A Class of Exactly-Solvable Eigenvalue Problems

The class of differential-equation eigenvalue problems −y′′(x)+x2N+2y(x) = xNEy(x) (N = −1, 0, 1, 2, 3, . . .) on the interval −∞ < x < ∞ 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 → ±∞. 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 x3 multiplying Airy functions of x2. The properties of the polynomials for all N are described in detail. Typeset using REVTEX 1