Born Expansion Research Articles

Analiticity and Padé method in the bound-state problem

Recently the method of Pad~ approximants has proved to yield satisfactory resul~ in strong-interaction physics (1-4). When it is applied to the partial-wave projection of the T or S matrix Born series, in the framework of the usual Lagrangian approach or the theory of germs, the uni tar i ty condition turns out to be automatically satisfied (for diagonal approximants). The analytic properties in the square of c.m. energy are globally correct; however the Pad6 denominator can exhibit some spurious singularities, that prevent us from obtaining actual quanti tat ive information about bound states (5). The purpose of this note is to improve the content of the Padg method, concerning bound states, by giving a prescription to treat the analiticity troubles. We focus our at tention on potential theory where the validity of any approximation method can be tested with some care. & large class of regular potentials has a partialwave Born expansion exhibiting the same analytic properties as field-theory models; so the same features are present in the search of bound states through Pad6 approximants. Potential theory gives some further advantages: the convergence of the Pad6 approximants to the exact amplitude and of their normalized denominators t o the ;lost function has been proved (e-s). This suggests that the spurious singularities mentioned above should disappear by increasing the order of approximation; an explicit test of this cancellation has been made for the exponential potential (g). However, from a practical point of view it may be very laborious and difficult to calculate higher Born terms. For these reasons, in order to have an amenable method to calculate bound states, we stop with the first Pad6 approximant and improve it by a suitable recipe based on a standard technique for the analytic functions.

Diffusion élastique alpha aux grands angles à l'aide de potentiels non locaux séparables

The use of separable non-local potentials allows an exact calculation of the elastic scattering amplitude with only two terms in the Born expansion. The calculation is performed for elastic scattering of α-particles on 12C and 16O at about 20 MeV. At backward angles where the Coulomb effect is negligible, the fit obtained is very good with only two parameters and without any normalization.

Use of Iteration Procedures to Generate Scattering Approximations

Our subject is the class of iteration procedures which can be written as a functional $F(\ensuremath{\psi})$ used, via the rule ${\ensuremath{\psi}}_{n+1}=F({\ensuremath{\psi}}_{n})$, to generate a sequence ${\ensuremath{\psi}}_{0}, {\ensuremath{\psi}}_{1}, {\ensuremath{\psi}}_{2}, \ensuremath{\cdots}$ desired to converge quickly to the wave function of a quantum-mechanical system. Given such a functional ${F}_{1}(\ensuremath{\psi})$, we seek methods of manipulating it to obtain an improvement ${F}_{2}(\ensuremath{\psi})$; the starting point is the functional which generates the Born expansion. Four such methods are developed which, if applied serially, are capable of generating many likely iteration procedures. Only simple potential scattering is considered, but our methods seem to promise extension to cases much more complex. In these methods a central role is played by matrix elements $J$ and $T$ which measure the behavior of a functional for small and large distances from the scattering center. The resulting $F(\ensuremath{\psi})$ are highly nonlinear; the ${\ensuremath{\psi}}_{n}$ they generate are rational functions of the coupling constant with coefficients given by various combinations of $J$- and $T$-matrix elements. Numerical results for the first two orders of approximations are presented for square-well and exponential potentials. One of the procedures $F(\ensuremath{\psi})$ gives the Fredholm determinantal expansion, and in a form more easily calculated than the expressions usually employed; this expansion is shown to be unitary, and a relation is obtained connecting the Fredholm determinant and a dispersion integral of the $T$ matrix off the energy shell. Another of the improvements generates the inverse $T$-matrix expansion; this is shown to converge for coupling constant less than the smallest value which puts the phase shift at \ifmmode\pm\else\textpm\fi{} 180\ifmmode^\circ\else\textdegree\fi{}. In all of the above, the iteration sequence was started with ${\ensuremath{\psi}}_{0}$ being the free-wave solution corresponding to the scattering energy: An important advantage of iteration procedures is that, in constrast with the perturbation expansion, they are not pinned to this choice of ${\ensuremath{\psi}}_{0}$. We go on to select as input the free-wave solution of a different energy which better represents the average wavelength in the scattering region. The above iteration procedures then become much more powerful. When applied to the square-well potential, they yield the exact solution on the first iteration.

Error and Convergence Bounds for the Born Expansion

Error and Convergence Bounds for the Born Expansion

Effective Nonlocal Energy-Independent Potential and the Possibility of a Repulsive Core at Small Distances

An effective nonlocal potential is constructed by requiring that its Born expansion give the same amplitude as the peripheral-interaction expansion of Cutkosky. The latter is a modified perturbation expansion based on dispersion theory in which each line of a Feynman graph represents not only discrete states, but also all states lying in the continuum which have a limited angular momentum; in higher order diagrams one also has to subtract out certain contributions which are already included in lower order diagrams, so as to avoid double counting. It can then be argued that, at least if we go up to fourth order, there should be a repulsive core in the potential. The Schr\"odinger equation is thus solvable without cutoff in configuration space. Because of the difficulties of such a program, only a rough momentum-space calculation was carried out for $\ensuremath{\pi}\ensuremath{\pi}$ scattering with $\ensuremath{\rho}$ exchange.

Error and Convergence Bounds for the Born Expansion

For scattering in nonrelativistic quantum mechanics, the range of validity of the perturbation expansion is studied by viewing the expansion as being generated by an iteration procedure, and applying to it a fundamental theorem for iteration processes. The theorem provides a sufficient condition for convergence and, at the same time, gives upper bounds on the error generated in truncating the expansion at a given number of terms. These error bounds for the wave function are in turn used to find upper bounds on the truncation error for the $S$-matrix expansion. Bounds for the exact $S$ matrix are also given. These results are illustrated by applying them to the simple case of one-particle potential scattering, for both the plane-wave and partial-wave analyses.

Method of Orthogonality Constraint and Rearrangement Collisions

In treating scattering problems involving a bound state, the method of orthogonality constraint modifies the potential by projecting out the "binding" effect of the potential strong enough to support a bound state, and considers only its "scattering" effect as the perturbation on a distorted wave which is orthogonal to the bound state. The "modified" Born series obtained by iterating the relevant Lippmann-Schwinger integral equation is proved to be convergent, provided resonances not associated with the bound state are absent. The convergence proof is given for one-dimensional even waves and for a separable potential. However, the proof is also valid for the $s$ waves. We believe, on the basis of our simplified analysis, that the method will work for a wide class of physically interesting potentials, though a rigorous proof for it is not available. The convergence of the Born expansion for rearrangement collisions is discussed in the context of a one-dimensional model for a three-body problem in which exchange scattering takes place. It is seen in the example that the kind of divergence pointed out by Aaron, Amado, and Lee may be removed by this method.

Nonrelativistic High-Energy Photoionization Cross Section

It is shown that the use of the dipole acceleration form of the photoionization cross section in a high-energy Born expansion gains one iteration of the Born series over the corresponding expansion of the dipole-velocity cross section. Some comparisons with Hartree-Fock indicate the domain of validity of the acceleration Born approximation.

Distorted-Wave Born Expansions

A successive approximation procedure is presented for calculating transition amplitudes for direct and rearrangement collision processes. The essential feature of the method is that the base problem about which the solution is expanded exhibits all the bound states (of subsystems of particles) which appear in initial or final scattering states. While no proof is given that our expansion converges, arguments which have been presented indicating that the ordinary Born expansion diverges for a wide class of problems no longer apply to the expansion proposed here. A particular choice of base problem leads directly to the well-known distorted-wave Born approximation. The variational nature of this approximation is exhibited and, as a consequence of the general formalism, a procedure for systematic improvement is presented. Circumstances are described under which the first term in the modified Born expansion has an error of known sign. The use of separable potentials to generate solutions to the base problem is discussed and is shown, in the three-body case, to lead to a model proposed recently by Amado. As a by-product of our work a variational principle for transition amplitudes is developed which is a generalization of the Kohn principle for the two-body elastic amplitude and is valid for any scattering process described by the Schr\odinger equation.

Systematic Approximations for the Single-Channel Scattering Amplitude

The scattering amplitude for two-particle single-channel scattering may be computed by inverting a Hilbert-space operator $1\ensuremath{-}K$, where $K$ is compleely continuous and depends on the real energy as a parameter. This inversion can always be accomplished by obtaining a finite number of eigenvalues and eigen-functions of $K$ together with a modified Born expansion which is guaranteed to converge. Another method consists in the inversion of a finite matrix, the elements of which are computed by a convergent perturbation series.

Continuation and optimization of the Born expansion in nonrelativistic quantum theory

Methods of von Neumann, Lanczos, and others, are applied to the non-relativistic scattering problem. It is shown that if the Born series is modified suitably, it will converge for almost any strength of interaction. Further, if suitable linear combinations of the modified Born approximations are used in the usual iteration process, the rate of convergence can be improved. The methods also apply to systems of integral equations which describe scattering of a particle from a complex system.

Propriétés analytiques de la matrice S dans la diffusion par un potentiel, relativement à la force de ce potentiel

Analytical properties of the S matrix are obtained with respect to the strength of any potential in the radial Schrodinger equation. Some properties of the analogs of Regge trajectories are derived in the same way, which should give information on the radius of convergence of Born expansion. Enough information is obtained by restricting the study to potentials decreasing faster than an exponential at infinite range. Results are derived for the branches of the trajectories relative to anti-bound states. Restriction to superpositions of Yulkawa potentials gives information on some points. In the case of nucleon- nucleon scattering, strengihs of both singlet and triplet potentials belong to the same trajectory. As an example, they are calculated in the case of a Yukawa potential from the energies of the deuteron and of the antibound /sup 1/S/sub O/ state. (auth)

Impulse Approximation for Stripping Reactions

The impulse approximation expansion is applied to stripping processes. We find that the first term of the impulse expansion for the stripping problem is identical with the first term of the Born expansion. It is suggested that the impulse approximation argument may provide a better justification for the usual treatment of stripping and pickup reactions than does the Born approximation argument.

On the Scattering of a Particle by a Static Potential

The behavior of the solutions of scattering integral equations is studied as a function of the potential strength $\ensuremath{\lambda}$. From an analysis of the second Born approximation for a Yukawa potential it seems indicated that the Born expansion for a nuclear potential has no useful domain of applicability. The convergence of the Born expansion is discussed. It is shown that the Fredholm theory of integral equations enables one to express the solutions as a quotient of infinite power series in $\ensuremath{\lambda}$ which still converge when the Born expansion breaks down. Only in exceptional cases can this method be used for obtaining rapid numerical estimates.

On the Absence of Polarization in Electron Scattering

In an attempt to explain the discrepancy between Mott's theory and the observational results on the polarization of electrons in the double scattering experiment, we have investigated the depolarizing effect of (1) multiple elastic scattering, (2) inelastic scattering with spin change of the incident electron and (3) exchange scattering in which the exchanged electrons have opposite spins. Although multiple scattering actually takes place in the Au foils used in the experiments, the depolarization caused by such scattering is negligible. For a screened Coulomb field and electrons of 100 kv the depolarization in a 2.5 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}5}$ cm Au foil is only 2.2 percent whereas a 90-100 percent depolarization is needed to reconcile theory and experiment. The depolarization caused by inelastic scattering is about 100 times smaller than that caused by elastic scattering. The small depolarization in these two cases is due to the fact that while there are many elastic and inelastic collisions, most of these collisions take place with small scattering angle $\ensuremath{\vartheta}$ and for small angles the depolarization per collision is small (of order ${\ensuremath{\vartheta}}^{2}$). The main contribution to the depolarization by inelastic collisions comes then from large scattering angles at which inelastic scattering is relatively infrequent. The exchange scattering contributes practically no depolarization (\ensuremath{\sim}${10}^{\ensuremath{-}12}$). While the depolarization per collision in each exchange process is relatively complete, the number of exchange processes is extremely small due to interference (\ensuremath{\sim}${10}^{\ensuremath{-}14}$ times the number of elastic collisions). Therefore the absence of polarization in electron scattering cannot be explained on the basis of the processes considered here.A discussion of the validity of the first Born approximation is given (Appendix \textsection{}2). It is shown that in the case that the Born expansion parameter $\frac{{e}^{2}Z}{\ensuremath{\hbar}v}$ is of order unity, the first Born approximation gives a result for the scattered intensity which is accurate to within 20 percent or better. The depolarization is given with an accuracy of 18 percent or better.