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A Generalization of the Direct and Inverse Problem for the Radial Schrödinger Equation
Author(s) -
Moses Harry E.
Publication year - 1978
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1978583187
Subject(s) - mathematics , linear subspace , mathematical physics , hilbert space , self adjoint operator , operator (biology) , hermitian matrix , mathematical analysis , differentiable function , boundary value problem , inverse scattering problem , combinatorics , pure mathematics , inverse problem , biochemistry , chemistry , repressor , transcription factor , gene
We consider the operators H 0 = − d 2 / dr 2 and H 1 = − d 2 / dr 2 + 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 ) = α f (0). H 1 and H 0 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 H 0 and the spectral measure function for the spectrum of H 1 through the use of an appropriate Gelfand‐Levitan equation. It is shown that generally the value of α associated with H 1 differs from that for H 0 , i.e., H 1 and H 0 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 H 1 is the same as before [i.e., α and V ( r ) are the same] from the operator H 0 , which uses the same value of α as H 1 . Thus H 1 and the new H 0 operate in the same subspace of Hilbert space, and scattering can be defined. The process of obtaining the new H 0 after finding H 1 from the old H 0 is somewhat analogous to renormalization in field theory, where a new H 0 is picked to have properties compatible with H 1 . A necessary and sufficient condition on the spectral data is given which makes the domains of H 0 and H 1 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 H 0 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.