A study of the angular momentum dependence of the phase shift for finite range and coulomb potentials

Valluri, S. R.; Romo, W.J.

doi:10.1016/0375-9474(89)90410-7

Cited by 8 publications

(6 citation statements)

References 11 publications

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“…The calculation of the partial derivative ∂θ/∂ h has an interesting parallel to the partial derivatives of the potential scattering phase shift with respect to physical quantities such as linear and orbital angular momentum, and which has possible application in a variety of fields that range from electron scattering from ions, heavy‐ion scattering in nuclear physics and resonance scattering to scalar waves scattering in a gravitational field by rotating BHs. Much of the physical information contained in the scattering phase shifts δ( h , l ) can be extracted from their partial derivatives with respect to such physical quantities as linear momentum (Romo & Valluri 1998) and angular momentum, associated with time delay and the deflection function (Valluri & Romo 1989, 1994) and other potential parameters. The deflection function 2 ∂δ( l , k )/∂ l is of relevance for rainbow and glory scattering in semiclassical collisions in a variety of problems that include scattering in BH gravitational fields.…”

Section: Introductionmentioning

confidence: 99%

An extension of Newton's apsidal precession theorem

Valluri¹,

Yu²,

Smith³

et al. 2005

Monthly Notices of the Royal Astronomical Society

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Newton's apsidal precession theorem in Proposition 45 of Book I of the ‘Principia’ has great mathematical, physical, astronomical and historical interest. The lunar theory and the precession of the perihelion of the planet Mercury are but two examples of the applications of this theorem. We have examined the precession of orbits under varying force laws as measured by the apsidal angle θ(N, e), where N is the index for the centripetal force law, for varying eccentricity e. The paper derives a general function for the apsidal angle, dependent only on e and N as the potential is spherically symmetric. Further, we explore approximate ways of the solution of this equation, in the neighbourhood of N= 2 which happens to be the case of greatest historical interest. Exact solutions are derived where they are possible. The first derivatives ∂θ/∂N and ∂θ/∂h[where h(N, e) is the angular momentum] are analytically expressed in the neighbourhood of N= 2 (case of the inverse square law). The value of ∂θ/∂N is computed numerically as well for 1 ≤N < 3. The resulting integrals are interesting improper integrals with singularities at both limits. Some of the integrals, especially for N= 2, can be given in closed form in terms of generalized hypergeometric functions which are reducible in terms of algebraic and logarithmic functions. No evidence was found for isolated cases of zero precession as e was increased. The N= 1 case of the logarithmic potential is also briefly discussed in view of its interest for the dynamics of eccentric orbits and its relevance to realistic galaxy models. The possibility of apsidal precession was also examined for a few cases of high‐eccentricity asteroids and extrasolar planets. We find that these systems may provide interesting new laboratories for studies of gravity.

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Section: Introductionmentioning

confidence: 99%

An extension of Newton's apsidal precession theorem

Valluri¹,

Yu²,

Smith³

et al. 2005

Monthly Notices of the Royal Astronomical Society

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“…In two recent studies we have investigated the dependence of the scattering phaseshift S,(k) on various parameters appearing in the Hamiltonian such as the masses and charges of the projectile or the target (Romo and Valluri 1987), and their relative orbital angular momentum (Valluri and Romo 1989). In all cases it was possible to display the dependence of S,(k) on the parameter of interest by an explicit formula for the partial derivative of S,(k) with respect to that parameter.…”

Section: Introductionmentioning

confidence: 99%

A study of the Pais variational phaseshift approximation

Romo¹,

Valluri²

1990

J. Phys. B: At. Mol. Opt. Phys.

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The accuracy of phaseshifts calculated by means of the Pais variational procedure is investigated and the method is employed to calculate partial derivatives of the phase shifts with respect to a number of potential parameters. One of these, the partial derivative of the phase shift delta L with respect to the orbital angular momentum L, plays an essential role in the semiclassical approach to scattering. Another is the partial derivative delta delta L(k)/ delta E, where E is the energy, which is related to physical quantities like the time delay, virial coefficients and resonance widths. The relative simplicity of the numerical calculations involved in the Pais approach and the fact that it gives accurate results for the range of values of L that are important in semiclassical calculations makes the use of Pais phaseshifts an attractive possibility in such calculations. The formalism that results has possible application to both electron scattering from atoms and molecules and to heavy ion scattering in nuclear physics.

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“…To evaluate the integral on the right hand side of eq. ( 23) we make use of some results obtained in an earlier work [24]. If one applies eq.…”

Section: Coulomb Extended Pais Methodsmentioning

confidence: 99%

“…For such cases also, our method would be helpful. Much of the physical information contained in the phase shifts can be extracted from their partial derivatives with respect to such physical quantities as linear momentum [23] and angular momentum, associated with time delay and the deflection function, [24] [25] and other potential parameters [26]. The modified Pais formalism also enables one to calculate partial derivatives of the phase shift with respect to such variables with only a minor additional effort.…”

Section: Introductionmentioning

confidence: 99%

An Extension of the Pais Variational Phase Shift Approximation

Romo

Valluri

2005

Phys. Scr.

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The Pais variational phase shift approximation has been modified so that it can be applied to systems in which either long range interactions are present or potential resonances exist, or both. Numerical calculations arising from the modified Pais formalism are relatively simple and lead to accurate results even for the lowest partial waves. Short-range and Coulomb potentials as well as their combinations have been considered in the applications of the extended Pais method. The formalism also enables the calculation of partial derivatives of the phase shift with respect to physical quantities such as linear and orbital angular momentum with only minor additional effort. It has possible application to both electron scattering from ions, heavy-ion scattering in nuclear physics and resonance scattering. The generalized Coulomb potential (GCP) has supersymmetry and shape invariance in the r and θ dimensions, and the energy eigenvalues, the Pais variational phase shifts and corresponding wavefunctions can also be obtained by the method of supersymmetric (SUSYQM) quantum mechanics, and shape invariance. The partial derivatives of the Pais phase shifts are also relevant for orbit and glory scattering, and rainbows in collision theory and for the case of scalar waves scattering in a gravitational field by rotating black holes.

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A study of the angular momentum dependence of the phase shift for finite range and coulomb potentials

Cited by 8 publications

References 11 publications

An extension of Newton's apsidal precession theorem

An extension of Newton's apsidal precession theorem

A study of the Pais variational phaseshift approximation

An Extension of the Pais Variational Phase Shift Approximation

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