Representation of free solutions for Schr\"odinger equations with strongly singular concentrated potentials

Yu. M. Shirokov

Teoretičeskaâ i matematičeskaâ fizika, Tome 46 (1981) no. 3, pp. 291-299

Résumé

A new representation is obtained for the Schrödinger equation constructed and solved in the author's earlier paper [1] for three-dimensional motion of a particle in the field of a strongly singular concentrated potential. In the new representation, the Schrödinger equation becomes free but the wave functions of the bound states have exponential growth at infinity. The transition to the new representation is linear but contains a procedure of analytic continuation, which makes it a transformation that does not possess a kernel and does not exist in the complete Hilbert space. It is shown that, using the new representation, one can readily obtain the complete solution of the original Schrödinger equation. The new “free-solution representation” is used to obtain the complete solution to the quantum problem of the motion of a particle in the field of a centrally symmetric concentrated potential that acts in states with $l\ne0$. Positivity of the metric has not been verified for the obtained solution. The possibility of applying the method to the quantum problem of several bodies with concentrated two-body interactions is noted.