Some remarks on the generalized Gelfand-Levitan equation

Abstract

The Gelfand-Levitan (GL) and MarEenko (M) equations arise in quantum scattering theory as an important part of the machinery used to recover the potential q(x) from a Sturm-Liouville operator D2 -q(x) via inverse methods (cf. [32, 39, 551). Modifications of this procedure for handling inverse problems also arise in other applied fields (see, e.g., [21-24,32,61]) and lead in particular to various types of GL equations. In all of this work the transmutation concept plays an important role (although often disguised). Now in earlier papers we developed an extensive theory and framework for transmuting an operator of the form J% = (dPu’)‘/dP + pi (pP = f lim A;/AP as x -+ co) into another operator of the same type (cf. [6-9, 12-16, 18-20, 231). In particular this theory applies to situations where P = P -pi is modeled on the radial Laplace-Beltrami operator in a rank one noncompact Riemannian symmetric space and the spherical functions arising from equations &: = -I’p<, (Q<(O) = 1, D,pl;(O) = 0 involve various special functions of interest in diverse contexts. In [8,9] we derived a general GL and M equation for transmutations D* + p (p as above). Given the fundamental role of such equations in the inverse scattering and inverse Sturm-Liouville problems and the nature of their derivation in [8,9] from eigenfunction kernel expressions, it is clear that the version of [8,9] contains valuable information about special functions (cf. also [ 28-3 1, 44-461, where some relations between discrete forms of the GL and M equations of quantum scattering theory and orthogonal polynomials have been studied-although some version of this approach can be envisioned for singular operators of the type P above, we shall not pursue this here). In this paper we shall first extend (in Section 2) the GL equation of [8, 91 to the context of transmutations $-+ $ and clarify its structure. In particular we shall represent it as a formula in spherical functions. Then (in Section 3) we take a model operator P, = D2 + ((2m + l)/x)D (A, =x2”‘+‘) and give 410 0022-247x/83 $3.00