Non-Relativistic Dispersion Relations for a Two Channel Scattering Process

1962 ◽  
Vol 58 (2) ◽  
pp. 363-376 ◽  
Author(s):  
J. Underhill
Keyword(s):  
Fourier Transform ◽  
Dispersion Relation ◽  
Scattering Amplitude ◽  
Bound State ◽  
Potential Scattering ◽  
Scattering Process ◽  
Relativistic Dispersion ◽  
The Fourier Transform ◽  
Image Position ◽  
Relativistic Potential

In (l) Khuri has proved the validity of a dispersion relation for non-relativistic potential scattering. More precisely, he has shown that if the potential V(r) is central and satisfies: then the scattering amplitude f(k, τ) = M(E, τ) (where E = k2 is the energy) satisfies the following dispersion relation for fixed momentum transfer τ ≤ 2α: In (2), Rj(τ) is the (real) residue of the scattering amplitude at the bound state Ej and is the Fourier transform of the potential .

scholarly journals On dilation equations and the Hölder continuity of the de Rham functions

1994 ◽  
Vol 36 (3) ◽  
pp. 309-311
Author(s):  
Yibiao Pan
Keyword(s):  
Fourier Transform ◽  
Approximation Method ◽  
Hölder Continuity ◽  
Simple Approximation ◽  
Holder Continuity ◽  
Integrable Solution ◽  
De Rham ◽  
The Fourier Transform ◽  
Image Position ◽  
Dilation Equations

AbstractWe use a simple approximation method to prove the Holder continuity of the generalized de Rham functions.1. Consider the following dilatation equationwhere |α|<l/2. Suppose that f is an integrable solution of (1); then f must satisfywhere is the Fourier transform of f, andwhich immediately leads to

Fourier restriction theorems for curves with affine and Euclidean arclengths

1985 ◽  
Vol 97 (1) ◽  
pp. 111-125 ◽  
Author(s):  
S. W. Drury ◽  
B. P. Marshall
Keyword(s):  
Fourier Transform ◽  
Smooth Manifold ◽  
Fourier Restriction ◽  
Survey Article ◽  
Restriction Theorems ◽  
Schwartz Class ◽  
The Fourier Transform ◽  
Image Position

Let M be a smooth manifold in . One may ask whether , the restriction of the Fourier transform of f to M makes sense for every f in . Since, for does not make sense pointwise, it is natural to introduce a measure σ on M and ask for an inequalityfor every f in (say) the Schwartz class. Results of this kind are called restriction theorems. An excellent survey article on this subject is to be found in Tomas[13].

The a-Local Operator Problem

1959 ◽  
Vol 11 ◽  
pp. 583-592 ◽  
Author(s):  
Louis de Branges
Keyword(s):  
Fourier Transform ◽  
Total Variation ◽  
Convergence Theorem ◽  
Borel Measure ◽  
Local Operator ◽  
Operator Problem ◽  
Dominated Convergence ◽  
The Fourier Transform ◽  
Image Position ◽  
The Right

Let be the Fourier transform of a Borel measure of finite total variation. The formulacan be justified if the integral on the right converges absolutely. ForwhereNow let h → 0 in both sides of this equation and use the Lebesgue dominated convergence theorem.

A note on the Fourier transform of generalized functions

1971 ◽  
Vol 70 (1) ◽  
pp. 49-51
Author(s):  
B. Fisher
Keyword(s):  
Fourier Transform ◽  
Generalized Functions ◽  
Generalized Function ◽  
Convolution Product ◽  
Counter Example ◽  
The Fourier Transform ◽  
Image Position

If F(f) denotes the Fourier transform of a generalized function f and f * g denotes the convolution product of two generalized functions f and g then it is known that under certain conditionsJones (2) states that this is not true in general and gives as a counter-example the case when f = g = H, H denoting Heaviside's function. In this caseand the product (x−1 – iπδ)2 is not defined in his development of the product of generalized functions.

scholarly journals On fourier transforms of radial functions

1965 ◽  
Vol 5 (3) ◽  
pp. 289-298 ◽  
Author(s):  
James L. Griffith
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Radial Functions ◽  
Cartesian Space ◽  
The Fourier Transform ◽  
Image Position

The Fourier transformF(y) of a functionf(t) inL1(Ek) whereEkis thek-dimensional cartesian space will be defined by.

Fourier Transforms of Unbounded Measures

1979 ◽  
Vol 31 (6) ◽  
pp. 1281-1292 ◽  
Author(s):  
James Stewart
Keyword(s):  
Fourier Transform ◽  
Real Line ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Stieltjes Transform ◽  
The Real ◽  
Bounded Complex ◽  
The Fourier Transform ◽  
Image Position

1. Introduction. One of the basic objects of study in harmonic analysis is the Fourier transform (or Fourier-Stieltjes transform) μ of a bounded (complex) measure μ on the real line R:(1.1)More generally, if μ is a bounded measure on a locally compact abelian group G, then its Fourier transform is the function(1.2)where Ĝ is the dual group of G and One answer to the question “Which functions can be represented as Fourier transforms of bounded measures?” was given by the following criterion due to Schoenberg [11] for the real line and Eberlein [5] in general: f is a Fourier transform of a bounded measure if and only if there is a constant M such that(1.3)for all ϕ ∈ L1(G) where

Comment on “Fourier transform of hydrogen-type atomic orbitals”

2019 ◽  
Vol 97 (12) ◽  
pp. 1349-1360 ◽  
Author(s):  
Ernst Joachim Weniger
Keyword(s):  
Fourier Transform ◽  
Bessel Functions ◽  
Fourier Transforms ◽  
Laguerre Polynomials ◽  
Bound State ◽  
Atomic Orbitals ◽  
Generalized Laguerre Polynomials ◽  
Classic Result ◽  
The Fourier Transform ◽  
Finite Sum

Podolsky and Pauling (Phys. Rev. 34, 109 (1929) doi: 10.1103/PhysRev.34.109 ) were the first ones to derive an explicit expression for the Fourier transform of a bound-state hydrogen eigenfunction. Yükçü and Yükçü (Can. J. Phys. 96, 724 (2018) doi: 10.1139/cjp-2017-0728 ), who were apparently unaware of the work of Podolsky and Pauling or of the numerous other earlier references on this Fourier transform, proceeded differently. They expressed a generalized Laguerre polynomial as a finite sum of powers, or equivalently, they expressed a bound-state hydrogen eigenfunction as a finite sum of Slater-type functions. This approach looks very simple, but it leads to comparatively complicated expressions that cannot match the simplicity of the classic result obtained by Podolsky and Pauling. It is, however, possible to reproduce not only Podolsky and Pauling’s formula for the bound-state hydrogen eigenfunction, but to obtain results of similar quality also for the Fourier transforms of other, closely related, functions, such as Sturmians, Lambda functions, or Guseinov’s functions, by expanding generalized Laguerre polynomials in terms of so-called reduced Bessel functions.

The Fourier Transforms of Smooth Measures on Hypersurfaces of Rn + 1

1986 ◽  
Vol 38 (2) ◽  
pp. 328-359 ◽  
Author(s):  
Bernard Marshall
Keyword(s):  
Fourier Transform ◽  
Asymptotic Expansion ◽  
Unit Sphere ◽  
Gaussian Curvature ◽  
Fourier Transforms ◽  
Surface Measure ◽  
Positive Gaussian Curvature ◽  
Smooth Measures ◽  
The Fourier Transform ◽  
Image Position

The Fourier transform of the surface measure on the unit sphere in Rn + 1, as is well-known, equals the Bessel functionIts behaviour at infinity is described by an asymptotic expansionThe purpose of this paper is to obtain such an expression for surfaces Σ other than the unit sphere. If the surface Σ is a sufficiently smooth compact n-surface in Rn + 1 with strictly positive Gaussian curvature everywhere then with only minor changes in the main term, such an asymptotic expansion exists. This result was proved by E. Hlawka in [3]. A similar result concerned with the minimal smoothness of Σ was later obtained by C. Herz [2].

scholarly journals Electron and Positron Scattering by Non-Central Potentials: Matrix Elements and Symmetry Properties in the First Born Approximation

2021 ◽  
Author(s):  
Marcos Barp ◽  
Felipe Arretche
Keyword(s):  
Fourier Transform ◽  
Born Approximation ◽  
Scattering Amplitude ◽  
Point Group ◽  
Computational Effort ◽  
Positron Scattering ◽  
Symmetry Properties ◽  
Linear Molecules ◽  
The Fourier Transform ◽  
Symmetry Relations

The Fourier transform of Cartesian Gaussian functions product is presented in the light of positron scattering. The calculation of this class of integrals is crucial in order to obtain the scattering amplitude in the first Born approximation framework for an ab initio method recently proposed. A general solution to the scattering amplitude is given to a molecular target with no restriction due to symmetry. Moreover, symmetry relations are presented with the purpose of identifying terms that do not contribute to the calculation for the molecules in the D∞h point group optimizing the computational effort. Keywords — Positron and electron scattering, Fourier transform of the Gaussian product theorem, McMurchie-Davidson procedure, Obara-Saika procedure, linear molecules .

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