A Generalization of the Direct and Inverse Problem for the Radial Schrödinger Equation

Abstract

We consider the operators H0= −d2/dr2 and H1 = −d2/dr2 + V(r) (0< r< ∞) acting on a Hilbert space of complex functions f(r) such that the subspaces in which the operators are defined consist of twice differentiable functions which satisfy the boundary condition (d/dr)f(0) = αf(0). H1 and H0 are Hermitian in this subspace. Assuming V(r)→0 as r→∞ sufficiently rapidly, the scattering operator formalism is set up for the direct scattering problem. Next we consider the inverse problem of determining V(r) from H0 and the spectral measure function for the spectrum of H1 through the use of an appropriate Gelfand‐Levitan equation. It is shown that generally the value of α associated with H1 differs from that for H0, i.e., H1 and H0 generally operate in different subspaces. Thus scattering cannot be defined. However, by changing the spectral measure function, one obtains a new Gelfand‐Levitan equation such that H1 is the same as before [i.e., α and V(r) are the same] from the operator H0, which uses the same value of α as H1. Thus H1 and the new H0 operate in the same subspace of Hilbert space, and scattering can be defined. The process of obtaining the new H0 after finding H1 from the old H0 is somewhat analogous to renormalization in field theory, where a new H0 is picked to have properties compatible with H1. A necessary and sufficient condition on the spectral data is given which makes the domains of H0 and H1 coincide and thus makes “renormalization” unnecessary. The direct problem is a generalization of the usual l=0 radial Schrödinger equation. The inverse problem is a generalization of the corresponding inverse problem. It is also a generalization of the case α=0 for H0 considered by Gelfand and Levitan in their early work on the inverse spectral problem. An incompletely understood connection of the inverse problem for the radial equation to solutions of the Korteweg‐deVries equation in the half space is discussed. The existence of such a connection is one of the motivations for studying the generalized radial Schrodinger equation.