High-order symplectic FDTD scheme for solving a time-dependent Schrödinger equation

Shen, Jing; Sha, Wei E. I.; Huang, Zhixiang; Chen, Mingsheng; Wu, Xianliang

doi:10.1016/j.cpc.2012.09.032

Cited by 21 publications

(19 citation statements)

References 28 publications

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“…In [19], when a is not a radical potential, using variational method, the authors studied the existences and multiplicity results for the following critical Schödinger equation (2) (−△) s u + a(P )u(P ) = 0…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Existence of weak solutions for the Schrödinger equation and its application

Huang¹

2019

Ann. Acad. Sci. Fenn. Math.

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In this paper, we are concerned with the existence of weak solutions for the Schrödinger equation with sign-changing potential in a smooth cone. For solving the Dirichlet boundary-value problem with respect to the Schrödinger operator, we prove the existence of at least one weak solution using changes of Schrödingerean harmonic measure, the energy estimate method and refined inequality technique. Due to the fact that the nonlinearity is allowed to change sign in our formulation, and the novelty of the boundary conditions, these results are new for discrete and arbitrary time scales. As an application, concentration results are also investigated.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Let (ϕ k , µ k ) be the eigenfunctions and eigenvalues of −△ in Ω with Dirichlet boundary data. According to [2, Lemmas 3.4 and 3.5], it holds (19) (…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Existence of weak solutions for the Schrödinger equation and its application

Huang¹

2019

Ann. Acad. Sci. Fenn. Math.

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“…So the S n (λ) requires 3 n/2 evaluations of S 2 , in the worst case. For completeness we note, that using the complex roots of (46) in (45) is also a viable approach in some numerical applications [59,61]. This scheme was already generalized for time-dependent Hamiltonians of the form H(t) = A(t) + B(t) in [60,62] as follows.…”

Section: Operator Splitting Formulaementioning

confidence: 99%

Fourth order real space solver for the time-dependent Schrödinger equation with singular Coulomb potential

Majorosi

Czirják

2016

Computer Physics Communications

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We present a novel numerical method and algorithm for the solution of the 3D axially symmetric timedependent Schrödinger equation in cylindrical coordinates, involving singular Coulomb potential terms besides a smooth time-dependent potential. We use fourth order finite difference real space discretization, with special formulae for the arising Neumann and Robin boundary conditions along the symmetry axis. Our propagation algorithm is based on merging the method of the split-operator approximation of the exponential operator with the implicit equations of second order cylindrical 2D Crank-Nicolson scheme. We call this method hybrid splitting scheme because it inherits both the speed of the split step finite difference schemes and the robustness of the full Crank-Nicolson scheme. Based on a thorough error analysis, we verified both the fourth order accuracy of the spatial discretization in the optimal spatial step size range, and the fourth order scaling with the time step in the case of proper high order expressions of the split-operator. We demonstrate the performance and high accuracy of our hybrid splitting scheme by simulating optical tunneling from a hydrogen atom due to a few-cycle laser pulse with linear polarization.

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“…For the Maxwell equations, many numerical methods, such as the finite-difference time-domain method has been developed [50][51][52]. For the Schrödinger equation, unitary algorithm has been proposed [49,[53][54][55]. Recently, a class of structure-preserving geometric algorithms have been developed for simulating classical particle-field interactions described by the Vlasov-Maxwell (VM) equations.…”

Section: Introductionmentioning

confidence: 99%

Canonical symplectic structure and structure-preserving geometric algorithms for Schrödinger–Maxwell systems

Chen

Qin

Liu

et al. 2017

Journal of Computational Physics

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An infinite dimensional canonical symplectic structure and structure-preserving geometric algorithms are developed for the photon-matter interactions described by the Schrödinger-Maxwell equations. The algorithms preserve the symplectic structure of the system and the unitary nature of the wavefunctions, and bound the energy error of the simulation for all time-steps. This new numerical capability enables us to carry out firstprinciple based simulation study of important photon-matter interactions, such as the high harmonic generation and stabilization of ionization, with long-term accuracy and fidelity.

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High-order symplectic FDTD scheme for solving a time-dependent Schrödinger equation

Cited by 21 publications

References 28 publications

Existence of weak solutions for the Schrödinger equation and its application

Existence of weak solutions for the Schrödinger equation and its application

Fourth order real space solver for the time-dependent Schrödinger equation with singular Coulomb potential

Canonical symplectic structure and structure-preserving geometric algorithms for Schrödinger–Maxwell systems

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