Black-Hole Approach to the Singular Problem of Quantum Mechanics. II

Abstract

A new approach is proposed for the quantum mechanical problem of the falling of a particle to a singularly attracting center, basing on a black-hole concept of the latter. The singularity r^{-2} in the potential of the radial Schroedinger equation is considered as an emitting/absorbing center. The two solutions oscillating in the origin are treated as asymptotically free particles, which implies that the singular point r=0 in the Schroedinger equation is treated on the same physical ground as the singular point r=infinity. To make this interpretation possible, it is needed that the norm squared of the wave function should diverge when r tends to zero, in other words, the measure used in definition of scalar products should be singular in the origin. Such measure comes into play if the Schroedinger equation is written in the form of the generalized (Kamke) eigenvalue problem for either of two - chosen differently depending on the sign of the energy E - operators, other than Hamiltonian. The Hilbert spaces where these two operators act are used to classify physical states, which are: i) states of "confinement"- continuum of solutions localized near the origin, E<0 - and ii) the states corresponding to the inelastic process of reflection/transmission, i.e. to transitions in-between states localized near the origin and in the infinitely remote region, E>0. The corresponding unitary 2x2 S-matrix is written in terms of the Jost functions. The complete orthonormal sets of eigen-solutions of the two operators are found using "quantization in a box" (a,b), followed by the transition to the limit a=0, b=infinity. The corresponding expansions of the unity are written.