On Schrodinger Operators with Inverse Square Potentials on the Half-Line

This expression is homogeneous of degree minus 2. However, when we try to realize it as a self-adjoint operator for real , or closed operator for complex , we find that this homogeneity can be broken. This leads to a definition of two holomorphic families of closed operators on , which we denote and , with , , and where 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 . 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 is the maximal region of parameters for which the operators can be defined within the framework of the Hilbert space .