre type differential operator \[ M_ k[y]=[(1-x^ 2)^ 2y'']''-[8+4\alpha(1-x^ 2)y']'+ky \] defined in the Hilbert function space \[ H=\{f: [-1,1]\to\mathbb{C}\mid f\in AC[-1,1]; \qquad f'\in AC_{loc}(-1,1); \qquad f',(1-x^ 2)f''\in L^ 2(-1,1)\} \] endowed with a particular inner product. Here \(\alpha>0\) and \(k\geq 0\) are given constants. The study of this operator is called a ``left-definite boundary value problem''. The main aim of the paper is to show that the self-adjoint left-definite operator \(S_ k[. ]\) associated with \(M_ k[. ]\) has a pure point spectrum. To this end they prove first a rather general theorem which claims that the space \(C^ 2[-1,1]\) is dense in the above-mentioned Hilbert space (weighted Sobolev space). The proof of this theorem, given by the authors, is constructive and can be generalized to other types of weighted Sobolev spaces.