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In the widely used Hartree-Fock procedure for atomic structure calculations, trial functions in the form of linear combinations of Slater determinants are constructed and the Rayleigh-Ritz minimum principle is applied to determine the best in that class. A generalization of this approach, applicable to low-energy electron-atom scattering, is developed here. The method is based on a unique decomposition of the scattering wave function into open- and closed-channel components, so chosen that an approximation to the closed-channel component may be obtained by adopting it as a trial function in a minimum principle, whose rigor can be maintained even when the target wave functions are imprecisely known. Given a closed-channel trial function, the full scattering function may be determined from the solution of an effective one-body Schr{umlt o}dinger equation. Alternatively, in a generalized Hartree-Fock approach, the minimum principle leads to coupled integrodifferential equations to be satisfied by the basis functions appearing in a Slater-determinant representation of the closed-channel wave function; it also provides a procedure for optimizing the choice of nonlinear parameters in a variational determination of these basis functions. Inclusion of additional Slater determinants in the closed-channel trial function allows for systematic improvement of that function, as well as the calculatedmore » scattering parameters, with the possibility of spurious singularities avoided. Electron-electron correlations can be important in accounting for long-range forces and resonances. These correlation effects can be included explicitly by suitable choice of one component of the closed-channel wave function; the remaining component may then be determined by the generalized Hartree-Fock procedure. As a simple test, the method is applied to s-wave scattering of positrons by hydrogen. {copyright} {ital 1997} {ital The American Physical Society}« less

An effective-potential formalism, previously developed for electron scattering by a neutral target, is extended to apply to electron-ion scattering, with the requirement of antisymmetrization now accounted for explicitly. A minimum principle for the effective potential is derived, valid for scattering below the ionization threshold and applicable when, as is usually the case, the target wave functions are imprecisely known. The basis for the minimum principle is the Rayleigh-Ritz property that is satisfied by the modified Hamiltonian in terms of which the effective potential is defined. An analysis of single-channel, zero-energy scattering for a particular partial wave is presented; it is based on the effective-potential formalism and leads to an absolute definition of the zero-energy phase shift \ensuremath{\delta}(0) of the form \ensuremath{\delta}(0)=\ensuremath{\mu}(\ensuremath{\infty})\ensuremath{\pi}, where \ensuremath{\mu}(n) is the quantum defect of the nth energy level. This result may be thought of as an extension of Levinson's theorem for scattering by short-range potentials. \textcopyright{} 1996 The American Physical Society.

The Volkov wave function describing the motion of a charged particle in a laser field serves as a modified plane wave in the formulation of external-field collision theory and is widely used in applications. Exact solutions are unavailable for those cases in which the target, as well as the projectile, carries a net charge. It is shown that this difficulty may be overcome through the adoption of a variational formulation of the theory in configuration space

Variational principles for three-body reaction amplitudes are derived which allow for colliding systems that are charged and composite and are applicable for energies lying below the threshold for breakup into four subsystems. The starting point of the analysis is a formulation of the collision dynamics based on coupled integral equations of the Faddeev type that are applicable in the presence of long-range Coulomb interactions. A variational identity (which becomes a variational approximation with the introduction of trial functions) for the amplitude for the breakup of a bound pair by a third particle is derived within the integral equation formulation. The expression is then converted to a differential form of the Kohn type involving wave functions in configuration space. Knowledge of the asymptotic form of the wave function representing the time-reversed final state, in which three unbound particles are incident, is not required in performing this derivation. The variational principle is enhanced by the existence of a subsidiary minimum principle satisfied by that component of the wave function describing virtual excitations of one or more of the three clusters that make up the scattering system.

A modified perturbation theory is developed for the calculation of the time-dependent wave function describing the propagation of an atom in a two-mode laser field. With the mode frequencies nearly equal, a first approximation is obtained that reproduces and extends known results, for both resonant and nonresonant transitions. This lowest-order solution sums terms that describe the absorption of a photon from one mode and emission into the other, leaving the total number of photons unchanged and shifting the momentum of the atom accordingly. Corrections are generated by expansion of a resolvent from which the nearly degenerate states have been projected out. These approximate solutions are used to construct asymptotic states in a formulation of atom-atom scattering in the two-mode field. In close analogy with the soft-photon approximation, a ``soft-pair'' approximation is developed based on the dominance of those asymptotic interactions in which the transfer of energy from the field to the atom is small, allowing one to relate the scattering amplitude in the presence of the field to the field-free amplitude. When the atomic ground state and an excited state are closely coupled by the field, the asymptotic wave function can have mixed parity allowing for long-range contributions to the effective potential. In addition to an enhancement of forward-scattering probabilities, this leads to a threshold anomaly that is analyzed here, in the resonant case, with the aid of a modified effective-range theory.

Levinson{close_quote}s theorem relates the zero-energy phase shift {delta} for potential scattering in a given partial wave {ital l}, by a spherically symmetric potential that falls off sufficiently rapidly, to the number of bound states of that {ital l} supported by the potential. An extension of this theorem is presented that applies to single-channel scattering by a compound system initially in its ground state. As suggested by Swan [Proc. R. Soc. London Ser. A {bold 228}, 10 (1955)], the extended theorem differs from that derived for potential scattering; even in the absence of composite bound states {delta} may differ from zero as a consequence of the Pauli principle. The derivation given here is based on the introduction of a continuous auxiliary {open_quote}{open_quote}length phase{close_quote}{close_quote} {eta}, defined modulo {pi} for {ital l}=0 by expressing the scattering length as {ital A}={ital a}cot{eta}, where {ital a} is a characteristic length of the target. Application of the minimum principle for the scattering length determines the branch of the cotangent curve on which {eta} lies and, by relating {eta} to {delta}, an absolute determination of {delta} is made. The theorem is applicable, in principle, to single-channel scattering in any partial wave for {ital e}{sup {plus_minus}}-atom and nucleon-nucleusmore » systems. In addition to a knowledge of the number of composite bound states, information (which can be rather incomplete) concerning the structure of the target ground-state wave function is required for an explicit, absolute, determination of the phase shift {delta}. As for Levinson{close_quote}s original theorem for potential scattering, {ital no} {ital additional} {ital information} {ital concerning} {ital the} {ital scattering} {ital wave} {ital function} {ital or} {ital scattering} {ital dynamics} {ital is} {ital required}. {copyright} {ital 1996 The American Physical Society.}« less

The zero-energy scattering in a particular partial wave by a potential V=${\mathit{V}}_{\mathit{s}}$+${\mathit{V}}_{\mathit{c}}$ that is a superposition of short range and attractive Coulomb components is characterized by the additional phase shift ${\mathrm{\ensuremath{\delta}}}_{\mathit{s}}$(0), due to ${\mathit{V}}_{\mathit{s}}$. It has been known for many years that ${\mathrm{\ensuremath{\delta}}}_{\mathit{s}}$(0)(mod\ensuremath{\pi})=\ensuremath{\mu}(\ensuremath{\infty})\ensuremath{\pi}, where \ensuremath{\mu}(n) is the quantum defect of the nth energy level. In analogy with Levinson's theorem for short-range potentials, one might expect that a more precise statement, based on an absolute definition of the phase shift, would be ${\mathrm{\ensuremath{\delta}}}_{\mathit{s}}$(0)=\ensuremath{\mu}(\ensuremath{\infty})\ensuremath{\pi}, with the value of the largest integer contained in \ensuremath{\mu}(\ensuremath{\infty}) representing the number of additional bound states due to ${\mathit{V}}_{\mathit{s}}$. A simple derivation of this relation is presented here, based on variational principles for the binding energies and phase shifts, and on the property (fundamental to quantum-defect theory) that appropriately normalized bound-state wave functions for n\ensuremath{\rightarrow}\ensuremath{\infty} merge smoothly into the energy-normalized regular continuum solutions at the continuum threshold.

A minimum principle for the nonrelativistic scattering length was derived some years ago under the assumption that only a finite number of discrete states exist below the scattering threshold. The variational bound is applicable even when the bound-state wave functions are imprecisely known\char21{}they need only be accurate enough to give binding in a Rayleigh-Ritz calculation. The method is generalized here to apply to potential scattering described by the Dirac equation. An apparent difficulty associated with the existence of a continuum of negative-energy states, that is, the problem of ``variational collapse,'' is removed through the inclusion of second-order terms in the variational expression involving matrix elements of the square of the Dirac Hamiltonian. In the course of the derivation a relativistic version of the Hylleraas-Undheim theorem is developed. Applications of this theorem are described that provide a sufficient condition for the existence of a given number of bound states, lower bounds on the energy eigenvalues, and a systematic procedure for improving the accuracy of trial bound-state wave functions. A very simple model calculation was performed to illustrate the minimum property and the stability of the numerical procedure.

Wave equations---nonrelativistic and relativistic---describing the motion of a charged particle in a multimode laser field are considered and solutions are derived that may serve as asymptotic wave functions in the evaluation of amplitudes for laser-assisted scattering and multiphoton ionization. Unlike the well-known Volkov solutions that can be used only for fields of the plane-wave type, the wave functions obtained here are appropriate for the wider class of fields for which the different modes need not all have the same direction of propagation. This generalization allows in principle for the construction of wave packets, fields that are localized in space as well as time and hence capable of providing a more realistic description. The treatment of the nonrelativistic case is essentially exact, in the sense that errors introduced, of order (v/c${)}^{2}$, are comparable to those inherent in the nonrelativistic approximation. As an illustration, a low-frequency approximation for potential scattering in a multimode field is derived based on the use of the generalized Volkov solutions as asymptotic wave functions. Generalized Volkov solutions of the Klein-Gordon and Dirac equations are derived as well, though here in an approximation requiring a limitation on the strength of the external field.

A relativistic version of an earlier, non-relativistic, formulation of the theory of ionization of an atomic system by electron impact is presented. With a time-independent resolvent operator taken as the basis for the dynamics, a wave equation is derived for a system with open channels consisting of two positive-energy electrons in an external field generated by the residual ion. Virtual intermediate states can be accounted for by the effective Hamiltonian that appears in the wave equation and which in principle may be constructed perturbatively. The asymptotic form of the wavefunction, modified by the effects of the long-range Coulomb interactions of the two electrons in the external field, is derived. These electrons are constrained, by projection operators which appear naturally in the theory, to propagate in positive-energy states only. The long-range Coulomb effects take the form of phase factors similar to those that are found in the non-relativistic version of the theory. With the boundary conditions established, an integral identity for the ionization amplitude is derived, and used to set up a distorted-wave Born expansion for the transition amplitude involving Coulomb-modified propagating waves.

A variational principle is applied to the study of the relativistic scattering of a charged particle from a center of force in the presence of a strong, slowly varying external electromagnetic field. The analysis is first given in terms of a spin-zero wave equation and then modified to describe the scattering of a Dirac particle. In contrast to previous treatments of problems of this nature, the field is not assumed to be of the plane-wave type. More realistically, it is modeled as a superposition of modes, each with a different frequency and direction of propagation. Exact solutions of the wave equation describing the asymptotic motion of the particle in the presence of the field are not available for fields of this type. Nevertheless, the variational principle allows for a formulation of the scattering problem and, by a suitable choice of trial functions, generates a gauge-invariant low-frequency approximation for the transition amplitude. The error introduced by the inaccuracy in the asymptotic wave function is compensated for by the inclusion of a variational correction term which properly accounts for the effect of stimulated Compton scattering in initial and final states. It is verified that previously derived low-frequency approximations for one- and two-photon spontaneous bremsstrahlung are reproduced by taking the weak-field limit of the approximate stimulated-bremsstrahlung amplitude.

While the exact amplitude for scattering in the presence of an external electromagnetic field is invariant under a change of gauge, the invariance property will not necessarily be preserved in approximations, and this may lead to inaccuracies and ambiguities in the results of numerical calculations. A method for removing this undesirable feature through a reformulation of the scattering problem is proposed here. The equations which determine the time evolution of the system are rewritten in a form which involves not the (gauge-dependent) vector and scalar potentials associated with the external field but rather effective potentials which are gauge independent; this allows for the introduction of gauge-invariant approximations in a systematic way. To focus on the essential features the method is described in the context of the relatively simple problem of nonrelativistic potential scattering but greater generality is possible. As an illustration of the method the problem of scattering in a slowly varying external field is considered and an approximation of the ''low-frequency'' type is obtained. The result, whose gauge invariance is verified explicitly, represents an improvement over earlier versions, having been obtained through a more accurate treatment of the interaction of the projectile with the external field in intermediate statesmore » of the scattering process.« less

Variational expressions for arbitrary quantum-mechanical matrix elements involve not only trial wave functions but also trial auxiliary Lagrange multipliers, which can be constants, functions, Green's functions, etc. Recently an extremum principle has been proposed as a useful tool for the approximate determination of any Langrange functions that appear; it is the first such tool of general applicability. This extremum principle has here been applied to the variational determination of the diamagnetic susceptibility and form factor of helium; each requires the approximate determination of one Lagrange function. The results, reported on in this paper, demonstrate that the extremum principle can in fact be used in the sense that no singularities appear and the functional that determines the trial Lagrange function L/sub t/ converges monotonically. The results indicate that provided L/sub t/ is roughly as accurate as the trial wave function, the variational estimate of a matrix element can be a significant improvement over the first-order estimate. The present method provides an alternative to the direct approach in which the trial bound-state wave functions are obtained in the course of a Rayleigh-Ritz calculation of the energy. Further studies are needed to determine the relative advantages of the two approaches. (AIP)

Variational principles are considered for the approximate evaluation of the diagonal matrix elements of an arbitrary known linear Hermitian operator. A method is derived that is immediately applicable to the variational determination of both the off-diagonal and diagonal matrix elements of normal and modified Green's functions.

It is well known that minimum principles for energy eigenvalues can be used not only for calculational purposes but also formally as the basis for studies of the nodal structure of the associated eigenfunctions. This formal feature is shown here to have a natural generalization to the zero-energy scattering problem, the scattering length playing the role of the energy eigenvalue in the formulation of the minimum principle. The number of nodes in the zero-energy wave function in a given partial wave for nonrelativistic potential scattering is shown to be equal to the number of negative-energy bound states of the same angular momentum L. Potentials with Coulomb tails are excluded from the present analysis, but will be treated elsewhere. (The connection with classical Sturm-Liouville theory comes perhaps as less of a surprise if one views the zero-energy state as being at the top of the discrete spectrum as well as at the bottom of the continuum.) The result derived here, when combined with the nodal definition of the phase shift along with some information on its threshold behavior, provides us with an alternative derivation of Levinson's theorem relating the zero-energy phase shift for orbital angular momentum L, ${\ensuremath{\delta}}_{L}$(0), to the number of bound states of the same L. (Very similar results can be derived for potential scattering as described by the Dirac and Klein-Gordon equations.) It may well be possible to extend some of the results obtained here to a number of single-channel multiparticle scattering problems, but this will be discussed elsewhere.

A minimum principle for the calculation of the scattering length, applicable when the ground-state wave function of the target system is known precisely, has been available for some time. When, as is almost always the case, the target wave function is imprecisely known, a minimum principle is available but the simple minimum principle noted above is not applicable. Further, as recent calculations show, numerical instabilities usually arise which severely limit the utility of even an ordinary variational approach. The difficulty, which can be traced to the appearance of singularities in the variational construction, is here removed through the introduction of a minimum principle, not for the true scattering length, but for one associated with a closely connected problem. This guarantees that no instability difficulties can arise as the trial scattering wave function and the trial target wave function are improved. The calculations are little different from those required when the target ground-state wave function is known, and, in fact, the original version of the minimum principle is recovered as the trial target wave function becomes exact. A careful discussion is given of the types of problems to which the method can be applied. In particular, the effects of the Paulimore » principle, and the existence of a finite number of composite bound states, can be accounted for.« less

In recent work it was shown how a rigorous subsidiary minimum principle of the Rayleigh-Ritz type could be used as an aid in the construction of the closed-channel part of the scattering wave function, thereby making available a potentially powerful new tool in the variational approach to multiparticle scattering problems. The earlier discussion, which was restricted to scattering below the threshold for target breakup, is generalized here to the case where both two-body and three-body channels are open. The scattering problem is formally reduced to an equivalent three-body problem. Effective two-body and three-body potentials are defined explicitly (without the use of projection operators) and integral equations of the Faddeev type are derived. This analysis, which suggests a variety of cluster approximations, is used here as the basis for a decomposition of the wave function into an open-channel part, which contains the two-body and three-body outgoing scattered waves, and a decaying closed-channel part. The closed-channel part is shown to satisfy a minimum principle whose rigor can be maintained even when the target bound-state wave functions are imprecisely known. A calculational procedure which combines this minimum principle with the Kohn variational construction of the scattering amplitude is described.NUCLEAR REACTIONS Scattering theory. Effective three-body formulation. Derivation of extremum principle for the wave function.

In a previous study of bound-state matrix elements of a Hermitian operator $W$, it was possible to obtain at most an upper (or lower) stationary bound. The possibility arose only for the diagonal matrix element case, and only for $W$ nonpositive (or nonnegative). In the present treatment, both upper and lower stationary bounds are obtained, for diagonal and off-diagonal matrix elements, and, though some restrictions on $W$ remain, the requirement that $W$ be of well-defined sign can be dropped. The derivation also improves upon that given previously in that the possibility of any difficulty with near singularities in the equation defining the trial auxiliary (or Lagrange) function is unambiguously avoided. As an example, the method is applied to the problem of the zero-energy scattering of positrons by atoms or ions, and an expression is derived which provides a rigorous stationary upper bound on the scattering length; the target ground-state wave function need not be known exactly. Crude but rigorous numerical results are obtained quite simply in the Born approximation.

We denote as a "primary minimum principle" one in which a quantity $B$ of physical interest is represented as the minimum value with respect to variations in a trial function ${\ensuremath{\psi}}_{t}$ of a functional $F({\ensuremath{\psi}}_{t})$; $F$ then provides a variational upper bound on $B$. (The Rayleigh-Ritz principle for the ground-state energy of a system is a familiar example.) If $F$ is quadratic in ${\ensuremath{\psi}}_{t}$, the variational property of $F$ enables one to determine the linear parameters relatively easily, but the minimum property is required if the nonlinear parameters are to be determined in a way which allows for systematic improvement of ${\ensuremath{\psi}}_{t}$. We show here that for a wide class of problems for which primary minimum principles do not exist, useful and rigorous secondary or "subsidiary minimum principles" are available. That is, we construct a functional ${F}^{\ensuremath{'}}({\ensuremath{\psi}}_{t})$ whose minimum value is reached for ${\ensuremath{\psi}}_{t}$ equal to some function $\ensuremath{\psi}$ of dynamical interest. (The Rayleigh-Ritz method provides a subsidiary minimum principle for the approximate determination of the ground-state wave function of a system.) If $B=B(\ensuremath{\psi})$, then a study of ${F}^{\ensuremath{'}}({\ensuremath{\psi}}_{t})$ provides a powerful tool for the estimation of $\ensuremath{\psi}$ and therefore $B$, though $B({\ensuremath{\psi}}_{t})$ is not normally a variational bound on $B(\ensuremath{\psi})$. Subsidiary minimum principles have recently been obtained for the approximation of the auxiliary functions that appear in the variational principle for the matrix element (${\ensuremath{\chi}}_{n}$, $W{\ensuremath{\chi}}_{m}$), where ${\ensuremath{\chi}}_{n}$ and ${\ensuremath{\chi}}_{m}$ are bound-state wave functions and $W$ is an arbitrary operator. Here we extend the method to the estimation of matrix elements of the Green's function $g(\ensuremath{\epsilon})$ of a bound system with $\ensuremath{\epsilon}$ below the continuum threshold energy. The response of the system to an external perturbation can be represented by matrix elements of this type. While no new results on the bound-state problem are obtained, our formulation is a convenient starting point for the further extension of the method to continuum problems. The new result obtained here is the derivation of a subsidiary minimum principle for the problem of scattering of a projectile by a target whose bound-state wave function is only imprecisely known. The subsidiary minimum principle allows for systematic improvement of the closed-channel component of the trial scattering wave function that appears in a Kohn-type variational calculation of the scattering amplitude.

A variational principle is formulated as a means for obtaining approximate solutions to the Faddeev integral equations for the nonrelativistic three-body scattering problem. Bound states are described by the homogeneous version of the Faddeev equations. In this case a minimum principle for the binding energies is obtained in close analogy with the familiar Rayleigh-Ritz method. An effective potential is defined as the solution of a set of modified Faddeev equations. The variational principle therefore applies to the effective potential. This is shown to be a minimum principle for the eigenphase shifts for energies below the three-particle breakup threshold. Some formal applications of the variational principle are described. These include derivations, within the Faddeev formalism, of a normalization condition for bound-state wave functions, a distorted-wave theory, and a perturbation theory for discrete states.

In the preceding paper we derived a Kohn-type variational principle for the scattering amplitude for the breakup of a bound pair by an incident third particle. In this paper an alternative derivation is presented. The new derivation, based on the Faddeev equations, though rather formal, is rigorous; the preceding derivation was not. This procedure sheds some light on the manipulations in the earlier derivation. We also derive an "adjoint" Kohn-type variational principle in which the operator $H\ensuremath{-}E$ acts on the final-state wave function. The "adjoint" result has some convergence difficulties but these have been overcome by techniques described. Finally, a Schwinger-type variational principle is derived and its connection with the recently proposed variational principle of Pieper, Schlessinger, and Wright is discussed.

A Kohn-type variational principle normally requires knowledge of the asymptotic form of the time-reversed final-state wave function. For the breakup collision of a bound pair of particles by a third particle, the time-reversed final state is a state in which three free particles are incident, and the asymptotic form of the associated wave function is only incompletely understood. We have nevertheless been able to obtain a Kohn-type variational principle for the breakup scattering amplitude (for short-range potentials). The final expression involves only well-defined integrals, though in the course of the derivation we are forced to separate finite integrals into separately divergent components. These divergences, which also occur elsewhere in scattering theory, are handled by a redefinition of the integrals which is the analog for integrals of Ces\`aro summation.

Levinson's theorem relation phase shifts and bound states, developed for a short-range potential ${V}_{\mathrm{sh}}$, is extended to scattering by ${V}_{\mathrm{sh}}+{V}_{\mathrm{C}}$, where ${V}_{\mathrm{C}}$ is a repulsive Coulomb potential. The subtleties associated with $L=0$, $E=0$ bound states for ${V}_{\mathrm{sh}}$ alone are not present for ${V}_{\mathrm{sh}}+{V}_{\mathrm{C}}$. Information is also obtained on the relation between the nodal structure of the zero-incident-energy wave function and the number of bound states; some extensions to scattering by a compound target are possible.

For potential scattering, with ${\ensuremath{\delta}}_{L}$(k) the phase shift modulo \ensuremath{\pi} for an incident wave number k, Levinson's theorem gives ${\ensuremath{\delta}}_{L}$(0)-${\ensuremath{\delta}}_{L}$(\ensuremath{\infty}) in terms of ${N}_{L}$, the number of bound states of angular momentum L, for ${\ensuremath{\delta}}_{L}$(k) assumed to be a continuous function of k. ${N}_{L}$ also determines the number of nodes of the zero-energy wave function ${u}_{L}$(r). A knowledge of the nodal structure and of the absolute value of ${\ensuremath{\delta}}_{L}$(0) is very useful in theoretical studies of low-energy potential scattering. Two preliminary attempts, one formal and one ``physical,'' are made to extend the above results to single-channel scattering by a compound system initially in its ground state. The nodal structure will be of greater interest to us here than an extension of Levinson's theorem.The formal approach is applied to ${e}^{+}$-H and ${e}^{+}$-He scattering. Both H and He have zero orbital angular momentum and a nodeless ground-state wave function ${\ensuremath{\psi}}_{T}$. An effective one-body wave function ${u}_{L}$ for the positron incident with zero kinetic energy can be constructed by factoring out the spin and Euler-angle dependence of the full scattering wave function \ensuremath{\Psi} and projecting the remaining ``radial'' function ${R}_{L}$ onto ${\ensuremath{\psi}}_{T}$.The nodal surfaces of ${R}_{L}$ are shown to divide configuration space into at most ${N}_{L}$+1 subdomains, where ${N}_{L}$ is the number of composite bound states of the given L. Since ${N}_{L}$=0 for ${e}^{+}$-H and ${e}^{+}$-He, it follows that ${u}_{L}$ is nodeless and that ${\ensuremath{\delta}}_{L}$(0)=0, for all L. Partial but useful information on the nodal structure of \ensuremath{\Psi} for ${e}^{\mathrm{\ensuremath{-}}}$-H scattering is also deduced. Interestingly, nodal surfaces exist which are not consistent with a naive generalization of (bound-state) one-dimensional Sturm-Liouville theory.The physical arguments, based on the qualitative concept of an effective central potential seen by each target particle and by the incident particle P, strengthen a previous surmise on the value of ${\ensuremath{\delta}}_{\mathrm{LJ}}$(0) for ${e}^{\ifmmode\pm\else\textpm\fi{}}$-atom scattering and for the scattering of neutrons or protons by a heavy nucleus; L and J are the quantum numbers of the incident P, and ${\ensuremath{\psi}}_{T}$ is assumed to have zero spin and zero orbital angular momentum, but ${\ensuremath{\psi}}_{T}$ need not be nodeless, and P need not be distinguishable. Roughly, the surmise is that ${\ensuremath{\delta}}_{\mathrm{LJ}}$(0)=${K}_{\mathrm{LJ}}$\ensuremath{\pi}, where, for the given L and J, ${K}_{\mathrm{LJ}}$ is the number of composite bound states plus the number of one-particle states excluded by the Pauli principle.

A low-frequency approximation for the bremsstrahlung transition amplitude is derived for a relativistic electron scattered by a central potential which is Coulombic at great distances. The Coulomb tail is shown here to have the effect of introducing correction terms, not present in previous relativistic treatments, which depend logarithmically on the photon frequency. The approximation procedure, in the form presented here, is based on the fact that at low frequencies the dominant contribution to the matrix element comes from the domain where r, the electron distance from the scattering center, is large. An asymptotic representation of the Dirac wave function is derived which includes a correction term of relative order 1/r compared to the leading term. This enables us to include a correction to the matrix element of relative order k, where k is the photon wave number. The essential feature of the low-frequency approximation, present in previous versions, is preserved here. That is, the result is expressed in terms of the physical amplitude, A, for scattering in the absence of any radiative interaction. Another potentially useful feature of the present version is that it involves angular derivatives of A, not energy derivatives, and should therefore remain valid in a regionmore » where A varies rapidly with energy.« less

Using projection operators, a variational-bound formulation is presented which, together with some previous work, can provide both upper and lower variational bounds on the phase shifts or equivalent network elements associated with obstacles located in waveguides. The scattering parameters for dielectric rectangular-parallelepiped obstacles in single and multimode waveguides serve as numerical examples to demonstrate the applicability of the method.