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 physics. A, Proceedings of the Physical Society. General

Complementary bounds on phase shifts

N. Anderson, A. M. Arthurs, Peter D. Robinson

Published Nov 1, 1970• 5 citations
Publisher

Abstract

Upper and lower bounds on phase shifts for scattering by short-range potentials are presented. Their derivation is based on complementary variational principles for a certain class of linear operator equations. The well-known Schwinger bound is obtained from this approach together with its complementary bound which appears to be new. The results are illustrated with calculations for a positive step potential.

DOI
10.1088/0305-4470/3/6/001

Related papers

Complementary variational principles in perturbation theory

A. M. Arthurs, Peter D. Robinson

Abstract Complementary upper and lower bounds are derived for second-order quantum-mechanical perturbation energies. The upper bound is equivalent to that of Hylleraas. The lower bound appears to be new, but reduces to that of Prager & Hirschfelder if a certain constraint is applied. A simple application to a perturbed harmonic oscillator is presented.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Mar 19, 1968•9 citations

Adiabatic optical potential and variational lower bounds on scattering parameters near threshold

Yukap Hahn

The exact elastic optical potential is compared with an analogous potential in the adiabatic approximation, which is shown to give a lower bound on the scattering length. A variational-bound procedure is formulated to estimate the scattering length and also the scattering phase shift near the elastic threshold.

Physical Review A•Dec 1, 1975•6 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

Calculating energy shifts in terms of phase shifts

Zhenhua Yu, Gordon Baym, C. J. Pethick

To clarify the relation of energy shifts to scattering phase shifts in one-body and many-body problems, we examine their relation in a number of different situations. We derive, for a particle in a container of arbitrary shape with a short-range scattering center, a general result for the energy eigenvalues in terms of the s-wave scattering phase shift and the eigenstates in the absence of the scatterer. We show that, while the energy shifts for a spherical container are proportional to the phase shift over large ranges, those for a cubic container have a more complicated behavior. We connect our result to the description of energy shifts in terms of the scattering T-matrix. The general relation is extended to problems of particles in traps with smoothly varying potentials, including, e.g., the interaction of a small neutral atom with a Rydberg atom. We then consider the many-body problem for particles with a two-body interaction and show that the free energy change due to the interaction is proportional to an average of a generalized phase shift that includes the effects of the medium. Finally, we discuss why, even though individual energy levels are very sensitive to boundary conditions, the energy of a many-body system is not.

arXiv:0911.2853•Journal of Physics B Atomic Molecular and Optical Physics•Sep 13, 2011•5 citations

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

Theory of low-energy scattering by a long-range potential

Vishwamittar

The present theory of low-energy scattering by a long-range potential has been examined by omitting the assumptions about the wavefunctions and taking the phase to be a function of the position of the scattered particle. The resulting integro-differential equation for the phase shift has been simplified for a particular case and it is found that the magnitude of the correction terms so obtained is quite large. It is shown that the two-potential formalism can be applied only to light pairs of scattering atoms.

Journal of Physics B Atomic and Molecular Physics•Jun 15, 1978•2 citations
Upper and lower bounds on phase shifts for scattering by short-range potentials are presented. Their derivation is based on complementary variational principles for a certain class of linear operator equations. The well-known Schwinger bound is obtained from this approach together with its complementary bound which appears to be new. The results are illustrated with calculations for a positive step potential.
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.
To clarify the relation of energy shifts to scattering phase shifts in one-body and many-body problems, we examine their relation in a number of different situations. We derive, for a particle in a container of arbitrary shape with a short-range scattering center, a general result for the energy eigenvalues in terms of the s-wave scattering phase shift and the eigenstates in the absence of the scatterer. We show that, while the energy shifts for a spherical container are proportional to the phase shift over large ranges, those for a cubic container have a more complicated behavior. We connect our result to the description of energy shifts in terms of the scattering T-matrix. The general relation is extended to problems of particles in traps with smoothly varying potentials, including, e.g., the interaction of a small neutral atom with a Rydberg atom. We then consider the many-body problem for particles with a two-body interaction and show that the free energy change due to the interaction is proportional to an average of a generalized phase shift that includes the effects of the medium. Finally, we discuss why, even though individual energy levels are very sensitive to boundary conditions, the energy of a many-body system is not.
The present theory of low-energy scattering by a long-range potential has been examined by omitting the assumptions about the wavefunctions and taking the phase to be a function of the position of the scattered particle. The resulting integro-differential equation for the phase shift has been simplified for a particular case and it is found that the magnitude of the correction terms so obtained is quite large. It is shown that the two-potential formalism can be applied only to light pairs of scattering atoms.