Unified Algorithm of Factorization Method for Derivation of Exact Solutions from Schrödinger Equation with Potentials Constructed from a Set of Functions

Abstract

We extend the scope of the unified factorization method to the solution of conditionally and unconditionally exactly solvable models of quantum mechanics, proposed in a previous paper [R.R. Nigmatullin, A.A. Khamzin, D. Baleanu, Results in Physics 41 (2022) 105945]. The possibilities of applying the unified approach in the factorization method are demonstrated by calculating the energy spectrum of a potential constructed in the form of a second-order polynomial in many of the linearly independent functions. We analyze the solutions in detail when the potential is constructed from two linearly independent functions. We show that in the general case, such kinds of potentials are conditionally exactly solvable. To verify the novel approach, we consider several known potentials. We show that the shape of the energy spectrum is invariant to the number of functions from which the potential is formed and is determined by the type of differential equations that the potential-generating functions obey.

Keywords

• exactly solvable models
• conditionally exactly solvable models
• Schrödinger equation
• factorization method
• superpotential