Exact and approximate solutions to Schroedinger's equation with decatic potentials

TL;DR

This paper investigates exact and approximate solutions to the one-dimensional Schrödinger equation with decatic polynomial potentials, identifying conditions for exact solvability and employing the asymptotic iteration method for general cases.

Contribution

It provides new conditions for exact solutions of decatic potentials and applies the asymptotic iteration method for broader parameter analysis.

Findings

Exact solutions exist under specific parameter conditions.

Polynomial solutions satisfy a four-term recurrence relation.

Asymptotic iteration method effectively approximates solutions for arbitrary parameters.

Abstract

The one-dimensional Schroedinger's equation is analysed with regard to the existence of exact solutions for decatic polynomial potentials. Under certain conditions on the potential's parameters, we show that the decatic polynomial potential $V(x)=ax^{10}+bx^8+cx^6+dx^4+ex^2$, $a>0$ is exactly solvable. By examining the polynomial solutions of certain linear differential equations with polynomial coefficients, the necessary and sufficient conditions for corresponding energy-dependent polynomial solutions are given in detail. It is also shown that these polynomials satisfy a four-term recurrence relation, whose real roots are the exact energy eigenvalues. Further, it is shown that these polynomials generate the eigenfunction solutions of the corresponding Schroedinger equation. Further analysis for arbitrary values of the potential parameters using the asymptotic iteration method is also presented.