TY - JOUR

T1 - Efficient representation of nonreflecting boundary conditions for the time-dependent Schrödinger equation in two dimensions

AU - Jiang, Shidong

AU - Greengard, Leslie

PY - 2008/2

Y1 - 2008/2

N2 - We present a fast algorithm for the evaluation of exact, nonreflecting boundary conditions for the time-dependent Schrödinger equation in two dimensions on the unit circle. After separation of variables, the exact outgoing condition for each Fourier mode contains a nonlocal term that is a convolution integral in time. The kernel for that convolution is the inverse Laplace transform of the logarithmic derivative of a modified Bessel function, and the convolution integral can be split into two parts: a local part and a history part, which can be treated separately. The local part is easily handled by an appropriate quadrature. For the history part, we show that the convolution kernel can be well approximated by a sum of exponentials. Once such a representation is available, the convolution integrals can be evaluated recursively, reducing the cost from O (N2) work to O(N), where N is the number of time steps. The main technical development lies in the uniform rational approximation of the logarithmic derivative of the modified Bessel function Kv(√is).

AB - We present a fast algorithm for the evaluation of exact, nonreflecting boundary conditions for the time-dependent Schrödinger equation in two dimensions on the unit circle. After separation of variables, the exact outgoing condition for each Fourier mode contains a nonlocal term that is a convolution integral in time. The kernel for that convolution is the inverse Laplace transform of the logarithmic derivative of a modified Bessel function, and the convolution integral can be split into two parts: a local part and a history part, which can be treated separately. The local part is easily handled by an appropriate quadrature. For the history part, we show that the convolution kernel can be well approximated by a sum of exponentials. Once such a representation is available, the convolution integrals can be evaluated recursively, reducing the cost from O (N2) work to O(N), where N is the number of time steps. The main technical development lies in the uniform rational approximation of the logarithmic derivative of the modified Bessel function Kv(√is).

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U2 - 10.1002/cpa.20200

DO - 10.1002/cpa.20200

M3 - Article

AN - SCOPUS:38049144584

SN - 0010-3640

VL - 61

SP - 261

EP - 288

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

IS - 2

ER -