Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Equivalent variational functionals in potential scattering

Peter D. Robinson

Published Aug 1, 1973• 2 citations
Publisher

Abstract

The equivalence is established between dual pairs of variational functionals (Kohn type) derived from the differential equation description of potential scattering and other dual pairs of variational functions (Schwinger type) derived from the integral equation description. Apparent differences are resolved by suitable choice of trial vector. The Kohn functional and minus the Schwinger functional are shown to be duals of each other.

DOI
10.1063/1.1666442

Related papers

Variational Principle for Potential Scattering

G. H. Goedecke, E. Brown

Potential scattering is treated by a minimum variational principle. The method, known as the error method, is applicable to problems which can be cast as linear inhomogeneous equations. The method makes use of a non-negative functional which reduces to zero for the exact solution. The magnitude of the functional for approximate solutions provides an indication of the accuracy of the wave function. The method is illustrated by some examples, and compared with the Schwinger and Kohn principles.

Physical Review•Mar 15, 1962•2 citations

Variational approach to the quantum theory of scattering. II. The Kohn method and Kato eigenvalue problem

H. J. Kolker

Certain similarities between the equations arising in scattering theory and the variational approach to time dependent perturbation theory are used to show that the Kohn variational technique for solving scattering problems is equivalent to a variational method for calculating a special set of eigenfunctions, in terms of which the solution to the scattering problem is expanded. Contrary to the results of previous studies, the error term in the scattering problem is bound from below, and the Kohn method is equivalent to a minimum principle. Levinson's theorem is rederived, and extended to higher energies. On the basis of the theory presented in this study, a new method for calculating upper and lower bounds on the energies of resonance and Ramsauer-Townsend states is suggested.

The Journal of Chemical Physics•Mar 15, 1973•6 citations

Integral Equations for the Schrödinger Wave Function

William Tobocman, Leslie L. Foldy

For many problems in quantum mechanics it proves to be useful to write the Schrödinger equation as an integral equation rather than as a differential equation. In this paper we describe how the transformation of the Schrödinger equation from differential form to integral form may be accomplished. Two methods are described, one appropriate to bound-state theory and one appropriate to scattering theory. It is shown that the bound-state wave function is also a solution of the integral equation (Lippman-Schwinger equation) appropriate to scattering theory.

American Journal of Physics•Oct 1, 1959•6 citations

Integral equations of scattering in one dimension

Vania Elisabeth Barlette, Marcelo M. Leite, Sadhan K. Adhikari

A self-contained discussion of integral equations of scattering is presented in the case of centrally symmetric potentials in one dimension, which will facilitate the understanding of more complex scattering integral equations in two and three dimensions. The present discussion illustrates in a simple fashion the concept of partial-wave decomposition, Green’s function, Lippmann–Schwinger integral equations of scattering for wave function and transition operator, optical theorem, and unitarity relation. We illustrate the present approach with a Dirac delta potential.

arXiv:quant-ph/0103018•American Journal of Physics•Sep 1, 2001•41 citations

Variational principles and second-order Lagrangian densities

G J N Brown, S F C O'Rourke, Derrick S. F. Crothers

The functionals of the Kohn and Sil variational principles of atomic collision theory contain second-order spatial derivatives of the wave-function. Ipso facto the "second-order" generalised Euler-Lagrange differential equation is a necessary condition for these functionals to yield a stationary value. To illustrate this we provide a classical prescription for the time-dependent and time-independent quantum potential scattering problems, generalising the latter to the three-body problem. As practical examples where the presence of second-order derivatives in the Lagrangian density makes a non-trivial contribution to the equations of motion, we consider the optimisation of translation factors in a trial wave-function using the Sil variational principle and the optimisation of a variable charge using the Kohn variational principle. As further examples of Lagrangian densities where second-order derivatives may occur we briefly mention electromagnetism and relativistic quantum mechanics.

Physica Scripta•Feb 1, 1994•3 citations

Potential Scattering

K. T. Tang

The well-known expression of the scattering amplitude is rederived from the integral representation. The exact equivalence between the integral and differential formulation is explicitly demonstrated.

American Journal of Physics•Feb 1, 1966•0 citations
The equivalence is established between dual pairs of variational functionals (Kohn type) derived from the differential equation description of potential scattering and other dual pairs of variational functions (Schwinger type) derived from the integral equation description. Apparent differences are resolved by suitable choice of trial vector. The Kohn functional and minus the Schwinger functional are shown to be duals of each other.
Certain similarities between the equations arising in scattering theory and the variational approach to time dependent perturbation theory are used to show that the Kohn variational technique for solving scattering problems is equivalent to a variational method for calculating a special set of eigenfunctions, in terms of which the solution to the scattering problem is expanded. Contrary to the results of previous studies, the error term in the scattering problem is bound from below, and the Kohn method is equivalent to a minimum principle. Levinson's theorem is rederived, and extended to higher energies. On the basis of the theory presented in this study, a new method for calculating upper and lower bounds on the energies of resonance and Ramsauer-Townsend states is suggested.
For many problems in quantum mechanics it proves to be useful to write the Schrödinger equation as an integral equation rather than as a differential equation. In this paper we describe how the transformation of the Schrödinger equation from differential form to integral form may be accomplished. Two methods are described, one appropriate to bound-state theory and one appropriate to scattering theory. It is shown that the bound-state wave function is also a solution of the integral equation (Lippman-Schwinger equation) appropriate to scattering theory.
A self-contained discussion of integral equations of scattering is presented in the case of centrally symmetric potentials in one dimension, which will facilitate the understanding of more complex scattering integral equations in two and three dimensions. The present discussion illustrates in a simple fashion the concept of partial-wave decomposition, Green’s function, Lippmann–Schwinger integral equations of scattering for wave function and transition operator, optical theorem, and unitarity relation. We illustrate the present approach with a Dirac delta potential.
The functionals of the Kohn and Sil variational principles of atomic collision theory contain second-order spatial derivatives of the wave-function. Ipso facto the "second-order" generalised Euler-Lagrange differential equation is a necessary condition for these functionals to yield a stationary value. To illustrate this we provide a classical prescription for the time-dependent and time-independent quantum potential scattering problems, generalising the latter to the three-body problem. As practical examples where the presence of second-order derivatives in the Lagrangian density makes a non-trivial contribution to the equations of motion, we consider the optimisation of translation factors in a trial wave-function using the Sil variational principle and the optimisation of a variable charge using the Kohn variational principle. As further examples of Lagrangian densities where second-order derivatives may occur we briefly mention electromagnetism and relativistic quantum mechanics.