EIGENFUNCTION EXPANSIONS FOR THE SCHRODINGER EQUATION WITH AN INVERSE-SQUARE POTENTIAL

We consider the one-dimensional Schrodinger equation - f '' + q(kappa)f = Ef on the positive half-axis with the potential q(kappa)(r) = (kappa(2) - 1/4)r(-2). For each complex number v, we construct a solution u(nu)(kappa)(E) of this equation that is analytic in kappa in a complex neighborhood of the interval (-1, 1) and, in particular, at the "singular" point kappa = 0. For-1 < kappa < 1 and real nu(,) the solutions u(nu)(kappa)(E) determine a unitary eigenfunction expansion operator U-kappa,U-nu: L-2(0,infinity) -> L-2(R,V kappa,nu), where V kappa,nu is a positive measure on R. We show that every self-adjoint realization of the formal differential expression -partial derivative(2)(r) + q(kappa)(r) for the Hamiltonian is diagonalized by the operator U-kappa,U-nu for some nu a R. Using suitable singular Titchmarsh-Weyl m-functions, we explicitly find the measures V-kappa,V-nu and prove their continuity in kappa and nu.