Exact polynomial solutions of second order differential equations and their applications

TL;DR

This paper systematically finds all polynomial solutions to a class of second-order differential equations with polynomial coefficients and applies these results to obtain exact solutions for various physical quantum systems.

Contribution

It provides a complete characterization of polynomial solutions for a specific class of second-order differential equations with polynomial coefficients, including explicit algebraic conditions for roots and special cases.

Findings

Derived algebraic equations for roots of polynomial solutions.

Classified all polynomial solutions for equations with algebraically dependent coefficients.

Obtained exact solutions for several quantum mechanical models.

Abstract

We find all polynomials $Z(z)$ such that the differential equation $${X(z)\frac{d^2}{dz^2}+Y(z)\frac{d}{dz}+Z(z)}S(z)=0,$$ where $X(z), Y(z), Z(z)$ are polynomials of degree at most 4, 3, 2 respectively, has polynomial solutions $S(z)=\prod_{i=1}^n(z-z_i)$ of degree $n$ with distinct roots $z_i$. We derive a set of $n$ algebraic equations which determine these roots. We also find all polynomials $Z(z)$ which give polynomial solutions to the differential equation when the coefficients of X(z) and Y(z) are algebraically dependent. As applications to our general results, we obtain the exact (closed-form) solutions of the Schr\"odinger type differential equations describing: 1) Two Coulombically repelling electrons on a sphere; 2) Schr\"odinger equation from kink stability analysis of $\phi^6$-type field theory; 3) Static perturbations for the non-extremal Reissner-Nordstr\"om solution; 4) Planar Dirac electron in Coulomb and magnetic fields; and 5) O(N) invariant decatic anharmonic oscillator.