On Schroedinger operators with inverse square potentials on the half-line

Abstract

The paper is devoted to operators given formally by the expression \begin{equation*} -\partial_x^2+\big(\alpha-\frac14\big)x^{-2}. \end{equation*} 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 $m^2=\alpha$, $-1<\Re(m)<1$, 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 $-1<\Re(m)<1$ is the maximal region of parameters for which the operators $H_{m,\kappa}$ can be defined within the framework of the Hilbert space $L^2({\mathbb R}_+)$.