One-dimensional Coulomb-like problem in general case of deformed space with minimal length

Abstract

In general case of deformed Heisenberg algebra leading to the minimal length we present a definition of the square inverse position operator. Our proposal is based on the functional analysis of the square position operator. Using this definition a particle in the field of the square inverse position potential is studied. We have obtained analytical and numerical solutions for the energy spectrum of the considerable problem in different cases of deformation function. We find that the energy spectrum slightly depends on the choice of deformation function.

Exact solutions for two-body problems in 1D deformed space with minimal length

“…Schrödinger equation in momentum representation for two-particle system can be written in a similar way as it was done for one-particle systems [15,16]:…”

“…The kernel of the potential energy operator in momentum representation in undeformed case has the following form [16]…”

“…a particle in delta potential and double delta potential [15], one-dimensional Coulomb-like problem in general case of deformation [16,17] and a particle in the singular inverse square potential [18,19].…”

Scattering solution of Schrödinger equation with $δ$-potential in deformed space with minimal length

“…The influence of the minimum length has been studied in the context of the following problems with singularity in potential energy: hydrogen atom [9][10][11][12][13][14][15], gravitational quantum well [16][17][18], a particle in delta potential and double delta potential [19,20], one-dimensional Coulomb-like problem [20][21][22], particle in the singular inverse square potential [23][24][25][26].…”

Regularization of 1/X2 potential in general case of deformed space with minimal length

“…The study of the effect of the minimal length on systems with singular potentials or point interactions is of particular interest, since such systems are expected to have a nontrivial sensitivity to minimal length. The impact of the minimum length has been studied in the context of the following problems with singularity in potential energy: hydrogen atom [9][10][11][12][13][14][15][16], gravitational quantum well [17][18][19], a particle in delta potential and double delta potential [20,21], one-dimensional Coulomb-like problem [21][22][23], particle in the singular inverse square potential [24][25][26][27], two-body problems with delta and Coulomb-like interactions [28].…”