Resonant equations and special functions

摘要

Differential equations of the form Lf=g, where L is a differential operator, are termed Resonant if g satisfies Lg=0. In the case when L represents a linear harmonic oscillator, resonance occurs when the forcing term g has the same frequency as that of the unperturbed system. Resonance is associated with a transition from boundedness to unboundedness of the solution. We study the cases where L is the Legendre or Hermite operator. The first case arose in the context of supersymmetric Casimir operators for the di-spin algebra, and has solutions expressible in terms of singular functions, Legendre functions and polylogarithms. The non-singular polynomial parts of a certain class of solutions exhibit interesting properties. The non-resonant Hermite equation supports the theory of the quantum mechanical harmonic oscillator. A standard technique for its solution involves a Darboux/Infeld-Hull factorization of the Hamiltonian as a product of two first-order linear operators. The algebra of these operators can also be used to study the solutions of the resonant Hermite equation. A lowest order solution is found by elementary means, and then higher order solutions are generated by the repeated application of a ladder operator.