One-dimensional generalised Coulomb problem - Lavrenov Аlexandre N.; Lavrenov, Ivan A. (Journal of the Belarusian State University. Physics, 2024)
The quantum-mechanical Coulomb problem, complicated in two directions, is considered in this article. The first generalisation is associated with the transition from Euclidean space to one-dimensional Cayley – Klein geometries, and the second one is connected with the addition of a singular term g/x2 to the Coulomb potential. It can be considered as a Calogerо – Sutherland potential, which is used to describe anyons, magnetic monopoles, dyons, etc. In addition to the methodological aspect, the problem under consideration will also be useful as a special case of the so-called coordinate-dependent mass model when describing nanostructures in quantum dots or on a plane, metamaterials and astronomical objects in strong magnetic fields. On the positive coordinate semiaxis, it turns into a certain generalisation of the model with the Kratzer potential, which is used to describe molecular energy and structure, interactions between molecules and non-bounded atoms. Using the factorisation method, the energy spectrum and wave functions of stationary states are found, having the curvature of space as a parameter. The formula for energy levels contains two terms. The first term gives the energy spectrum of the one-dimensional Coulomb problem, and the second term explicitly depends on the presence of curvature and is responsible for the spectrum of the particle on the circle S1(j). The coupling constant g of Calogero – Sutherland potential is non-linearly contained in both terms through a variable β0 (g2 = β0(β0 - 1)) representing an additive correction to the number of the energy level. In the special case of a purely Coulomb field, the results obtained coincide with the results published earlier.