In this review we discuss techniques to solve numerically the time-dependent linear Schrödinger equation on unbounded domains. We present some recent approaches and describe alternative ideas pointing out the relations between these works

Antoine, Xavier

Arnold, Anton

Besse, Christophe

Ehrhardt, Matthias

Schädle, Achim

DepositOnce

Proceedings in Applied Mathematics and Mechanics, 1 October 2007A Review of Artificial Boundary Conditions for the Schr ödinger EquationXavier Antoine1,, Anton Arnold 2, ∗, Christophe Besse3,, Matthias Ehrhardt 4, ∗∗, andAchim Schädle5, ∗∗∗1 Institut Elie Cartan Nancy, Université Henri Poincaŕe Nancy 1, UMR 7502, INRIA Corida, Nancy-Université, France2 Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8, A–1040 Wien, Austria3 INRIA Futurs – Simpaf Team, Laboratoire Paul Painlevé UMR CNRS 8524 Univ. des Sciences et Technologies Lille, France4 Institut für Mathematik, MA 3–3, Technische Universität Berlin, Strasse des 17. Juni 136, D–10623 Berlin, Germany5 Zuse-Institut Berlin, Takustrasse 7, D–14195 Berlin, GermanyIn this review we discuss techniques to solve numerically the time–dependent linear Schrödinger equation on unboundeddomains. We present some recent approaches and describe alternative ideas pointing out the relations between these works.Copyright line will be provided by the publisher1 IntroductionThe equation under consideration is the 1D Schrödinger equation given on the unbounded domainΩ = {(x, t) ∈ R × R+}:i∂tu = −∂2xu+ V (x, t)u, x ∈ R, t > 0,u(x, 0) = uI(x), lim|x|→∞u(x, t) = 0,(1)whereV denotes a given real potential. We assume that the initial dat is compactly supported, i.e.supp(uI) ⊂ [xl, xr].Furthermore, we assume thatV is constant outside an interval[xl, xr], i.e.V (x) = Vl for x < xl, V (x) = Vr for x > xr.Equation (1) is one of the basic equations of quantum mechanis d it arises in many areas of physical and technologicalinterest, e.g. in quantum semiconductors, in electromagnetic wave propagation, and in seismic migration. The Schrödingerequation is the lowest order one-way approximation (paraxial wave equation) to the Helmholtz equation and is calledFresnelequation in optics , orstandard parabolic equation in underwater acoustics.If one wants to solve such a whole space evolution problem numerically, one has to restrict the computational (interior)domainΩint = {(x, t) ∈]xl, xr[×R+} by introducing artificial boundary conditions or absorbinglayers. Artificial boundaryconditions are constructed with the objective to approximate the exact solution of the whole–space problem, restricted toΩint.Such BCs are calledabsorbing boundary conditions (ABCs) if they yield a well–posed initial boundary value problem (IBVP),where some ‘energy functional’ is absorbed at the boundary.If this approximate solution actually coincides onΩint with theexact solution of the whole–space problem, one refers to these BCs astransparent boundary conditions (TBCs). While TBCsfor the Schr̈odinger equation are nonlocal in time (and space for multi-dimensional cases), it is often desirable to constructABCs that arelocal in space and/or time to allow for an efficient numerical implementation.2 Transparent boundary conditions for the Schr̈odinger equationHere we sketch the different ways of deriving transparent boundary conditions (TBCs). We start with the classical derivationof theanalytic TBC for the IBVP (1). Secondly, we shall mimic this procedurefor the time–discrete Schrödinger equation toderive temporally discrete TBCs. Note that the Schrödinger equation discretized in time is also the starting point f r TBCsbased on theperfectly matched layer method and thepole condition. Finally, we shall considerfully discrete TBCs.2.1 Analytic TBCsAnalytic TBCs for the Schr̈odinger equation (1) were independently derived by severalauthors from various application fieldsThey are non–local int and connect∂xv(xl,r, t) with v(xl,r, t). As a Dirichlet-to-Neumann (DtN) map they read∂nv(x, t) = −e−π4i√πe−iVl,rtddt∫ t0v(x, τ)eiVl,rτ√t− τ dτ atx = xl, xr, (2)wheren denotes the outwardly unit normal vector atxl, xr.∗ Research supported by the DFG under Grant-No. AR 277/3-3∗∗ Corresponding author, supported by the DFG Research CenterMATHEON “Mathematics for key technologies” in Berlin, E-mail: ehrhardt@math.tu-berlin.de, Phone: +0049 30 314 29294, Fax: +0049 30 314 29621∗∗∗ Supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin.Copyright line will be provided by the publisher2 Xavier Antoine et al.: A Review of ABCs for the Schrödinger EquationThese TBCs may be derived by the followinggeneral procedure. First, split original problem into coupled equations,interior and exterior problems. Secondly, apply a Laplace transformation in timet and solve the resulting ordinary differentialequations inx. Allow only ‘outgoing’ waves by selecting the decaying solution asx → ±∞. Finally, match Dirichlet andNeumann values atx = xl, x = xr. and apply the inverse Laplace transformation to obtain theTBCs (2) .2.2 Temporally discrete TBCsWe consider the problem (1) discretized uniformly in time with the step size∆t by thetrapezoidal ruleiun+1 − un∆t= −∂2xun+1 + un2+V n+1(x)un+1 + V n(x)un2, x ∈ R, ∀n ∈ N0,u0 = uI(x) given forx ∈ R, lim|x|→∞un(x) = 0, ∀n ∈ N0,(3)Instead of a Laplace transformation w.r.t.t o (1), we apply aZ-transformationZ(un) = û(z) := ∑∞n=0 un z−n, z ∈ C andobtain by the above described procedure:ivn+1 − vn∆t= −∂2xvn+1 + vn2+V n+1(x)vn+1 + V n(x)vn2, x ∈ Ω, ∀n ∈ N0,v0(x) = uI(x), x ∈ Ω,∂nvn+1 =n+1∑k=0ψ(l,r)k vn+1−k, atx = xl, xr,(4)where the weightsψ(l,r)n are given (in case of a vanishing potential) byψk = −e−iπ4√2√∆t(−1)kψ̃k, k ∈ N0, (ψ̃0, ψ̃1, ψ̃2, ψ̃3, ψ̃4, ψ̃5, . . . ) =(1, 1,12,12,1 · 32 · 4 ,1 · 32 · 4 , . . .). (5)2.3 Fully discrete TBCsFully discrete TBCs are obtained if we discretize (3) additionally in space, e.g. using the uniform gridxj = xl + j∆x, j ∈ Z:iun+1j − unj∆t= −D2xun+1j + unj2+V n+1j un+1j + Vnj unj2, j ∈ Z, n ∈ N0,lim|j|→∞unj = 0, n ∈ N0,u0j = uI(j∆x), j ∈ Z,(6)whereD2x denotes the standard second order difference quotient. Theright artificial boundary is located atxJ = xl +J∆x =xr and the left boundary atx0 = xl. The right discrete TBC is obtained analogously and reads (written as DtN map):unJ − unJ−1 = −n∑k=1sn−kukJ + un−1J−1, n ∈ N, (7)with the explicitly calculated convolution weights:sn = (−iR+ σ) δ0n + (1 + iR + σ) δ1n + γ e−inϕPn(µ) − Pn−2(µ)2n− 1 , ϕ = arctan2R(σ + 1)R2 − 2σ − σ2 ,σ = ∆x2Vr, γ = i4√(R2 + σ2) (R2 + (σ + 2)2) eiϕ/2, µ =R2 + 2σ + σ2√(R2 + σ2) (R2 + (σ + 2)2).(8)Pn denotes the Legendre polynomials (P−1 ≡ P−2 ≡ 0), δkn is the Kronecker symbol andR = ∆x2/∆t is the mesh ratio.3 Further ReadingFor a much more detailed description of the previous derivations and a concise discussion of the situation for multidimensionaland nonlinear (cubic) Schrödinger equations, efficient approximations and numericalex mples we refer the reader to [1].Moreover, there exists supplementary MATLAB software witha graphical user interface to compare the different approaches.References[1] X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, and A. Schädle, Preprint No. 18–2007 of the Institute of Mathematics,TU Berlin, May2007. (submitted to: Applied Numerical Mathematics) see:http://www.tbc-review.de.vuCopyright line will be provided by the publisher

A Review of Artificial Boundary Conditions for the Schrödinger Equation

https://depositonce.tu-berlin.de/bitstream/11303/16911/1/Report-37-2007.pdf

A Review of Artificial Boundary Conditions for the Schrödinger Equation

Abstract

Similar works

Full text

Available Versions

DepositOnce