Singular Centre in Quantum Mechanics as a Black Hole

Abstract

We consider the radial Schroedinger equation with an attractive potential singular in the origin. The additional continuum of states caused by the singularity, that usually remain nontreatable, are shown to correspond to particles, asymptotically free near the singularity (in the inner channel). Depending on kinematics, they are either confined by the centre or may escape to infinity (to the outer channel). The orthonormality within the continuum of confined states is established and the scattering phase of the particle emitted by the centre and then reflected back to it is found. For the deconfinement case a unitary 2x2 S-matrix is found in terms of the Jost functions, and describes transitions within and between the two channels. The volume elements in the two channels are different. The two-channel situation is analogous to the known behaviour of radiation in the black hole metrics. We discuss the black hole essence of singularly attracting centre for classical motion and the relativity of time inherent to this problem.