Codiagonal Perturbations

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Abstract
Matrix methods are developed for calculating the eigenvalues and eigenvectors of a large class of quantum‐mechanical operators which may be regarded as perturbed forms of special‐function operators. Specific representations are obtained for the latter, including all of the important cases treated by Infeld and Hull. To these are added representations for terms sufficient to generate forms corresponding to the Mathieu equation, the Lamé equation, and others. A rapidly convergent computational scheme applicable to asymmetric matrices, which retains its stability even when the perturbing terms become large, is described; and its use is illustrated by application to the operator p(1 − q2)p − α2q2, corresponding to the Legendre‐like form (d/dx)(1 − x2)(d/dx) + α2x2. Though group‐theoretic considerations are stressed, appropriate correlations with differential and integral equations are presented throughout.

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