The research of the rst author was supported in part by an Information Technology

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1 Tecnical Report Absorbing Boundary Conditions for te Scrodinger Equation Toas Fevens Hong Jiang February 16, 1995 Te researc of te rst autor was supported in part by an Inforation Tecnology Researc Centre bursary. Researc of te second autor was supported in part by te Natural Sciences and Engineering Researc Council of Canada.

2 Abstract. A large nuber of dierential equation probles wic adit traveling waves ave very large (typically innite) naturally dened doains, wit boundary conditions dened at te doain boundary. To be able to nuerically solve tese probles in saller subdoains of te original doain, articial boundary conditions ust be dened for tese subdoains. One suc artical boundary conditions wic can iniize te size of suc subdoains are absorbing boundary conditions. A tecnique used to reduce te necessary spatial doain wen nuerically solving partial dierential equations tat adit traveling waves is te iposition of absorbing boundary conditions. Suc absorbing boundary conditions ave been extensively studied in te context of yperbolic wave equations. A general absorbing boundary condition will be developed for te Scrodinger equation wit one spatial diension, using group velocity considerations. Previously publised absorbing boundary conditions will be sown to reduce to special cases of tis absorbing boundary condition. Te well-posedness of te Initial Boundary Value Proble of te absorbing boundary condition, coupled to te interior Scrodinger equation, will also be discussed. Extension of te general absorbing boundary condition to iger spatial diensions is deonstrated. Nuerical siulations using initial single Gaussian, double Gaussian and Pseudo-delta function distributions will be given, wit coparision to exact solutions, to deonstrate te reectivity properties of various orders of te absorbing boundary condition. Key words. Scrodinger Equation, Absorbing Boundary Conditions, Radiation Boundary Conditions, Initial-Boundary Value Probles AMS subject classications. 35L35, 65M99, 35A40 2

3 Absorbing BCs for te Scrodinger Equation 3 1. Introduction. A large variety ofnuerical calculations involving te solutions to partial dierential equations require te iposition of articial boundary conditions to deliit te coputational doain to a anageable size. Tis often appens wen te natural doain for te proble being solved is innite and tus te natural boundary conditions for te proble are dened at innity. But if we desire te nuerical solution on only a nite section of te doain, te use of articial boundary conditions is necessitated. It is a requireent ofsuc articial boundary conditions to not adversely aect te nuerical calculation in te interior doain. Specically, we will consider probles were traveling waves are present. If standard Diriclet or Neuann boundary conditions are used for our articial boundary conditions, ten in any cases, a traveling wave evolved via a wave equation will view te boundary condition as an ipenetrable barrier and te wave would be copletely reected back into te interior doain. Obviously, tis boundary condition would not serve our purposes since te reected wave would disrupt te interior solution. Te only way tat suc a boundary condition could be used would be to place te boundary condition at a large distance fro te relevant interior solution, suc tat te reected wave would not eect te interior solution until a large nuber of tie steps (before wic te solution would be obtained). Tis approac would be costly for ulti-diensional probles or probles evolving over any tie steps. It would be preferable to use articial boundary conditions wic do not aect te interior solution but wic don't ave to be reoved to a large distance fro te relevant interior solution. Since te boundary condition ust be coupled wit te interior solution, te boundary condition ust be well-posed wit respect to te interior solution, and te boundary condition ust be stable, suc tat te nuerical solution will reain bounded. Also te articial boundary condition sould anniilate all incident waves suc as to produce no reections wic will ten propagate into te interior doain. Boundary conditions wic satisfy all tese conditions are called absorbing (or open, or radiation, or transparent) boundary conditions. Te use of absorbing boundary conditions allows for te nuerical solution of probles involving traveling waves wit a inial nuber of spatial points wile aintaining te accuracy desired for te solution. Tis can result in probles being solved ore quickly, and allow for te solution of ore coplex probles, especially in iger diensions. In tis paper, we will review te previous work tat as been done wit respect to absorbing boundary conditions for wave equations and siilar dierential equations. Ten we will introduce te Scrodinger equation for wic we will develop absorbing boundary conditions. We will discuss previously considered absorbing boundary conditions for te Scrodinger equation and ten derive a new absorbing boundary condition. We will sow tat te previously publised absorbing boundary conditions reduce to special cases of te new absorbing boundary condition. We will consider te well-posedness properties of te Initial Boundary Value Proble of te Scrodinger equation coupled to te absorbing boundary condition. Also, we will outline ow te general absorbing boundary condition can be extended iger diensional probles. Finally, we will use a nite dierence scee to solve te Scrodinger equation, and consider te properties of several nuerical siulations, using various orders of te absorbing boundary condition. 2. Review of Absorbing Boundary Conditions. In tis section, we will consider previous work tat as been done to devise absorbing boundary conditions for various wave equations. Absorbing boundary conditions ay be divided into

4 4 T. Fevens and H. Jiang boundary conditions for dispersive or non-dispersive equations. A dispersive equation is one tat adits plane wave solutions of te for e,i(!t,kx) ; and te speed of propagation of te wave is partially, or copletely, a function of te wave nuber k. For solutions for a given wave equation,! is a function of k and is called te dispersion relation for te dierential equation. Te dispersion relation allows us to dene te pase speed, c(k) =!(k) ;of individual waves, and te group velocity, k C(k) = d! (k) ofwave packets. Energy, for instance, travels wit group velocity. dk Altoug a dierential equation ay be non-dispersive (for exaple, te scalar wave equation) its discretization will nearly always be dispersive [41], so we will consider only dispersive equations. We will review work tat as been done to devise absorbing boundary conditions for particular dierential equations, exploiting tese properties and oter properties of dispersive equations Absorbing Boundary Conditions for Wave Equations. A fundaental requireent of an absorbing boundary condition is tat te interior solution tat is generated is close to te sae unique solution as tat produced if te boundary conditions were placed at a large distance (say, innity) fro te interior region. For interior scees involving traveling waves, ten te absorbing boundary condition ust ave te ability to absorb waves incident on it rater tan reecting te back into te interior of te doain Daping Regions. Te earliest approaces to developing suc boundaries used a narrow region extended past te required boundary were dissipation is added to te wave equation [25]. Ten te wave ipinging on te boundary is daped on te way into te region, reected by a conventional boundary condition at te end of te appended region, and furter daped on te way out. Te iniu widt of te region ust be te widt of several of te longest wavelengts to be effective. Terefore, tis etod can be costly in ters of space and tie requireents to ipleent, especially in iger diensional probles Soerfeld Radiation Boundary Condition. In 1949, Soerfeld dened te condition of radiation as \te sources ust be sources, not sinks of energy. Te energy wic is radiated fro tese sources ust scatter to innity; no energy ay be radiated fro innity into... te eld" [40,p.189], tus dening a \radiation boundary condition". Anuber of researcers ave put Soerfeld's absorbing boundary condition in ateatical for for wave =0; were u is te solution we are seeking in te interior of te coputational doain, c is soe eective pase velocity, and (1) is applied at te x = L rigt-and boundary. Anuber of approaces were considered to deterine te optial value for c to iniise reections. Pearson considered gravity wave propagation in stratied ow were c is a function of wavelengt [35]. Pearson suggested xing c to te Dopplersifted pase speed of te doinant vertical ode. Anoter approac was suggested by Orlanski of calculating c fro a point just witin te boundary at eac tie step, using a \oating pase velocity" approac [34]. \Floating" iplies tat te c used canges value wit respect to te easured pase speed of te incident waveatte boundary. Miller and Torpe expanded on tis \oating pase velocity" approac wit iger order approxiations to c [32] wereas Hedley and Yau used Orlanski's

5 Absorbing BCs for te Scrodinger Equation 5 original calculation of te oating pase velocity but added constraints on te value of c to avoid instabilities [13]. Te Soerfeld radiation boundary condition was expanded to two diensions by Rayond and Kuo wit + c c =0; were c x and c y are te x and y coponents of te pase velocity [36]. Also Lick et al. generalised te Soerfeld boundary condition to allow for partial reection and incoing disturbances (fro exterior to te doain) for wave equations in one [28] and two diensions [29]. Lindan considered a furter variation of te Soerfeld radiation boundary condition of te + c c x x on te x = 0 boundary were S is a source function wic generates waves into te doain and te n 's are correction functions to te absorbing boundary condition wic are functions of past data on te boundary, tailored to iniise reections for incident waves at dierent angles to te noral of te boundary [30]. Tis necessitates te updating of up to N functions on te boundary at eac tie step Engquist and Majda Approac. In teir paper [5], Engquist and Majda proposed a pseudo-dierential operator wic acts as a perfectly absorbing boundary condition for te scalar wave equation, 2 2 wit te related dispersion relation (4) NX 2 u 2 =0;! 2 = k 2 + l 2 : Te exact absorbing boundary condition is obtained by inverting tis dispersion relation to get an expression for k; (5) n ; k = p! 2, l 2 =!p 1, l 2 =! 2 : If te positive branc of tis equation is coosen and a apping is ade between te dual of a variable and its related dierential operator (via a Fourier transfor wic involves integrating over all possible values of te duals), te result is a pseudodierential equation wic applied to te x = L boundary wic would perfectly absorb all rigt-traveling waves ipinging on te boundary. But since te pseudodierential for of te absorbing boundary condition is non-local and tus not directly ipleentable in a nite-dierent scee, Engquist and Majda derive local approxiations to Equation (5) by expanding out te square root into ters of te Pade series, to various orders of accuracy. For exaple, using te approxiation q 1, l2 = 1 + O(! 2 l2 ) for te square root in Equation (5), and apping (via a! 2 Fourier transfor) te duals to teir respective dierential fors (6) uj x=0

6 6 T. Fevens and H. Jiang as a rst approxiation to te perfectly absorbing boundary condition wose sybol 1 is given in (5). Note te equivalence of tis absorbing boundary condition to tose of te Soerfeld radiation boundary condition (1) wit c =,1: A ierarcy of local absorbing boundary conditions ay derived using iger order approxiations. In a second paper, Engquist and Majda introduce a two-diensional version of teir approxiations based on an expression in te angle of te wave easured wit respect to te noral of te boundary [6]. Durran et al. [4] recently copared te oating pase velocity approac of Orlanski [34] and Hedley and Yau [13] wit tat of Engquist and Majda [5], and found tat te latter gave better results for te siulations of a one-diensional sallow-water ow odel and a two-level sallow-water odel. Clayton and Engquist consider absorbing boundary conditions for te acoustic wave equation. Tey consider an interpolation of te dispersion relation for te acoustic wave equation to develop rational expressions wic can be applied at te boundary [3]. Te use of a nuber of dierent interpolation points in te approxiation of te dispersion relation perits te better absorption of ore coplex ipinging waves coposed of a nuber of doinant pase velocities to te boundary as copared wit te Pade series using a single interpolation point, wic will ideally absorb only one coponent. Approxiating te duals (k x ;k z ;!)by teir corresponding dierential operators leads to an explicit dierential equation wic can be discretized and applied to te boundary, wit good results. Altoug Clayton and Engquist's approac allows for exible interpolation of te dispersion relation, teir solution lacks a general approac of derivation. Israeli and Orszag consider te idea of ixing daping regions wit absorbing boundary conditions [21]. Te daping regions act to reduce te aplitude of te outgoing waves as well as any waves tat are reected by te absorbing boundary condition at te end of te daping region. Altoug tis approac cobines te general reduction properties of te daping region, wit te ore specic eliination properties of te absorbing boundary conditions, but tere is still a trade-o wit respect to te extra grid points wic ave to be solved at eac tie step Bayliss and Turkel Approac. Bayliss and Turkel consider anoter approac todevelop ig order absorbing boundary conditions [1]. Bayliss and Turkel dene te following operator (7) were B = Y l=1 L + 2l, 1 r ; (8) @r : Ten it follows tat B p = O(1=r 2+1 ); leading te to dene te following absorbing boundary condition B p =0:Note tat tis boundary condition becoes asyptotically ore accurate as r!1:in two-diensional Cartesian coordinates, (8) as te for ;a2 +b 2 =1;a>0;analogous to te radiation boundary condition of Equation (2). 1 A sybol is te dual for of a dierential equation for te absorbing boundary condition, or equivalently, its dispersion relation.

7 Absorbing BCs for te Scrodinger Equation Canonical Absorbing Boundary Conditions. Higdon sowed tat te iger order approxiations to (5) wit teir accopanying substitutions of corresponding dierential etc.), ay be expressed in te following canonical for [14] [16], 2 (9) 4 p ujx=0 =0: Tis canonical for reduces to Engquist and Majda's boundary conditions, based on Pade approxiations, wen j =0:Furter, te relationsip between Engquist and Majda's iger-order approxiations and Bayliss and Turkel's general boundary condition (7) is revealed by te product for of (9). Boundary conditions of tis for were also derived independently by Keys [24]. Higdon was able to generalise Engquist and Majda's approxiations of te exact absorbing boundary condition (5) in two iportant ways. First, e sowed tat Engquist and Majda's approxiations could be factorized into rst order dierential operators, siilar to approxiation (6). Furter, e generalised te factors suc tat tey would anniilate waves incident on te boundary at any angle, rater tat optially at te noral. Tis ore general for greatly siplies ipleentation and stability analysis. Anoter approac, using group velocity, to derive absorbing boundary conditions, was proposed by Jiang and Wong [22]. Teir global absorbing boundary condition applies to any linear yperbolic equation wit constant coecients were te dispersion relation is known (for exaple, te wave equation or Klein-Gordon equation). Jiang and Wong's approac considers te group velocity, C(k), of te solution at te boundaries. Reeber tat te ow of energy propagates at te group velocity. Ifwe again consider te x = 0 boundary, any coponent of te solution wic as a positive group velocity would obviously be a coponent of a reected wave. Terefore, tis boundary condition can be expressed in te following anner, (10) C(k)j x=0 =,jc(k)j x=0 j : Unfortunately, like Engquist and Majda's perfectly absorbing boundary condition (5), tis absorbing boundary condition, wen apped into dierential for, is non-local, due to te absolute value function, and tus a rational approxiation is necessary. To do tis, Jiang and Wong use an approac siilar to tat utilized by Higdon [14]. First, a rst order approxiation is developed fro te exact boundary condition (10) by assuing tat te incident wave as a certain group velocity, b; wic is ten absorbed at te x = 0 boundary by (11) C(k)j x=0 + b =0: Let us consider te wave equation (3) wose dispersion relation is given in Equation (4). In tis case, d!(k; l) C(k) = = k dk! : Terefore, boundary condition (11) is equivalent to te sybol k +b! = 0; or reebering te corresponding dierential uj x=0

8 8 T. Fevens and H. Jiang Terefore, taking Higdon's lead [14], te canonical for of tis absorbing boundary condition is 2 Y 3 (13) 5 j ujx=0 j=0 Tis is equivalent to Higdon's canonical for if b j =1=cos j. As before, (13) is perfectly absorbing for incident waves wit group velocities b j. Te advantage of tis approac is all we need to know is te dispersion relation, and we can derive te absorbing boundary conditions to any order by using te group velocity. In [20], Higdon developed canonical radiation boundary conditions for te dispersive wave equation. Higdon sowed tat te perforance of te boundary condition was not sensitive to te coice of paraeters for te boundary conditon. Furter, Higdon sowed tat anoter absorbing boundary condition developed for te dispersive wavewould eiter be equivalent to is canonical for, unstable, or not optial in te sense te absorbing boundary condition could be odied, witout increasing its order, to ake it ore eective. Tus, if a canonical absorbing boundary condition is found, eiter troug using pase velocity, group velocity, or anoter tecnique, it will reduce to one ideal canonical for Oter Wave Equations. In te eld of Optics, Hadley considers a transparent boundary condition for bea propagation [10] [11]. Te proble of interest considers te propagation of a single scalar coponent of te radiation eld, E; wic Hadley assues to ave te for (14) E = E 0 e ikxx ; at te boundary were E 0 and k x are coplex. Fro te interior equation wit diraction being ignored, it is straigtforward to sow tat te ux of E leaving te interior is F (E) = Re (kx)je(b)j2 k ; were b is te value of x at te boundary. Hadley discretizes Equation (14) as E n j+1 = Ej n e ikxx were k x is calculated fro te previous n, 1 z spatial step (z and x being ortonoral directions). Tis approac is siilar in nature to te oating pase velocity approac used by Orlanski [34] for gravity wave propagation. For te Heloltz and Laplacian equations, Keller and Givoli devise non-local absorbing boundary conditions were te nite eleent etod is used to solve te dierential equations [23]. Tey sow tat te non-locality of te boundary conditions does not aect te banded structure of te nite eleent atrix wic ust be solved (analogous to a nite dierence iplicit scee) and te absorbing boundary condition is exact. Unfortunately, te non-local nature of te absorbing boundary condition proposed by Keller and Givoli restricts its practical application to nite eleent scees. Higdon, building on is canonical for of Engquist and Majda's absorbing boundary conditions, as considered applications wit acoustic and elastic waves in two and tree diensions [17] [18], and elastic waves in stratied edia [19]. Tese waves are coon in geopysical probles. Oter tan te particular details in te ipleentation and due to te nature of te particular proble, te underlying teory wit respect to absorbing boundary conditions is te sae as in Higdon's previous papers [14] [16].

9 Absorbing BCs for te Scrodinger Equation Oter Absorbing Boundary Conditions Tecniques. An eective etod to iprove te eciency of standard absorbing boundary conditions was developed by Mei and Fang for te solution of Maxwell's equations [31]. Tey call teir etod, \superabsorption". Tey consider te evolution of te coupled electrical and agnetic coponents of a transverse agnetic (TM) wave in two diensions, were te coponents of te wave are calculated on alternative alf-tie steps and alf a spatial step apart. By coparing te errors for te dierent coponents produced by absorbing boundary conditions for te individual coponents, and since te coponents are coupled, Mei and Fang ay eliinate te coon error, leading to a ore accurate recalculation of te individual coponents. In nuerical experients, Mei and Fang sow tat tis tecnique can iprove absorbing boundary conditions by about an order of agnitude less reection. An interesting alternate approac as been considered by Van Daalen et al. [44]. Tey derive absorbing boundary conditions witout assuing tat solutions are available beforeand and tus witout knowledge of any dispersion relation. Instead, tey consider te energy transission at te boundaries, considering continuous systes were te syste is governed by a Lagrangian density, L: Te evolution of te syste is given by applying te variation principle to te action integral (te integral of te Lagrangian density over te spatial and teporal doains). Te vanising of tis variation produces a \natural t xi n i =0;were " n is te local ux density in te noral direction to te boundary and te second ter is te partial derivative of te Lagrangian density in te noral direction. In a second paper, Broeze and Van Daalen [2] consider te two diensional wave equation (3) as an exaple. Since te group velocity (i.e., te local ux density on te boundary) is `oating' in te ipleentations, te approac ofvan Daalen and Broeze is closely related to te 'oating pase velocity' approac wit te Soerfeld radiation boundary condition Stability and Well-Posedness of Absorbing Boundary Conditions. Of course, if a boundary condition is unstable or generates spurious solutions, it is useless. Issues related to stability and well-posedness of absorbing boundary conditions ave been considered by a nuber of researcers. Te ain teoretical work describing te well-posedness of initial boundary value probles as been done by Kreiss [26], Sakaoto [38], and oters. Te well-posedness of absorbing boundary conditions for te wave equation in particular as been considered by Trefeten and Halpern [43]. Te well-posedness properties of particular absorbing boundary conditions are considered in [1] [5] [15] [17] [22] [37] by teir autors. Te well-posedness teory is closely related to te stability of nite dierence approxiations of initial boundary value probles for yperbolic equations. Te stability criterion for yperbolic initial boundary value probles is outlined by Gustafsson et al. [9], wit a group velocity interpretation of teir rater abstract criterion given by Trefeten [42]. Higdon considers te teory related to well-posedness of initial boundary value probles for linear rst-order yperbolic systes [15]. 3. New Absorbing Boundary Conditions. In tis Section, we will develop exible absorbing boundary conditions for Scrodinger equation.

10 10 T. Fevens and H. Jiang 3.1. Te Scrodinger Equation. Te following is te one-diensional Scrodinger equation, wit related dispersion relation 2 (x; t)=, (x; t)+v(x) (x; t); 2@x2 (16) 2 k 2 =2[!,V]; were is te ass of te particle, V (x) is te potential, is te wave-function, = =2 were is Planck's constant, and i = p,1: Te Scrodinger equation is a fundaental equation in te eld of quantu pysics. It is used to describe te propagation of a quantu particle, suc as an electron, in a potential background described by V (x): If V (x) = 0; ten te particle is oving in a vacuu. Te square of te wave function, j j 2 ; describes te probability distribution for te position of te particle Previous Absorbing Boundary Conditions. Anuberoftecniques ave been considered for boundary conditions wic would reove spurious reections fro articial boundaries during te nuerical solution of te one-diensional Scrodinger equation. Koslo and Koslo [25] used an enlarged coputational doain and ten applied a daping (or penalty) function in te articial part of te doain to decrease te aplitude of outgoing waves. Altoug tis etod can produce good results, te enlarged doain is costly, especially for extensions to iger diensions. A related approac was considered by Neuasuer and Baer [33], were tey added a negative coplex sort-range potential to te potential in te asyptotic region outside te coputational doain to construct nearly perfect absorbing boundary conditions. Work by Sibata [39] and Kuska [27], wic is based priarily on te work of Engquist and Majda [5] [6], as lead to one-way absorbing boundary conditions for te Scrodinger equation. Teir approac is to invert Equation (16) to obtain an expression for te sign and agnitude of te wave nuber, (17) k = p 2[!, V ]; were te positive value for te square root corresponds to waves traveling in an increasingly positive x direction, and eventually ipinging on te rigt-and boundary (x = L): To obtain te expression for waves traveling te opposite direction, siply substitute k wit,k: A boundary condition of te for (17) is an exact absorbing boundary condition siilar to Engquist and Majda's boundary condition in Equation (5). To see tis, consider all waves incident on te x = L boundary satisfying (17). Ten tere are no waves were k =, p 2[!, V ] at te boundary, and tus tere are no left-traveling (ence reected) waves. But due to te square root function, (17) can not be ipleented directly in pysical space, but rater (17) ust be put in rational dierential for, and tus into a nite dierence for, to be ipleented on te boundary. In order to develop a dierential equation to create a boundary condition transparent towaves leaving te doain, te rigt and side of Equation (17) ust be approxiated by a rational expression. In ters of te one-diensional Scrodinger equation, two rational expressions for te square root function ave been considered

11 Absorbing BCs for te Scrodinger Equation 11 in te literature. Te rst was developed by Sibata [39] and as te following for: (18) k = p 22, p p p 2 1 [!, V ]+ 2 21, 1 22 ; 2, 1 2, 1 were 1 and 2 are adjustable paraeters. Equation (18) is a straigt line interpolation of Equation (17), wic intersects te dispersion relation at two points. A second approxiation to Equation (17) was developed by Kuska [27] and as te for: (19) k =k 0 1+3z 3+z ; wit z =2(!, V )= 2 k 0 2 : His absorbing boundary condition is based on an expansion of 17) about te value k 0 : Equation (19) is essentially an approxiation to Equation (17) wic intersects te square root function at only one point. Copared to Sibata's approxiation (18), (19) is a iger order approxiation over te lengt of te dispersion relation, but is liited by te single interpolation point, and is tus less exible. Tese absorbing boundary conditions developed by Sibata and Kuska are eiter liited in eiter order of eectiveness or in exibility to absorb dierent energies of incident waves New Absorbing Boundary Condition. Here, we will consider an alternate approac to tat used by Sibata and Kuska. First, we will assue tat te potential is constant in tie, suc tat V = fnite set of constant valuesg =@t = 0; and is a constant value (V = K) in te vicinity of te boundaries. Terefore, fro te dispersion relation, Equation (17), we can calculate te group velocity, (20) = k : Tis gives te group velocity ofawave packet as it is evolved by te Scrodinger equation. Te following approac of using te group velocity todevelop absorbing boundary conditions was rst used by Jiang and Wong for yperbolic dierential equations [22]. Forawave traveling to te rigt witin te doain and ipinging on te x = L boundary, te group velocity fro (20) ust be positive, since te energy of te wave propagates at group velocity. Tis iplies tat te energy associated wit k is leaving te interior doain. A negative group velocity would ean tat energy is entering te interior doain and ence is a reected wave. Put in ateatical for, te sybol for te boundary condition as te following for at te x = L boundary, (21) k = k For te x = 0 boundary, siply replace k wit,k: Te pseudo-dierential boundary condition tat could be developed fro tis sybol is an exact absorbing boundary condition if satised on te boundary since all te group velocities on te boundary are positive (no spurious reections o te boundary). But, like Equation (17), tis boundary condition cannot be realized in pysical space by dierential operators due :

12 12 T. Fevens and H. Jiang to te absolute value function, and tus we ust use an approxiation to obtain an explicit rational dierential for wic can be applied on te boundary. Since a single dierential equation can only absorb waves of a certain group velocity, let us consider an approxiation to (21) of te for of (22) k a; on te boundary, were a is positive and real. Using te correspondence between te dual k and te partial derivative inx; we obtain te following dierential operator relation fro (22), + a If tis dierential equation is satised on te boundary, ten waves traveling to te rigt wit group velocity a would be absorbed copletely leading to no reections o te boundary fro tat coponent of te nuerical solution for te wave. But in general waves are coposed of ore tan one coponent wit dierent group velocities. So, a generalization of te operator in Equation (23) is + a l (24) =0: l=1 Here, te group velocity values, a l ; are real, fro Equations (17) and (20), assuing! V: For waves traveling to te left and ipinging on te x = 0 boundary, a l would be substituted wit,a l in equation (24). Te otivation for tis generalization coes fro a siilar generalization tat was considered by Higdon [14] wic was sown to correspond to te known absorbing boundary conditions developed by Engquist and Majda [5] and oters, for wave equations, as discussed in Capter 2. If a k 6= a l ; k6= l; ten te eect of te dierential equation (24), wen applied to te boundary, would be to copletely absorb p dierent coponents of te coputed wave solution wit p dierent group velocities, eac being absorbed to te rst order. If a k = a l ; k6= l; ten te eect of tis dierential equation, wen applied to te boundary, would be to copletely absorb te coponent of te coputed wave solution wit group velocity a j to te pt order. Eac a i = can be considered as an interpolation point of te dispersion relation at te value k in Equation (16). Terefore, in (24) te a i = interpolates (16) at dierent values of k wit p being te order of te interpolation. If all te group velocities a i are te sae, ten (24) is essentially a series expansion of (16) to te pt order about te point a i =: 3.4. Coparison wit Previous Work. Equation (24) is a general absorbing boundary condition fro wic it is possible to derive te specic absorbing boundary conditions presented by Sibata and Kuska. To see tis, rst consider p = 2:Ten Equation (24) as te sybol (were we ave replaced i@=@x wit,k); (25),k + a 1 =0:,k+ a 2 If we ultiply out (25) and solve for k; ten we obtain (26) k = =0: 2 a 1 + a 2 (!, V )+ a 1a 2 a 1 + a 2 :

13 Absorbing BCs for te Scrodinger Equation 13 Here we ave used Equation (16) to substitute for k 2 : Note tat Equation (26) is syetric in a j 's: To obtain te equation for left-going waves, replace te left-and side of Equation (26) wit,k: Sibata's relationsip in Equation (18) reduces to Equation (26) via te following substitutions, 1 = a and 2 = a : Altoug Kuska states tat 1 and 2 as used by Sibata are two \unpysical paraeters" [27], we ay attac eaning to te in tat tey are kinetic energy paraeters, were te kinetic energy is propagating at group velocities a 1 and a 2 : Now, consider p =3:Ten Equation (24) as te sybol (27),k + a 1,k+ a 2,k+ a 3 were te sae substitution for i@=@x was used. Again, using (16) to substitute for k 2 ; tis siplies to te relation =0; (28) were k = 2 1(!, V )+ 3 2(!, V )+ 2 ; 1 = (a 1 + a 2 + a 3 ) ; 1 2 = 2 a 1 a 2 a a 1 a 2 a 3 3 = 3 a 1 a 2 a 3 : Again, to obtain te equation for left-going waves, ultiply te rigt-and side of Equation (28) by,1: We ay obtain Kuska's sybol for is absorbing boundary condition (19) by letting a 1 = a 2 = a 3 = k0 : Terefore, Sibata's relationsip for k is equivalent to our second-order (p = 2) absorbing boundary condition and Kuska's relationsip for k is equivalent toa special case of our tird-order (p = 3) boundary condition. Kuska's special case absorbing boundary condition (19) would be expected to work well if te incident wave on te boundary was coposed oogeneously of only one k 0 coponent. Tis absorbing boundary condition would be expected to reove te k 0 coponent of te reected wave to te tird order. But, if te incident wavewas coposed of a nuber of dierent group velocity coponents, a ore general absorbing boundary condition would be needed, wic could be `tuned' to reove te doinant reected wave coponents. Oterwise, te coponents of te wave coposed of wave packets traveling wit dierent group velocities oter tan tat associated wit k 0 would reected to a large degree (tis will be quantied in te next Section on reection). Considering tat Kuska's absorbing boundary condition is siply te tird-order for of (24) wit identical a i 's; we can derive te fourt-order absorbing boundary condition for coparison. Note tat part of te otivation in te developent of previous two absorbing boundary conditions was to develop a boundary condition wic is rst order in te boundary variable of interest, wic isxis tis case, and possibly of iger order in te oter spatial and teporal variables. Hence, we obtain a boundary condition wic is a rst order dierential on te incident boundary, in order to obtain an \interior-pointing" nite dierence scee. Continuing tis ;

14 14 T. Fevens and H. Jiang approac, let p = 4 in Equation (24), replacing te partial derivative in x wit its corresponding wave vector, (29) were 4 2 (!, V ) 2 +2(,g 1 k + g 2 )(!, V ), g 3 k + g 4 =0; g 1 = (a 1 + a 2 + a 3 + a 4 ) ; g 2 = 2 (a 1 a 2 + a 1 a 3 + a 1 a 4 + a 2 a 3 + a 2 a 4 + a 3 a 4 ) ; 1 g 3 = 3 a 1 a 2 a 3 a ; a 1 a 2 a 3 a 4 g 4 = 4 a 1 a 2 a 3 a 4 : Again, we ave replaced k 2 wit its lower order equivalent fro Equation (16). If we let all te interpolation points be identical, a 1 = a 2 = a 3 = a 4 = k0, ten we obtain k = k 0 z2 +6z+1 (30) ; 4 z+1 were z =2(!, V )= 2 k0 2. Tis would be te p = 4 absorbing boundary condition equivalent to Kuska's absorbing boundary condition Expression for Reection. We can also calculate an expression for te aount of reection we can expect o te absorbing boundary condition (24). Consider a wave ipinging on te x = L boundary, wit a reected coponent, (31) =e,i(!t,kx) + re,i(!t+kx) : In tis expression for te wave, te rst ter is te incident wave, and te second ter is te reected wave wit r as te reection coecient. If we apply our absorbing boundary condition (23) to tis wave, we obtain k + a (32) B =,k+ a Terefore, te reection coecient is (33) i e,i(!t,kx) + r R=jrj=,k+ a k + a : i e,i(!t+kx) =0: Note tat jrj is always less tan one. Te general for of te reection for te full absorbing boundary condition (24) is (34) R = py l=1,k + al k + al Tis expression for te reection sows tat were a l = = k; te absorbing boundary condition (24) is perfectly absorbing since R = 0:Oterwise, jrj < 1 and all te incident coponents of te wave are reduced in aplitude in te reected wave, iplying absorption of te incident wave. To iniize te reection produced by te absorbing boundary condition, we can do two tings. Since jrj is less tan one, te larger te value of p in (34), te saller te value of te reection. Hence, we seek a iger order absorbing boundary condition, were feasible, to iniize te reection. Also, were te incident wave is coposed of several wave packets wit dierent group velocities, coosing a l to coincide wit te incident group velocities will decrease te value of te reection given by (34). :

15 Absorbing BCs for te Scrodinger Equation Well-Posedness of te New Absorbing Boundary Condition. Of course, it is also very iportant to sow tat tese boundary conditions given by Equation (24) generate well-posed initial boundary value probles, wen coupled wit te Scrodinger equation. Awell-posed proble is one tat does not adit eiter solutions wit exponentially growing aplitudes anywere in te doain, or spurious solutions generated fro te boundary. We are liited in ow uc we can deterine regarding te well-posedness of absorbing boundary conditions for te Scrodinger equation, since tere is no teoretical proof, to our knowledge, of te well-posedness for te Scrodinger equation. In te absence of any foral teory, we will adapt te well-posedness teory tat as been developed for te wave equation. We ust keep in ind tat tis ay, oray not, be a valid assuption, and ence our proof will proceed wit te assuption tat it is a valid approac. Te Kreiss condition [26] for wave equations states tat for well-posedness, tat te proble ust i) not adit any eigenvalues, and ii) tat tere are no generalized eigenvalues. Eigenvalues are tose coplex values s tat siultaneously satisfy bot te dispersion relation of te interior dierential equation and te sybol of te boundary condition, suc tat Re(s) > 0. If suc eigenvalues exist, ten te initial boundary value proble adits a noral ode e st : If eigenvalues are aditted by te boundary condition, ten te solution on te boundary will grow unboundedly, and ence be unstable. Generalized eigenvalues are coplex values s tat also satisfy te dispersion relation and te sybol of te boundary condition, but were Re(s) = 0 and te group velocity of te noral ode is 0(0) on te left-and (rigt-and) boundary. If tere are any generalized eigenvalues, ten te boundary condition will adit a spurious traveling wave solution wic will propagate energy into te interior doain. Following te exaple worked out by Engquist and Majda [5] for bot constant coecient and variable coecient wave equations, we use te general algebraic noral ode analysis for cecking well-posedness, specialized for te Scrodinger equation: Proposition 3.1. Te initial boundary value proble for te Scrodinger equation is well-posed if tere are no solutions to te frozen coecient alf-space probles of te for (35) ~ (s) =e,st, p 2 2 [is+v ]x ; wit Re s 0 (were s is coplex). Here te alf-space weareconsidering is x 0 wit te boundary at x =0:A solution were Res>0would be an eigenvalue, or a solution were Re s=0would be generalized eigenvalue. Note tat te dispersion relation for te interior solutions as been substituted for te wave nuber in (35). Te solutions would ave to satisfy (36) and (37) py l=1 (s) 2 ~(s) =, + 2 V a l ~ (s) =0: Specically, te criterion corresponds to waves ipinging on te left-and boundary, but te result olds equally for te rigt-and boundary. Applying te boundary

16 16 T. Fevens and H. Jiang condition (37), wit p = 1;to ~ yields te following result, r 2 (38), i 2 [is + V ], a 1 =0: Taking te second ter to te rigt and side, and squaring bot sides, leads to te following relation, s = i 2 a1 (39) + V : 2 Fro (39), it is obvious tat (40) Re s =0; since s is wolely iaginary. Tis iplies tat tere are no eigenvalues. Also, since te boundary condition is constructed suc tat te group velocity is 0(0) on te left-and (rigt-and) boundary, tere are also no generalized eigenvalues wic will propagate waves into te interior. But tere is a generalized eigenvalue wit zero group velocity wic reains on te boundary. Terefore, if tere are any instabilities wic igt be aditted by te generalized eigenvalue of te absorbing boundary condition, tey would not propagate into te interior of te solution, and tus tey will not aect te interior solution. Terefore, te boundary condition is well-posed wit te except of te zero group velocity generalized eigenvalue for p =1:To see tat te boundary condition is also well-posed for p>1;consider te product for of te boundary condition (37). Since we did not specify any particular value for a 1 ; it is obvious tat if tere were eigenvalues or generalized eigenvalues wit non-zero group velocities tat violated te above criterion, tey would ave appeared in te analysis for p =1:Since tey did not, te results old for all p and a i ; and te boundary condition is well posed for all orders of (24). Recall tat tis conclusion is tepered by te assuption tat te initial boundary value proble well-posedness teory developed for te wave equation can applied to te Scrodinger equation Higer Diensions. It is straigt-forward to extend te general absorbing boundary condition (24) to two or tree diensions. Consider te Scrodinger equation in two diensions, (x; y; t) i =, 2 (x; y; t) (x; y; t) 2 V (x; y) (x; y; t); were ; V (x; y); and (x; y; t) are dened as before. Te interior nuerical solution for te two diensional Scrodinger equation is given by Galbrait et al. [7]. Kuska developed a two diensional version of is boundary condition as a straigtforward extension of is one diensional version (19) [27]. Following essentially te sae procedure as tat used to derive (24), consider a two diensional plane wave of te for (42) (x; t) =e,i(!t,kxx,kyy) ; were! is te frequency of te wave and k x and k y are te wave vectors in te x and y directions, respectively. Equivalently, k x = k cos and k y = k sin were is te angle of te direction of k easured fro te noral (pointing away fro

17 Absorbing BCs for te Scrodinger Equation 17 te interior) of te x = a boundary (a can be 0 or L): Ten we ave te following dispersion relation (43) 2 k 2 x + 2 k 2 y=2[!,v]; assuing tat te potential V is a constant in te neigbourood of te boundaries. Tis gives us te following group velocities, (44) (C x ;C y kx y ; k y Terefore, on te x = L boundary, te two diensional version of Equation (22) is, using te sae arguent as in Section 3.3, (45) k x a x; were a x is te x coponent ofatwo diensional group velocity suc tat a x = a cos : Te angle is te direction of group velocity easured fro te noral of te x = L boundary, asabove. Since te corresponding dierential operator to k x is,i@=@x; our general absorbing boundary condition in two diensions becoes + a lcos l (46) =0: l=1 Actually, cos l can be absorbed by a l leaving a l as te only necessary paraeter. Again, for waves traveling to te left and ipinging on te x = 0 boundary, a l would siply be substituted by,a l in equation (46). As before, we can derive te expected reection produced by te two-diensional absorbing boundary condition. If we consider an incident wave of te for : (47) =e,i(!t,kcosx,ksin y) + re,i(!t+kcos x,ksin y) ; were r is te aplitude of te reected coponent, ten absorbing boundary condition (46) would generate te following reection coecient. (48) R = py l=1,kcos + alcos l alcos l kcos + : We would coose cos l and a l to iniize te reection coecient. Let us consider a practical ipleentation of tis two diensional absorbing boundary condition wit p = 3:Ten, in wave vector forat, Equation (46) takes te following for,kx + ax1,k x + a x2,k x + a x3 (49) =0: Tis siplies to te relation, wit te substitution of ters of k 2 x wit (43), (50) k x = 2~ 1 (!, V ), 2 ~ 1 k 2 y + ~ 3 2(!, V ), 2 k 2 y + ~ 2 ;

18 18 T. Fevens and H. Jiang were ~ 1 = (a x1 + a x2 + a x3 ) ; ~ 2 = 2 a x1 a x2 a x3 1 a x1 + 1 a x2 + 1 a x3 ; ~ 3 = 3 a x1 a x2 a x3 : In explicit dierential for, were we ave substituted for te duals in te sybol (50) wit teir respective dierential operators, tis absorbing boundary condition evaluated at te x = L boundary would ave te for (51), @ +i(2v, ~ ~ 2 @x +(2 ~ 1 V,~ 3 ) x=l If a xj =k 0x =; ten we recover te absorbing boundary condition given by Kuska in is equation (11), wit k 0x being dened as te x coponent of te initial wave vector k 0 [27]. 4. Nuerical Tests of Absorbing Boundary Conditions. In tis Section, we will a nite-dierence scee to test te eectiveness of various orders of te general absorbing boundary condition (24). Tis scee will used wit several initial distributions of i) a single Gaussian distribution odulating a traveling plane wave, ii) te su of two Gaussian distributions odulating two waves traveling at dierent initial group velocities, and iii) a pseudo-delta function approxiated by a Gaussian distribution wit sall initial spread odulating a single plane wave. Te aount of reection generated by te absorbing boundary conditions will be copared at te dierent orders to deterine teir relative eectiveness Scrodinger Equation Iplicit Interior Scee. For te nuerical results, an iplicit nite-dierence interior scee will be used to nuerically solve te Scrodinger equation. Te spatial doain of te nuerical solution of Equation (15) is x = x j = j; wit j 2 [0; J ]; were is te spatial es widt. Terefore, te left-ost boundary is x = 0 and te rigt-ost boundary is x = J = L: Siilarly, te tie variable as te following range, t = t n = n; wit n =0;1;2; :::;N: (x j ;T = N) are te last calculated wave-function values. We will discuss later ow tocoose N: Along te sae lines, te discretization of te wave function is n j = (x j;t n ): We will use following iplicit scee [8], n+1 j+1 +,2+ 4i (52), n j+1 +, 22 V j i + 22 V j 2 =0: n+1 j + n+1 j,1 = n j, n j,1 ; were = = 2 : Te discretization V j iplies te value V (x j ): Te iplicit scee is based on te calculation of te next tie step of te integration using only te previous tie step. Tis scee preserves te unitarity of te wave-function, as well as eet te von Neuann test requireent [12,p.102].

19 Absorbing BCs for te Scrodinger Equation Nuerical Scees for te Absorbing Boundary Condition. To nuerically ipleent te absorbing boundary conditions, we need to discretize te dierential operators to produce nite dierence scees. We will consider a nuber of orders (p) of te general absorbing boundary condition (24) Te p = 2 Absorbing Boundary Condition. Tis boundary condition is based on Equation (26). Tis boundary condition is equivalent to te absorbing boundary condition presented by Sibata [39] in Equation (18). Translated into explicit dierential for, te absorbing boundary condition becoes (53) i x, ic 1 t +(c 1 V(x),c 2 ) = 0; ) were (54) c 1 = 2 a 1 + a 2 ; c 2 = a 1a 2 a 1 + a 2 : Te positive sign on te rst ter refers te boundary condition applied to te x =0 boundary and te negative sign refers to te x = L boundary. Te following nite-dierence discretizations [27] will be used for te dierential operators in (53). (55) (56) (57) (Z + I) (J + I) 2 2 x (Z+I) (J,I) 2 t (J + I) 2 (Z, I) were te top sign of a double-signed ter refers to te x = 0 boundary condition and te botto sign refers to te x = L boundary condition (fro te negative and positive values for wave vector k; denoting left and rigt-traveling waves, respectively). Tis abbreviation convention will be used trougout. In te above, te following sift operators were used, J n j = n j+1 ; I n j = n j ; J, n j = n j,1 :Siilar sift operators will also be used for tie operations, Z n j = n+1 j ; Z, n j = n,1 j : Using tese discretizations in (53) yields te following p = 2 absorbing boundary condition, (58) i 2, ic (c 1V (0;J ), c 2 ) 4 +, i =, i + n j ; n j ; n j ; n+1 (1;J,1) 2, ic (c 1V (0;J ), c 2 ) 4 2, ic 1 2, (c 1V (0;J ), c 2 ) 4 i 2, ic 1 2, (c 1V (0;J ), c 2 ) 4 n+1 (0;J ) n (1;J,1) n (0;J ) : Te abbreviation for te wave function, n+1 ; denotes tat te discrete scee (1;J,1) uses te value n+1 1 on te x = 0 boundary, and n+1 J,1 on te x = L boundary.

20 20 T. Fevens and H. Jiang Te p = 3 Absorbing Boundary Condition. Tis boundary condition is based on Equation (28). A special case of tis boundary condition would yield te absorbing boundary condition considered by Kuska [27] in Equation (19). Rewritten in explicit dierential for, Equation (28) as te for (59) i 2 2, V (x) x 2 tx, i 1 t, 3 2, 1V (x) were i are dened in te previous Section. Te nite dierence discretizations given in Equations (55) to (57), along wit te following discretization [27], will be used for te dierential operators in (59). =0; (60) tx (Z,I) (J,I) n j : Tese discretizations applied to Equation (59) yield te following p = 3 absorbing boundary condition, a (61) = 2, b, c 2, d 4, a 2, b, c 2 + d 4 n+1 +, a (1;J,1) 2 + b, c 2, d 4 n (1;J,1) + a 2 + b, c 2 + d 4, were a = i 2, 2, V (0;J ) ;b= 2 ;c=i1 ;d= 3 2, 1V (0;J ) : n+1 (0;J ) n (0;J ) ; Te p = 4 Absorbing Boundary Condition. Equation (29) for p = 4 leads to te following dierential absorbing boundary condition, (62) p 1 tt p 2 tx + p 3 t p 4 x + p 5 =0; were p 1 =,4 2 2 ;p 2 =2g 1 2 ;p 3 =2ig 2, 8 2 iv (0;J ) ;p 4 =2ig 1 V (0;J ), ig 3 ;p 5 =4 2 (V (0;J ) ) 2, 2g 2 V (0;J ) + g 4 ; were g i are dened as before, in Section 3. To discretize tis dierential boundary condition, we will use (63) tt, (Z,,I) (Z,I) n j : along wit te discretizations given in Equations (55) to (57) and Equation (60). Te result of tese discretizations is in Equation (62) yields te following p = 4 absorbing boundary condition. (64) p2 + p p p 5 4 p2 = + p 3 2, p 4 2, p 5 4 +, p 1 n,1 ) 2 (0;J ) n+1 (1;J,1) + p1, p p 3 2, p p 5 4 n + (1;J,1) 2 p 1, p 2 + p 3 + p n+1 (0;J ) 2, p 5 4 n (0;J ) Note tat two previous tie levels are needed to calculate te subsequent tie level in (64)

21 Absorbing BCs for te Scrodinger Equation Nuerical Results. Te initial conditions used for all nuerical calculations is a Gaussian distribution (eiter singularly or in cobinations), (65) 0 j =e,(xj,)2 =2 0 2e ik0xj : In all te following calculation results, unless oterwise stated, =0:5 and =1: Also, te potential V (x) will be set to zero for all calculations. Wat does tis coice for initial conditions tell us about te applicability of te boundary conditions wit respect to copletely general initial conditions? We note tat any general intial conditions can be expressed in ters of a Fourier series, and a single Fourier ode is essentially a plane wave. Terefore, since te Gaussian distribution's carrier wave is a plane wave, if te boundary conditions are well beaved for various frequencies of plane waves in our exaples, ten te boundary condition would be expected to be well-beaved for any arbitrary coice of initial conditions wose Fourier odes are doinant at te sae frequencies Tests of te Reection Properties of te General Absorbing Boundary Condition. We will copare te relative properties, in ters of reection, of dierent orders of te general absorbing boundary condition (24, wit respect to eac oter and wit respect to te exact solution. Also, te eectiveness of te ore general for of (24) will be considered in coparison wit te publised absorbing boundary conditions of Sibata [39] and Kuska [27]. Te reection ratio r at t n was calculated as [27] (66) r = P J j=0 j n j j2 P J j=0 j 0 j j2 Tis r is siilar to te reection coecient jrj in (33). Here, r is te ratio of te integration of te squared aplitude of te reected wave-function over te initial wave-function (essentially, te ratio of te reected wave wit respect to te initial wave). Since our wave-functions are discrete, te integration is a suation over te doain. Wen te wave is copletely reected ten r =1;wereas if te wave is copletely absorbed, ten r = 0;after te initial wave as passed troug te absorbing boundary condition. To copare te dierent scees, we will plot te reection ratio as a function of tie. Tis etod of coparison is ost useful wen only one wave is passing troug a boundary at a tie. Wen no waves are passing troug eiter boundary and any traveling waves are presently only in te interior of te doain, ten te reection ratio as a function of tie is a plateau wose value easures te total wave aplitude reaining in te interior of te doain. Waves sootly passing troug a boundary are represented by a sootly decreasing reection ratio as a function of tie. As te waves present in te interior doain are diinised by passing troug te absorbing boundary conditions, r eventually goes to zero. Te reection ratio values tat we are priarily interested in are te idpoints of te rst plateau, wen te Gaussian distribution as passed troug te x = L boundary and any waves reected are still in te interior of te doain and ave not yet reaced te x = 0 boundary. Obviously, for ore arbitrary wave solutions, tis etod of coparison would not be adequate on sall doains, since we would not be able to tell wen te particular waves we are interested in are passing troug te boundaries.