Polynomial-type eigenfunctions

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Abstract
The differential equation for the harmonic oscillator is generalised to include an interaction potential containing a positive definite quadratic denominator. Conditions are developed under which certain eigenfunctions take the form of an exponential function multiplied by a polynomial. The problem reduces to that of finding the eigenvalues of a certain matrix that is tridiagonal in form. Properties of the eigenvalues of this matrix are investigated, since they are functions of a parameter occurring in the positive definite quadratic form. The asymptotic forms of these eigenvalues are developed together with computed results expressed as curves showing the variation of the eigenvalues with respect to this parameter.

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