M. Amirfachrian | Searcharxiv

arXiv2011

On Schrödinger equation with potential U = - αr^{-1} + βr + kr^{2} and the bi-confluent Heun functions theory

It is shown that Schrödinger equation with combination of three potentials U = - α r^{-1} + β r + kr^{2}, Coulomb, linear and harmonic, the potential often used to describe quarkonium, is reduced to a bi-confluent Heun differential equation. The method to construct its solutions in the form of polynomials is developed, however with additional constraints in four parameters of the model, α, β, k, l. The energy spectrum looks as a modified combination of oscillator and Coulomb parts.

math-ph↗

arXiv2011

Wave functions of a particle with polarizability in the Coulomb potential

Quantum mechanical scalar particle with polarizability is considered in the presence of the Coulomb field. Separation of variables is performed with the use of Wigner $D$-functions, the radial system of 15 equations is reduced to a single second order differential equation, which among the Coulomb term includes an additional interaction term of the form σα^{2} / M^{2}r^{4}. Various physical regimes exist that is demonstrated by examining the behavior of the curves of generalized squared radial momentum P^{2}(r). Eigenstates of the equations can be constructed in terms of double confluent Heun functions. Numerical analysis proves the existence of the bound states in the system; the lowest energy level and corresponding solution are calculated based on generalization of Ritz variational procedure.

math-ph↗