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

The paper is devoted to operators given formally by the expression ... 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 ... 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 ...