Magnus-factorized method for numerical solving the time-dependent Schrödinger equation

Пузынин, И.В.; Selin, A.V.; Vinitsky, S. I.

doi:10.1016/s0010-4655(99)00225-8

Cited by 22 publications

(29 citation statements)

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“…In this section we extend the work of Puzynin et al [19,20]. The basic idea is to replace the exponential operator exp(−iH∆t) by the diagonal Padé approximant.…”

Section: Time Advancementioning

confidence: 97%

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Accurate numerical solutions of the time-dependent Schrödinger equation

Dijk

Toyama

2007

Phys. Rev. E

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We present a generalization of the often-used Crank-Nicolson (CN) method of obtaining numerical solutions of the time-dependent Schrödinger equation. The generalization yields numerical solutions accurate to order (∆x) 2r−1 in space and (∆t) 2M in time for any positive integers r and M , while CN employ r = M = 1. We note dramatic improvement in the attainable precision (circa 10 or greater orders of magnitude) along with several orders of magnitude reduction of computational time. The improved method is shown to lead to feasible studies of coherent-state oscillations with additional short-range interactions, wavepacket scattering, and long-time studies of decaying systems.

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“…In this section we extend the work of Puzynin et al [19,20]. The basic idea is to replace the exponential operator exp(−iH∆t) by the diagonal Padé approximant.…”

Section: Time Advancementioning

confidence: 97%

“…We have found these very useful for making long-time or large-space problems tractable [7], but we will not discuss such boundary conditions further in this paper. Puzynin et al [19,20] indicate how to generalize the time development to higher order, but do not discuss spatial integration.…”

Section: Introductionmentioning

confidence: 99%

Accurate numerical solutions of the time-dependent Schrödinger equation

Dijk

Toyama

2007

Phys. Rev. E

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“…The development of numerical procedures for approximating the action of f (A) to a vector has received considerable attention in the last two decades. This is possibly related to the significant increase of methods for the numerical solution of partial differential equations that either directly approximate the exact solution (see, e.g., [20], [47], [24], [23]), or employ matrix functional integrators; see, e.g., [30], [28], [29]. In addition, large-scale advanced scientific applications often require function evaluations of matrices; see, e.g., [5], [43], [55], [17].…”

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confidence: 99%

A new investigation of the extended Krylov subspace method for matrix function evaluations

Knizhnerman

Simoncini

2010

Numerical Linear Algebra App

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Abstract. For large square matrices A and functions f , the numerical approximation of the action of f (A) to a vector v has received considerable attention in the last two decades. In this paper we investigate the Extended Krylov subspace method, a technique that was recently proposed to approximate f (A)v for A symmetric. We provide a new theoretical analysis of the method, which improves the original result for A symmetric, and gives a new estimate for A nonsymmetric. Numerical experiments confirm that the new error estimates correctly capture the linear asymptotic convergence rate of the approximation. By using recent algorithmic improvements, we also show that the method is computationally competitive with respect to other enhancement techniques.

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“…Since this problem amounts to solving an inhomogeneous, time dependent Schrödinger equation with time dependent Hamiltonian, a number of numerical methods are available in the literature. We follow the method described by Puzynin et al [18]. It is a higher-order stable operator-difference scheme, generalizing the Crank-Nicolson scheme.…”

Section: Methods Of Solutionmentioning

confidence: 99%

“…We actually used the 2nd order variant described in detail in Ref. [18]. In order to solve the equation…”

Section: Methods Of Solutionmentioning

confidence: 99%

Beyond integrability: Baryon-baryon backward scattering in the massive Gross-Neveu model

Thies

2017

Phys. Rev. D

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Due to integrability, baryon-baryon scattering in the massless Gross-Neveu model at large N features only forward elastic scattering. A bare mass term breaks integrability and is therefore expected to induce backward elastic scattering as well as inelastic reactions. We confirm these expectations by a study of baryon-baryon scattering in the massive Gross-Neveu model near the non-relativistic limit. This restriction enables us to solve the time-dependent Hartree-Fock equations with controlled approximations, using a combination of analytical methods from an effective field theory and the numerical solution of partial differential equations.

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Magnus-factorized method for numerical solving the time-dependent Schrödinger equation

Cited by 22 publications

References 0 publications

Accurate numerical solutions of the time-dependent Schrödinger equation

Accurate numerical solutions of the time-dependent Schrödinger equation

A new investigation of the extended Krylov subspace method for matrix function evaluations

Beyond integrability: Baryon-baryon backward scattering in the massive Gross-Neveu model

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