A Class of Exactly-Solvable Eigenvalue Problems

TL;DR

This paper introduces a class of exactly solvable eigenvalue problems involving differential equations with polynomial potentials, providing explicit solutions, eigenvalues, and new orthogonal polynomials for all integer N.

Contribution

It presents closed-form solutions for a family of eigenvalue problems, revealing new orthogonal polynomials for odd N and detailed properties of these solutions.

Findings

Eigenvalues are all integers.

Eigenfunctions are confluent hypergeometric functions.

New classes of orthogonal polynomials for odd N.

Abstract

The class of differential-equation eigenvalue problems $-y''(x)+x^{2N+2}y(x)=x^N Ey(x)$ ($N=-1,0,1,2,3,...$) on the interval $-\infty<x<\infty$ can be solved in closed form for all the eigenvalues $E$ and the corresponding eigenfunctions $y(x)$. The eigenvalues are all integers and the eigenfunctions are all confluent hypergeometric functions. The eigenfunctions can be rewritten as products of polynomials and functions that decay exponentially as $x\to\pm \infty$. For odd $N$ the polynomials that are obtained in this way are new and interesting classes of orthogonal polynomials. For example, when N=1, the eigenfunctions are orthogonal polynomials in $x^3$ multiplying Airy functions of $x^2$. The properties of the polynomials for all $N$ are described in detail.