Coupling constant dependence for the Schrodinger equation with an inverse-square potential

摘要

We consider the one-dimensional Schrodinger equation -f ''+q(alpha)f = Ef on the positive half-axis with the potential q(alpha)(r) = (alpha-1/4)r(-2). It is known that the value alpha = 0 plays a special role in this problem: all self-adjoint realizations of the formal differential expression -partial derivative(2)(r) + q(alpha)(r) for the Hamiltonian have infinitely many eigenvalues for alpha = 0. We find a parametrization of self-adjoint boundary conditions and eigenfunction expansions that is analytic in alpha and, in particular, is not singular at alpha = 0. Employing suitable singular Titchmarsh-Weyl m-functions, we explicitly find the spectral measures for all self-adjoint Hamiltonians and prove their smooth dependence on alpha and the boundary condition. Using the formulas for the spectral measures, we analyze in detail how the "phase transition" through the point alpha = 0 occurs for both the eigenvalues and the continuous spectrum of the Hamiltonians.