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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Probing a nonperturbative method on theSmatrix of the nonlinear Schrödinger model in the strong-coupling regime

Ralph Girard, H. Kröger

Published Sep 15, 1986• 5 citations
Publisher

Abstract

A recently proposed nonperturbative scheme of calculation for the S matrix of field theories is tested for the nonlinear Schr\"odinger model. For the case of two-body scattering, we study the dependence of the approximate S matrix on the following approximation parameters: the momentum-space cutoff \ensuremath{\Lambda}, the number of nodes \ensuremath{\nu}, and the scattering time T. We find convergence towards the reference value, when the approximation parameters tend to infinity, in case of a weak as well as a strong coupling constant. Also in the case of overlapping wave packets, which corresponds to an infinite effective coupling constant, we obtain acceptable convergence.

DOI
10.1103/physrevd.34.1824

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A recently proposed nonperturbative scheme of calculation for the S matrix of field theories is tested for the nonlinear Schr"odinger model. For the case of two-body scattering, we study the dependence of the approximate S matrix on the following approximation parameters: the momentum-space cutoff \ensuremath{\Lambda}, the number of nodes \ensuremath{\nu}, and the scattering time T. We find convergence towards the reference value, when the approximation parameters tend to infinity, in case of a weak as well as a strong coupling constant. Also in the case of overlapping wave packets, which corresponds to an infinite effective coupling constant, we obtain acceptable convergence.
The nonperturbative Hamiltonian eigenvalue problem for bound states of a quantum field theory is formulated in terms of Dirac's light-front coordinates and then approximated by the exponential operator technique of the many-body coupled-cluster method. This approximation eliminates any need for the usual approximation of Fock-space truncation. Instead, the exponentiated operator is truncated, and the terms retained are determined by a set of nonlinear integral equations. These equations are solved simultaneously with an effective eigenvalue problem in the valence sector, where the number of constituents is small. Matrix elements can be calculated, with extensions of techniques from many-body coupled-cluster theory, to obtain form factors and other observables.
We investigate the scattering theory of two particles in a generic DDD-dimensional space. For the sss-wave problem, by adopting an on-shell approximation for the TTT-matrix equation, we derive analytical formulas which connect the Fourier transform $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)$ of the interaction potential to the sss-wave phase shift. In this way we obtain explicit expressions of the low-momentum parameters ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}$ of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}+{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}{k}^{2}+\ensuremath{\cdots}$ in terms of the sss-wave scattering length as{a}_{s}as​ and the sss-wave effective range rs{r}_{s}rs​ for D=3D=3D=3, D=2D=2D=2, and D=1D=1D=1. Our results, which are strongly dependent on the spatial dimension DDD, are a useful benchmark for few-body and many-body calculations. As a specific application, we derive the zero-temperature pressure of a two-dimensional uniform interacting Bose gas with a beyond-mean-field correction which includes both scattering length and effective range.
We introduce a nonperturbative approximation scheme for performing scattering calculations in two dimensions that involves neglecting the contribution of the evanescent waves to the scattering amplitude. This corresponds to replacing the interaction potential vvv with an associated energy-dependent nonlocal potential Vk{\mathcal{V}}_{k}Vk​ that does not couple to the evanescent waves. The scattering solutions $\ensuremath{\psi}(\mathbf{r})$ of the Schr\"odinger equation, $(\ensuremath{-}{\ensuremath{\nabla}}^{2}+{\mathcal{V}}_{k})\ensuremath{\psi}(\mathbf{r})={k}^{2}\ensuremath{\psi}(\mathbf{r})$, have the remarkable property that their Fourier transform $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\psi}}(\mathbf{p})$ vanishes unless p\mathbf{p}p corresponds to the momentum of a classical particle whose magnitude equals kkk. We construct a transfer matrix for this class of nonlocal potentials and explore its representation in terms of the evolution operator for an effective nonunitary quantum system. We show that the above approximation reduces to the first Born approximation for weak potentials and, similar to the semiclassical approximation, becomes valid at high energies. Furthermore, we identify an infinite class of complex potentials for which this approximation scheme is exact. We also discuss the appealing practical and mathematical aspects of this scheme.
A nonperturbative method is developed for calculating the excitation of a many-body system by a time-dependent Hamiltonian. The stationary-phase approximation to a functional-integral representation of the interaction-picture many-body evolution operator results in a mean-field approximation to the SSS matrix which is asymptotically time independent. A one-body temporally nonlocal evolution equation defines the stationary mean-field configurations. The general method and character of the stationary solutions are illustrated by application to the forced harmonic oscillator and forced Lipkin model. Potential applications to realistic nuclear and atomic scattering situations are discussed.NUCLEAR REACTIONS Mean-field approximation, many-body SSS matrix, time-dependent Hartree Fock.
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We introduce a non perturbative general approximation scheme (NGAS) that can handle interactions of any strength in quantum theory. This approach starts with an input Hamiltonian that can be solved exactly. The interaction effects are then built into this Hamiltonian through nonlinear feedback enforced by self consistency conditions. While the method itself is nonperturbative it can be systematically improved using a new perturbation method called 'mean field perturbation theory' which does not involve power series expansion in any small parameter. We put this scheme to the test on one dimensional anharmonic interactions using the harmonic approximation. The results are consistently accurate across various cases including quartic, sextic, and octic anharmonic oscillators, as well as the quartic double well oscillator (QDWO) even when the coupling strength varies widely. The flexibility of the method is demonstrated when we swap the input Hamiltonian for that of an infinite square well and still achieve comparable accuracy. When applied to the λϕ4 quantum field theory this approach aligns with the Gaussian effective potential method under the harmonic approximation. Beyond that it reveals the condensate structure of the effective vacuum and highlights the instability of the perturbative ground state. Notably, our ground-state energy results for the QDWO stand in stark contrast to those from standard perturbation theory where Borel summation fails regardless of coupling strength.