Showing papers on "Fourier transform published in 2009"

Classical Fourier Analysis

Abstract: Preface.- 1. Lp Spaces and Interpolation.- 2. Maximal Functions, Fourier Transform, and Distributions.- 3. Fourier Series.- 4. Topics on Fourier Series.- 5. Singular Integrals of Convolution Type.- 6. Littlewood-Paley Theory and Multipliers.- 7. Weighted Inequalities.- A. Gamma and Beta Functions.- B. Bessel Functions.- C. Rademacher Functions.- D. Spherical Coordinates.- E. Some Trigonometric Identities and Inequalities.- F. Summation by Parts.- G. Basic Functional Analysis.- H. The Minimax Lemma.- I. Taylor's and Mean Value Theorem in Several Variables.- J. The Whitney Decomposition of Open Sets in Rn.- Glossary.- References.- Index.

Spectral analysis of nonlinear flows

Abstract: We present a technique for describing the global behaviour of complex nonlinear flows by decomposing the flow into modes determined from spectral analysis of the Koopman operator, an infinite-dimensional linear operator associated with the full nonlinear system. These modes, referred to as Koopman modes, are associated with a particular observable, and may be determined directly from data (either numerical or experimental) using a variant of a standard Arnoldi method. They have an associated temporal frequency and growth rate and may be viewed as a nonlinear generalization of global eigenmodes of a linearized system. They provide an alternative to proper orthogonal decomposition, and in the case of periodic data the Koopman modes reduce to a discrete temporal Fourier transform. The Arnoldi method used for computations is identical to the dynamic mode decomposition recently proposed by Schmid & Sesterhenn (Sixty-First Annual Meeting of the APS Division of Fluid Dynamics, 2008), so dynamic mode decomposition can be thought of as an algorithm for finding Koopman modes. We illustrate the method on an example of a jet in crossflow, and show that the method captures the dominant frequencies and elucidates the associated spatial structures.

Linear and nonlinear waves

Abstract: The study of waves can be traced back to antiquity where philosophers, such as Pythagoras (c.560-480 BC), studied the relation of pitch and length of string in musical instruments. However, it was not until the work of Giovani Benedetti (1530-90), Isaac Beeckman (1588-1637) and Galileo (1564-1642) that the relationship between pitch and frequency was discovered. This started the science of acoustics, a term coined by Joseph Sauveur (1653-1716) who showed that strings can vibrate simultaneously at a fundamental frequency and at integral multiples that he called harmonics. Isaac Newton (1642-1727) was the first to calculate the speed of sound in his Principia. However, he assumed isothermal conditions so his value was too low compared with measured values. This discrepancy was resolved by Laplace (1749-1827) when he included adiabatic heating and cooling effects. The first analytical solution for a vibrating string was given by Brook Taylor (1685-1731). After this, advances were made by Daniel Bernoulli (1700-82), Leonard Euler (1707-83) and Jean d’Alembert (1717-83) who found the first solution to the linear wave equation, see section (3.2). Whilst others had shown that a wave can be represented as a sum of simple harmonic oscillations, it was Joseph Fourier (1768-1830) who conjectured that arbitrary functions can be represented by the superposition of an infinite sum of sines and cosines now known as the Fourier series. However, whilst his conjecture was controversial and not widely accepted at the time, Dirichlet subsequently provided a proof, in 1828, that all functions satisfying Dirichlet’s conditions (i.e. non-pathological piecewise continuous) could be represented by a convergent Fourier series. Finally, the subject of classical acoustics was laid down and presented as a coherent whole by John William Strutt (Lord Rayleigh, 1832-1901) in his treatise Theory of Sound. The science of modern acoustics has now moved into such diverse areas as sonar, auditoria, electronic amplifiers, etc.

Spectral analysis of nonlinear flows

Abstract: We present a technique for describing the global behaviour of complex nonlinear flows by decomposing the flow into modes determined from spectral analysis of the Koopman operator, an infinite-dimensional linear operator associated with the full nonlinear system. These modes, referred to as Koopman modes, are associated with a particular observable, and may be determined directly from data (either numerical or experimental) using a variant of a standard Arnoldi method. They have an associated temporal frequency and growth rate and may be viewed as a nonlinear generalization of global eigenmodes of a linearized system. They provide an alternative to proper orthogonal decomposition, and in the case of periodic data the Koopman modes reduce to a discrete temporal Fourier transform. The Arnoldi method used for computations is identical to the dynamic mode decomposition recently proposed by Schmid & Sesterhenn (Sixty-First Annual Meeting of the APS Division of Fluid Dynamics, 2008), so dynamic mode decomposition can be thought of as an algorithm for finding Koopman modes. We illustrate the method on an example of a jet in crossflow, and show that the method captures the dominant frequencies and elucidates the associated spatial structures.

Modern Fourier analysis

Abstract: Preface.- Smoothness and Function Spaces.- BMO and Carleson Measures.- Singular Integrals of Nonconvolution Type.- Weighted Inequalities.- Boundedness and Convergence of Fourier Integrals.- Time-Frequency Analysis and the Carleson-Hunt Theorem.- Multilinear Harmonic Analysis.- Glossary.- References.- Index.

Introduction to Nonlinear Dispersive Equations

Abstract: 1. The Fourier Transform.- 2. Interpolation of Operators.- 3. Sobolev Spaces and Pseudo-Differential Operators.- 4. The Linear Schrodinger Equation.- 5. The Non-Linear Schrodinger Equation.- 6. Asymptotic Behavior for NLS Equation.- 7. Korteweg-de Vries Equation.- 8. Asymptotic Behavior for k-gKdV Equations.- 9. Other Nonlinear Dispersive Models.- 10. General Quasilinear Schrodinger Equation.- Proof of Theorem 2.8.- Proof of Lemma 4.2.- References.- Index.

Compressive Sensing by Random Convolution

Abstract: This paper demonstrates that convolution with random waveform followed by random time-domain subsampling is a universally efficient compressive sensing strategy. We show that an $n$-dimensional signal which is $S$-sparse in any fixed orthonormal representation can be recovered from $m\gtrsim S\log n$ samples from its convolution with a pulse whose Fourier transform has unit magnitude and random phase at all frequencies. The time-domain subsampling can be done in one of two ways: in the first, we simply observe $m$ samples of the random convolution; in the second, we break the random convolution into $m$ blocks and summarize each with a single randomized sum. We also discuss several imaging applications where convolution with a random pulse allows us to superresolve fine-scale features, allowing us to recover high-resolution signals from low-resolution measurements.

Fourier transform spectroscopy with a laser frequency comb

Abstract: Molecular fingerprinting using absorption spectroscopy is a powerful analytical method, particularly in the infrared, the region of intense spectral signatures Fourier transform spectroscopy—the widely used and essential tool for broadband spectroscopy—enables the recording of multi-octave-spanning spectra, exhibiting 100 MHz resolution with an accuracy of 1 × 10−9 and 1 × 10−2 in wavenumber and intensity determination, respectively Typically, 1 × 106 independent spectral elements may be measured simultaneously within a few hours, with only average sensitivity Here, we show that by using laser frequency combs as the light source of Fourier transform spectroscopy it is possible to record well-resolved broadband absorption and dispersion spectra in a single experiment, from the beating signatures of neighbouring comb lines in the interferogram The sensitivity is thus expected to increase by several orders of magnitude Experimental proof of principle is here carried out on the 15-µm overtone bands of acetylene, spanning 80 nm with a resolution of 15 GHz Consequently, without any optical modification, the performance of Fourier spectrometers may be drastically boosted By using an optical frequency comb as the light source for Fourier transform spectroscopy, scientists show that well-resolved broadband absorption and dispersion spectra can be recorded in a single experiment, providing sensitive detection of multiple molecular species over a broad spectral window

Five-dimensional interpolation: Recovering from acquisition constraints

Abstract: Although 3D seismic data are being acquired in larger volumes than ever before, the spatial sampling of these volumes is not always adequate for certain seismic processes. This is especially true of marine and land wide-azimuth acquisitions, leading to the development of multidimensional data interpolation techniques. Simultaneous interpolation in all five seismic data dimensions (inline, crossline, offset, azimuth, and frequency) has great utility in predicting missing data with correct amplitude and phase variations. Although there are many techniques that can be implemented in five dimensions, this study focused on sparse Fourier reconstruction. The success of Fourier interpolation methods depends largely on two factors: (1) having efficient Fourier transform operators that permit the use of large multidimensional data windows and (2) constraining the spatial spectrum along dimensions where seismic amplitudes change slowly so that the sparseness and band limitation assumptions remain valid. Fourier reconstruction can be performed when enforcing a sparseness constraint on the 4D spatial spectrum obtained from frequency slices of five-dimensional windows. Binning spatial positions into a fine 4D grid facilitates the use of the FFT, which helps on the convergence of the inversion algorithm. This improves the results and computational efficiency. The 5D interpolation can successfully interpolate sparse data, improve AVO analysis, and reduce migration artifacts. Target geometries for optimal interpolation and regularization of land data can be classified in terms of whether they preserve the original data and whether they are designed to achieve surface or subsurface consistency.

3D Images of Materials Structures: Processing and Analysis

Abstract: PERFACE INTRODUCTION PRELIMINARIES General Notation Characteristics of Sets Random Sets Fourier Analysis LATTICES, ADJACENCY OF LATTICE POINTS, AND IMAGES Introduction Point Lattices, Digitizations and Pixel Configurations Adjacency and Euler Number The Euler Number of Microstructure Constituents Image Data Rendering IMAGE PROCESSING Fourier Transform of an Image Filtering Segmentation MEASUREMENT OF INTRINSIC VOLUMES AND RELATED QUANTITIES Introduction Intrinsic Volumes Intrinsic Volume Densities Directional Analysis Distances Between Random Sets and Distance Distributions SPECTRAL ANALYSIS Introduction Second-Order Characteristics of a Random Volume Measure Correlations Between Random Structures Second-Order Characteristics of Random Surfaces Second-Order Characteristics of Random Point Fields MODEL-BASED IMAGE ALANYSIS Introduction,Motivation Point Field Models MacroscopicallyHomogeneous Systems of Non-overlapping Particles Macroscopically Homogeneous Systems of Overlapping Particles Macroscopically Homogeneous Fibre Systems Tessellations SIMULATION OF MATERIAL PROPERTIES Introduction Effective Conductivity of Polycrystals by StochasticHomogenization Computation of Effective Elastic Moduli of Porous Media by FEM Simulation REFERENCES INDEX

Broadband CARS spectral phase retrieval using a time-domain Kramers-Kronig transform.

Abstract: We describe a closed-form approach for performing a Kramers–Kronig (KK) transform that can be used to rapidly and reliably retrieve the phase, and thus the resonant imaginary component, from a broadband coherent anti-Stokes Raman scattering (CARS) spectrum with a nonflat background. In this approach we transform the frequency-domain data to the time domain, perform an operation that ensures a causality criterion is met, then transform back to the frequency domain. The fact that this method handles causality in the time domain allows us to conveniently account for spectrally varying nonresonant background from CARS as a response function with a finite rise time. A phase error accompanies KK transform of data with finite frequency range. In examples shown here, that phase error leads to small (<1%) errors in the retrieved resonant spectra.

High-resolution, wide-field object reconstruction with synthetic aperture Fourier holographic optical microscopy

Abstract: We utilize synthetic-aperture Fourier holographic microscopy to resolve micrometer-scale microstructure over millimeter-scale fields of view. Multiple holograms are recorded, each registering a different, limited region of the sample object's Fourier spectrum. They are "stitched together" to generate the synthetic aperture. A low-numerical-aperture (NA) objective lens provides the wide field of view, and the additional advantages of a long working distance, no immersion fluids, and an inexpensive, simple optical system. Following the first theoretical treatment of the technique, we present images of a microchip target derived from an annular synthetic aperture (NA = 0.61) whose area is 15 times that due to a single hologram (NA = 0.13); they exhibit a corresponding qualitative improvement. We demonstrate that a high-quality reconstruction may be obtained from a limited sub-region of Fourier space, if the object's structural information is concentrated there.

A versatile ultrastable platform for optical multidimensional Fourier-transform spectroscopy

Abstract: The JILA multidimensional optical nonlinear spectrometer (JILA-MONSTR) is a robust, ultrastable platform consisting of nested and folded Michelson interferometers that can be actively phase stabilized. This platform generates a square of identical laser pulses that can be adjusted to have arbitrary time delay between them while maintaining phase stability. The JILA-MONSTR provides output pulses for nonlinear excitation of materials and phase-stabilized reference pulses for heterodyne detection of the induced signal. This arrangement is ideal for performing coherent optical experiments, such as multidimensional Fourier-transform spectroscopy, which records the phase of the nonlinear signal as a function of the time delay between several of the excitation pulses. The resulting multidimensional spectrum is obtained from a Fourier transform. This spectrum can resolve, separate, and isolate coherent contributions to the light-matter interactions associated with electronic excitation at optical frequencies. To show the versatility of the JILA-MONSTR, several demonstrations of two-dimensional Fourier-transform spectroscopy are presented, including an example of a phase-cycling scheme that reduces noise. Also shown is a spectrum that accesses two-quantum coherences, where all excitation pulses require phase locking for detection of the signal.

Direction-of-Arrival Estimation for Nonuniform Sensor Arrays: From Manifold Separation to Fourier Domain MUSIC Methods

Abstract: In this paper, the problem of spectral search-free direction-of-arrival (DOA) estimation in arbitrary nonuniform sensor arrays is addressed. In the first part of the paper, we present a finite-sample performance analysis of the well-known manifold separation (MS) based root-MUSIC technique. Then, we propose a new class of search-free DOA estimation methods applicable to arrays of arbitrary geometry and establish their relationship to the MS approach. Our first technique is referred to as Fourier-domain (FD) root-MUSIC and is based on the fact that the spectral MUSIC function is periodic in angle. It uses the Fourier series to expand this function and reformulate the underlying DOA estimation problem as an equivalent polynomial rooting problem. Our second approach applies the zero-padded inverse Fourier transform to the FD root-MUSIC polynomial to avoid the polynomial rooting step and replace it with a simple line search. Our third technique refines the FD root-MUSIC approach by using weighted least-squares approximation to compute the polynomial coefficients. The proposed techniques are shown to offer substantially improved performance-to-complexity tradeoffs as compared to the MS technique.

The Pseudo-analytical Method: Application of Pseudo-Laplacians to Acoustic And Acoustic Anisotropic Wave Propagation

Abstract: Summary We generalize the pseudo-spectral method for the acoustic wave equation to create analytical solutions to the constant velocity acoustic wave equation in an arbitrary number of space dimensions. We accomplish this by modifying the Fourier Transform of the Laplacian operator so that it compensates exactly for the error due to the second-order finite-difference time marching scheme used in the conventional pseudo-spectral method. Of more practical interest, we show that this modified or pseudo-Laplacian is a smoothly varying function of the parameters of the acoustic wave equation (velocity most importantly) and thus can be further generalized to create near-analyticallyaccurate solutions for the variable velocity case. We call this new method the pseudo-analytical method. We further show that by applying this approach to the concept of acoustic anisotropic wave propagation, we can create scalar-mode VTI and TTI wave equations that overcome the disadvantages of previously published methods for acoustic anisotropic wave propagation. These methods should be ideal for forward modeling and reverse time migration applications.

Making sense of the Legendre transform

Abstract: The Legendre transform is a powerful tool in theoretical physics and plays an important role in classical mechanics, statistical mechanics, and thermodynamics. In typical undergraduate and graduate courses the motivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by Fourier transforms. We review and modify the presentation of Legendre transforms in a way that explicates the formal mathematics, resulting in manifestly symmetric equations, thereby clarifying the structure of the transform. We then discuss examples to motivate the transform as a way of choosing independent variables that are more easily controlled. We demonstrate how the Legendre transform arises naturally from statistical mechanics and show how the use of dimensionless thermodynamic potentials leads to more natural and symmetric relations.

Fast Fourier transform telescope

Abstract: We propose an all-digital telescope for 21 cm tomography, which combines key advantages of both single dishes and interferometers. The electric field is digitized by antennas on a rectangular grid, after which a series of fast Fourier transforms recovers simultaneous multifrequency images of up to half the sky. Thanks to Moore's law, the bandwidth up to which this is feasible has now reached about 1 GHz, and will likely continue doubling every couple of years. The main advantages over a single dish telescope are cost and orders of magnitude larger field-of-view, translating into dramatically better sensitivity for large-area surveys. The key advantages over traditional interferometers are cost (the correlator computational cost for an N-element array scales as Nlog{sub 2}N rather than N{sup 2}) and a compact synthesized beam. We argue that 21 cm tomography could be an ideal first application of a very large fast Fourier transform telescope, which would provide both massive sensitivity improvements per dollar and mitigate the off-beam point source foreground problem with its clean beam. Another potentially interesting application is cosmic microwave background polarization.

The robustness of seismic attenuation measurements using fixed- and variable-window time-frequency transforms

Abstract: Frequency-based methods for measuring seismic attenuation are used commonly in exploration geophysics. To measure the spectrum of a nonstationary seismic signal, different methods are available, including transforms with time windows that are either fixed or systematically varying with the frequency being analyzed. We compare four time-frequency transforms and show that the choice of a fixed- or variable-window transform affects the robustness and accuracy of the resulting attenuation measurements. For fixed-window transforms, we use the short-time Fourier transform and Gabor transform. The S-transform and continuous wavelet transform are analyzed as the variable-length transforms. First we conduct a synthetic transmission experiment, and compare the frequency-dependent scattering attenuation to the theoretically predicted values. From this procedure, we find that variable-window transforms reduce the uncertainty and biasof the resulting attenuation estimate, specifically at the upper and lower ends of th...

Fourier Transform-Based Modified Phasor Estimation Method Immune to the Effect of the DC Offsets

Abstract: This paper proposes a Fourier transform-based modified phasor estimation method to eliminate the adverse influence of the exponentially decaying dc offsets when discrete Fourier transform (DFT) is used to calculate the phasor of the fundamental frequency component in a relaying signal. By subtracting the result of odd-sample-set DFT from the result of even-sample-set DFT, the information of dc offsets can be obtained. Two dc offsets in a secondary relaying signal are treated as one dc offset which is piecewise approximated in one cycle data window. The effect of the dc offsets can be eliminated by the approximated dc offset. The performance of the proposed algorithm is evaluated by using computer-simulated signals and Electromagnetic Transients Program-generated signals. The algorithm is also tested on a hardware board with TMS320C32 microprocessor. The evaluation results indicate that the proposed algorithm can estimate the accurate phasor of the fundamental frequency component regardless of not only the primary decaying dc offset but also the secondary decaying dc offset caused by CT circuit itself including its burden.

Spatial light interference microscopy and fourier transform light scattering for cell and tissue characterization

Abstract: Methods and apparatus for rendering quantitative phase maps across and through transparent samples. A broadband source is employed in conjunction with an objective, Fourier optics, and a programmable two-dimensional phase modulator to obtain amplitude and phase information in an image plane. Methods, referred to as Fourier transform light scattering (FTLS), measure the angular scattering spectrum of the sample. FTLS combines optical microscopy and light scattering for studying inhomogeneous and dynamic media. FTLS relies on quantifying the optical phase and amplitude associated with a coherent image field and propagating it numerically to the scattering plane. Full angular information, limited only by the microscope objective, is obtained from extremely weak scatterers, such as a single micron-sized particle. A flow cytometer may employ FTLS sorting.

Accurate estimators of power spectra in N‐body simulations

Abstract: A method to rapidly estimate the Fourier power spectrum of a point distribution is presented. This method relies on a Taylor expansion of the trigonometric functions. It yields the Fourier modes from a number of fast Fourier transforms (FFTs), which is controlled by the order N of the expansion and by the dimension D of the system. In three dimensions, for the practical value N= 3, the number of FFTs required is 20. We apply the method to the measurement of the power spectrum of a periodic point distribution that is a local Poisson realization of an underlying stationary field. We derive an explicit analytic expression for the spectrum, which allows us to quantify – and correct for – the biases induced by discreteness and by the truncation of the Taylor expansion, and to bound the unknown effects of aliasing of the power spectrum. We show that these aliasing effects decrease rapidly with the order N. For N= 3, they are expected to be, respectively, smaller than ∼10−4 and 0.02 at half the Nyquist frequency and at the Nyquist frequency of the grid used to perform the FFTs. The only remaining significant source of errors is reduced to the unavoidable cosmic/sample variance due to the finite size of the sample. The analytical calculations are successfully checked against a cosmological N-body experiment. We also consider the initial conditions of this simulation, which correspond to a perturbed grid. This allows us to test a case where the local Poisson assumption is incorrect. Even in that extreme situation, the third-order Fourier–Taylor estimator behaves well, with aliasing effects restrained to at most the per cent level at half the Nyquist frequency. We also show how to reach arbitrarily large dynamic range in Fourier space (i.e. high wavenumber), while keeping statistical errors in control, by appropriately ‘folding’ the particle distribution.

A Versatile Ultra-Stable Platform for Optical Multidimensional Fourier-Transform Spectroscopy

Abstract: The JILA Multidimensional Optical Nonlinear SpecTRometer (JILA-MONSTR) is a robust, ultra-stable platform consisting nested and folded Michelson interferometers that can be actively phase stabilized. This platform generates a square of identical laser pulses that can be adjusted to have arbitrary time delay between them, while maintaining phase stability. The JILA-MONSTR provides output pulses for nonlinear excitation of materials and phase-stabilized reference pulses for heterodyne detection of the induced signal. This arrangement is ideal for performing coherent optical experiments, such as multidimensional Fourier-transform spectroscopy, which records the phase of the nonlinear signal as a function of the time delay between several of the excitation pulses. The resulting multidimensional spectrum is obtained from a Fourier transform. This spectrum can resolve, separate and isolate coherent contributions to the light-matter interactions associated with electronic excitation at optical frequencies. To show the versatility of the JILA-MONSTR, several demonstrations of two-dimensional Fourier-transform spectroscopy are presented, including an example of a phase-cycling scheme that reduces noise.

Fourier transform measurements of SO2 absorption cross sections: II.: Temperature dependence in the 29 000–44 000 cm−1 (227–345 nm) region

Abstract: This paper is the second of a series that reports results on the measurements of the absorption cross section of SO2 in the UV/visible region at high resolution and that investigates high temperatures in support to planetary applications. Absorption cross sections of SO2 have been obtained in the 29 000–44 000 cm−1 spectral range (227–345 nm) with a Fourier transform spectrometer at a resolution of 2 cm−1 (0.4500 cm MOPD and boxcar apodisation). Pure SO2 samples were used and measurements were performed at room temperature (298 K) as well as at 318, 338 and 358 K. Temperature effects in this spectral region are investigated and are favorably compared to existing studies in the literature. Comparison of the absorption cross section at room temperature shows good agreement in intensity with most of the literature data, but shows that most of the latter suffer from inaccurate wavelength scale definition. Moreover, literature data are often given only on restricted spectral intervals. Combined with the data described in the first part of this series of papers on SO2, this new data set offers the considerable advantage of covering the large spectral interval extending from 24 000 to 44 000 cm−1 (227–420 nm), at the four temperatures investigated.

A Convolution and Product Theorem for the Linear Canonical Transform

Abstract: The linear canonical transform (LCT) plays an important role in many fields of optics and signal processing. Many properties for this transform are already known, however, the convolution theorems don't have the elegance and simplicity comparable to that of the Fourier transform (FT), which states that the Fourier transform of the convolution of two functions is the product of their Fourier transforms. The purpose of this letter is to introduce a new convolution structure for the LCT that preserves the convolution theorem for the Fourier transform and is also easy to implement in the designing of filters. Some of well-known results about the convolution theorem in FT domain, fractional Fourier transform (FRFT) domain are shown to be special cases of our achieved results.

A Fourier transform method for nonparametric estimation of multivariate volatility

Abstract: We provide a nonparametric method for the computation of instantaneous multivariate volatility for continuous semi-martingales, which is based on Fourier analysis. The co-volatility is reconstructed as a stochastic function of time by establishing a connection between the Fourier transform of the prices process and the Fourier transform of the co-volatility process. A nonparametric estimator is derived given a discrete unevenly spaced and asynchronously sampled observations of the asset price processes. The asymptotic properties of the random estimator are studied: namely, consistency in probability uniformly in time and convergence in law to a mixture of Gaussian distributions.

Matched coordinates and adaptive spatial resolution in the Fourier modal method.

Abstract: Several improvements have been introduced for the Fourier modal method in the last fifteen years. Among those, the formulation of the correct factorization rules and adaptive spatial resolution have been crucial steps towards a fast converging scheme, but an application to arbitrary two-dimensional shapes is quite complicated.We present a generalization of the scheme for non-trivial planar geometries using a covariant formulation of Maxwell's equations and a matched coordinate system aligned along the interfaces of the structure that can be easily combined with adaptive spatial resolution. In addition, a symmetric application of Fourier factorization is discussed.

Sommerfeld enhancement: general results from field theory diagrams

Abstract: Assuming that two incoming annihilating particles interact by a generally massive attractive vector potential,we find, by taking the non-relativistic limit of the field theory ladder diagrams, that the complete annihilation amplitude A is equal to: the convolution of a solution of the Schroedinger equation (including the vector potential) with the Fourier transform of the bare (i.e. ignoring the attraction) annihilation amplitude A0. The main novelty is that A0 is completely arbitrary. In particular for a massless vector potential we find for the l-partial-wave cross-section the Sommerfeld enhancement 2pi/(l!)^2 (alpha/ v)^{2l+1} (v relative velocity), e.g. for the P wave the enhancement 2pi(alpha/ v)^3.