Construction of nuclear potentials from phase shifts at different energies and angular momenta

1988 ◽  
Vol 66 (7) ◽  
pp. 618-621 ◽  
Author(s):  
M. A. Hooshyar ◽  
M. Razavy
Keyword(s):  
Inverse Problems ◽  
Approximate Method ◽  
Continued Fraction ◽  
Phase Shifts ◽  
Nuclear Potential ◽  
Matrix Elements ◽  
Fixed Energy ◽  
Angular Momenta ◽  
Partial Waves

This paper is concerned with an approximate method of construction of a central nuclear potential when [Formula: see text]-matrix elements or phase shifts for different partial waves are given at different energies. This is done by a generalization of the continued-fraction technique that was formulated for solving inverse problems at fixed energy.

scholarly journals SEMI-ANALYTIC EQUATIONS TO THE COX–THOMPSON INVERSE SCATTERING METHOD AT FIXED ENERGY FOR SPECIAL CASES

2008 ◽  
Vol 22 (23) ◽  
pp. 2191-2199 ◽  
Author(s):  
TAMÁS PÁLMAI ◽  
MIKLÓS HORVÁTH ◽  
BARNABÁS APAGYI
Keyword(s):  
Inverse Scattering ◽  
Scattering Problem ◽  
Identical Particle ◽  
Inverse Scattering Problem ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Fixed Energy ◽  
Angular Momenta ◽  
Partial Waves ◽  
Special Cases

Solution of the Cox–Thompson inverse scattering problem at fixed energy1–3 is reformulated resulting in semi-analytic equations. The new set of equations for the normalization constants and the nonphysical (shifted) angular momenta are free of matrix inversion operations. This simplification is a result of treating only the input phase shifts of partial waves of a given parity. Therefore, the proposed method can be applied for identical particle scattering of the bosonic type (or for certain cases of identical fermionic scattering). The new formulae are expected to be numerically more efficient than the previous ones. Based on the semi-analytic equations an approximate method is proposed for the generic inverse scattering problem, when partial waves of arbitrary parity are considered.

scholarly journals The Distribution of Phase Shifts for Semiclassical Potentials with Polynomial Decay

2018 ◽  
Vol 2020 (19) ◽  
pp. 6294-6346
Author(s):  
Jesse Gell-Redman ◽  
Andrew Hassell
Keyword(s):  
Asymptotic Formula ◽  
Unit Circle ◽  
Hamiltonian Dynamics ◽  
Semiclassical Limit ◽  
Scattering Matrix ◽  
Phase Shifts ◽  
Polynomial Decay ◽  
Homogeneous Distribution ◽  
Fixed Energy ◽  
Atomic Measure

Abstract This is the 3rd paper in a series [6, 9] analyzing the asymptotic distribution of the phase shifts in the semiclassical limit. We analyze the distribution of phase shifts, or equivalently, eigenvalues of the scattering matrix $S_h$, at some fixed energy $E$, for semiclassical Schrödinger operators on $\mathbb{R}^d$ that are perturbations of the free Hamiltonian $h^2 \Delta $ on $L^2(\mathbb{R}^d)$ by a potential $V$ with polynomial decay. Our assumption is that $V(x) \sim |x|^{-\alpha } v(\hat x)$ as $x \to \infty $, $\hat x = x/|x|$, for some $\alpha> d$, with corresponding derivative estimates. In the semiclassical limit $h \to 0$, we show that the atomic measure on the unit circle defined by these eigenvalues, after suitable scaling in $h$, tends to a measure $\mu $ on $\mathbb{S}^1$. Moreover, $\mu $ is the pushforward from $\mathbb{R}$ to $\mathbb{R} / 2 \pi \mathbb{Z} = \mathbb{S}^1$ of a homogeneous distribution. As a corollary we obtain an asymptotic formula for the accumulation of phase shifts in a sector of $\mathbb{S}^1$. The proof relies on an extension of results in [14] on the classical Hamiltonian dynamics and semiclassical Poisson operator to the larger class of potentials under consideration here.

scholarly journals TIME IN COSMOLOGY

1999 ◽  
Vol 08 (01) ◽  
pp. 1-22 ◽  
Author(s):  
R. BROUT ◽  
R. PARENTANI
Keyword(s):  
Total Energy ◽  
Scale Factor ◽  
Matrix Elements ◽  
Temporal Dependence ◽  
Radiative Processes ◽  
Fixed Energy ◽  
Transition Matrix Elements ◽  
In Virtue Of ◽  
The Universe ◽  
Reparametrization Invariance

The notion of time in cosmology is revealed through an examination of transition matrix elements of radiative processes occurring in the cosmos. To begin with, the very concept of time is delineated in classical physics in terms of correlations between the succession of configurations which describe a process and a standard trajectory called the clock. The total is an isolated system of fixed energy. This is relevant for cosmology in that the universe is an isolated system which we take to be homogeneous and isotropic. Furthermore, in virtue of the constraint which arises from reparametrization invariance of time, it has zero total energy. Therefore the momentum of the scale factor is determined from the energy of matter. In the quantum theory this is exploited through the use of WKB approximation for the wave function of the scale factor, justified for a large universe. The formalism then gives rise to matrix elements describing matter processes. These are shown to take on the form of usual time dependent quantum amplitudes wherein the temporal dependence is given by a background which is once more fixed by the total energy of matter.

NOTES ON THE DISTRIBUTION OF PHASE SHIFTS

2008 ◽  
Vol 22 (23) ◽  
pp. 2163-2175 ◽  
Author(s):  
MIKLÓS HORVÁTH
Keyword(s):  
Inverse Scattering ◽  
Lower Bounds ◽  
Three Dimensional ◽  
Variational Formula ◽  
Phase Shifts ◽  
Upper And Lower Bounds ◽  
One Dimensional ◽  
Fixed Energy ◽  
Symmetrical Potential ◽  
The One

We consider three-dimensional inverse scattering with fixed energy for which the spherically symmetrical potential is nonvanishing only in a ball. We give exact upper and lower bounds for the phase shifts. We provide a variational formula for the Weyl–Titchmarsh m-function of the one-dimensional Schrödinger operator defined on the half-line.

The vector-ladder operators for the rotation group O3

1993 ◽  
Vol 71 (3-4) ◽  
pp. 152-154
Author(s):  
J. E. Hardy
Keyword(s):  
Rotation Group ◽  
Matrix Elements ◽  
Ladder Operators ◽  
Straightforward Calculation ◽  
Angular Momenta ◽  
State Formula

The two vector-ladder operators, which step from any state [Formula: see text] in an irreducible multiplet in the space of product states of two commuting angular momenta, are defined and all their nonvanishing matrix elements are given, facilitating direct, straightforward calculation of the six nearby nonsibling states [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text].

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