A Class of Exactly-Solvable Eigenvalue Problems

Abstract

The class of differential-equation eigenvalue problems $-y^{\prime \prime }(x)+x^{2N+2}y(x)=x^{N}Ey(x)$ ($N=-1,0,1,2,3,...$) 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.

REVTeX, 16 pages, no figure

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