On Schr\xf6dinger Operators with Inverse Square Potentials on the Half-Line

Abstract

The paper is devoted to operators given formally by the expression -∂x2+α-141x2.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} -\\partial _x^2+\\left( \\alpha -\\frac{1}{4}\\right) \\frac{1}{x^{2}}. \\end{aligned}$$\\end{document}This expression is homogeneous of degree minus 2. However, when we try to realize it as a self-adjoint operator for real alpha , or closed operator for complex alpha , we find that this homogeneity can be broken. This leads to a definition of two holomorphic families of closed operators on L^2({mathbb {R}}_+), which we denote H_{m,kappa } and H_0^nu , with m^2=alpha , -1<mathrm{Re}(m)<1, and where kappa ,nu in {mathbb {C}}cup {infty } specify the boundary condition at 0. We study these operators using their explicit solvability in terms of Bessel-type functions and the Gamma function. In particular, we show that their point spectrum has a curious shape: a string of eigenvalues on a piece of a spiral. Their continuous spectrum is always [0,infty [. Restricted to their continuous spectrum, we diagonalize these operators using a generalization of the Hankel transformation. We also study their scattering theory. These operators are usually non-self-adjoint. Nevertheless, it is possible to use concepts typical for the self-adjoint case to study them. Let us also stress that -1<mathrm{Re}(m)<1 is the maximal region of parameters for which the operators H_{m,kappa } can be defined within the framework of the Hilbert space L^2({mathbb {R}}_+).