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Physical Review

Error and Convergence Bounds for the Born Expansion

Irwin Manning

Published Jul 26, 1965• 12 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.139.b495

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•Aug 14, 2016•8 citations
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 SSS-matrix expansion. Bounds for the exact SSS 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.
It is shown that if the wave function of a nonrelativistic particle satisfies the Schr\"odinger equation with a velocity-independent potential, then its scattering amplitudes (and the SSS matrix in general) satisfy the same dispersion formulas as those derived for the scattering of light. In the present derivation, the validity of a perturbation expansion and certain integrability of the potential are assumed, and the requirement of the outgoing wave Green's function replaces the condition of strict causality for the scattering of light. The scattering amplitude for fixed momentum transfer is also shown to satisfy the dispersion relations. Together with the unitarity of the SSS matrix, the complete SSS matrix is determined by the Fourier transform of the potential through an iterative procedure using the dispersion relations. If the potential possesses bound states, then all the dispersion formulas are modified to include residue terms corresponding to singularities on the positive imaginary axis of the momentum plane. The necessity of these modifications is related to the divergence of the perturbation series for small wave numbers. Essential singularities of the SSS matrix due to exponentially damped potentials give no additional contribution to the dispersion formulas.
I provide a straightforward proof that a simple harmonic oscillator perturbed by an (almost) arbitrary positive interaction has a perturbative expansion for any finite-time Euclidian transition amplitude which obeys the following result: the difference of the sum of the first N terms of the series and the exact result is bounded in absolute value by the next term in the series. The proof makes no assumptions about either the strength of the interactions or the convergence of the perturbation series. I then argue that the result generalizes immediately to a much broader class of quantum mechanical systems, including bare perturbation theory in quantum field theory. The case of renormalized perturbation theory is more subtle and remains open, as does the generalization to energy levels and connected S-matrix elements.
The Fredholm theory of integral equations is used to give a rigorous proof of the analyticity and boundedness of the ordinary nonrelativistic scattering amplitude for a fixed momentum transfer. The results follow from ordinary quantum mechanics and certain conditions on the potentials. These conditions are stated explicitly, and the bound states are treated with rigor. It is shown that the amplitude vanishes in the limit of large momenta, and thus simple dispersion relations are derived. Finally, it is proved that the partial-wave expansion is convergent in the unphysical region, provided the potentials satisfy the same conditions as above.
Nonrelativistic quantum chemistry is based on the Schrödinger equation in continuous, unbounded space. Computational approaches on classical and quantum computers often truncate the spatial domain and introduce a finite-dimensional basis, reducing the continuum problem to a finite-dimensional one. There is therefore a fundamental question: to what extent does this approximation capture the original continuum problem? In this work, we develop a general Sobolev-Galerkin framework for estimating spatial discretization errors in quantum simulation with unbounded Hamiltonians. We apply this framework to many-body Coulomb systems and establish both periodization and Galerkin error estimates for wave-function dynamics and ground-state energies. For dynamics, the finite-basis projection error of the exact wave function does not capture the full basis-discretization error: an additional dynamical projection error enters our bound at a different rate, and we introduce an inner-product argument to control this contribution. Interestingly, the spatial convergence rates for wave-function dynamics and ground-state energies differ quantitatively. For ground-state energies, we establish a sharp M−1M^{-1}M−1 convergence rate at fixed cell size, where MMM is the number of one-particle basis functions. We also show that a uniform M−1M^{-1}M−1 convergence rate is impossible for the corresponding real-time wave-function error with general H2H^2H2 initial data. The analysis relies on many-body Sobolev estimates with explicit polynomial dependence on system size. Combining these estimates with state-of-the-art quantum algorithms, we obtain, to our knowledge, the first rigorous end-to-end complexity bounds for quantum simulation of molecular and electronic dynamics with all spatial errors explicitly controlled. The number of basis functions is determined directly from the continuum problem, without regularization.
In this paper, the Born series for the s-wave scattering a0 is calculated for a class of central potentials V(r) up to sixth order in a dimensionless coupling strength g. Examples of exponentially decaying potentials as well as truncated potentials involving a single length-scale a are considered. In certain favorable cases, the exact result for the g-dependent s-wave scattering length a0=A0(g)a can be given in terms of special functions. The poles of A0(g) at increasing positive values of g correspond to the thresholds, where s-wave bound-states occur successively. Further scattering problems, where A0(g) is solvable in terms of elementary functions, are also presented.
Abstract.: Using a recent path integral representation for the T -matrix in nonrelativistic potential scattering we investigate new variational approximations in this framework. By means of the Feynman-Jensen variational principle and the most general ansatz quadratic in the velocity variables --over which one has to integrate functionally-- we obtain variational equations which contain classical elements (trajectories) as well as quantum-mechanical ones (wave spreading). We analyse these equations and solve them numerically by iteration, a procedure best suited at high energy. The first correction to the variational result arising from a cumulant expansion is also evaluated. Comparison is made with exact partial-wave results for scattering from a Gaussian potential and better agreement is found at large scattering angles where the standard eikonal-type approximations fail