Exponential integrators for stochastic Schr\"odinger equations driven by Ito noise

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by  Rikard Anton

EXPONENTIALINTEGRATORSFORSTOCHASTICSCHRÖDINGER
EQUATIONSDRIVENBYITONOISE
RIKARD ANTON∗AND DAVID COHEN†
Abstract. WestudyanexplicitexponentialschemeforthetimediscretisationofstochasticSchrödingerequa-
tionsdrivenbyadditiveormultiplicativeItonoise. Thenumericalschemeisshowntoconvergewithstrongorder
6 1ifthenoiseisadditiveandwithstrongorder1/2formultiplicativenoise. Inaddition,ifthenoiseisadditive,we
1 showthattheexactsolutionsofourproblemssatisfytraceformulasfortheexpectedmass,energy,andmomentum
0 (i.e.,lineardriftsinthesequantities).Furthermore,weinspectthebehaviourofthenumericalsolutionswithrespect
2 tothesetraceformulas.Severalnumericalsimulationsarepresentedandconfirmourtheoreticalresults.

n
Keywords. Stochasticpartial differential equations; StochasticSchrödinger equations; Numericalmethods;
a
Geometricnumericalintegration;Stochasticexponentialintegrators;Strongconvergence;Traceformulas
J

5
AMSsubjectclassifications.35Q55,60H15,65C20,65C30,65C50,65J08
2

] 1. Introduction. WeconsidertemporaldiscretisationsofnonlinearstochasticSchrödinger
A
equationsdrivenbyItonoise
N
. idu=D udt+F(x,u)dt+G(u)dW in Rd (0,¥ ),
h × (1.1)
at u(·,0)=u0 in Rd,
m
whereu=u(x,t),andi=√ 1. TheproductbetweenGanddW isofItotype,andfurther
[ −
details on F and G and on the dimension d will be specified later. The stochastic process

1 W(t) isasquareintegrablecomplex-valuedQ-Wienerprocesswithrespecttoanormal
t 0
v {filtratio}n≥ F onafilteredprobabilityspace(W ,F,P, F ). Theregularityoftheco-
t t 0 t t 0
3 varianceo{per}at≥orQwillbespecifiedlaterinthetext.Thein{itia}lv≥alueu isanF -measurable
2 0 0
complex-valuedfunction,whichwillbefurtherspecifiedbelow.
6
6 TheSchrödingerequationiswidelyusedwithinphysicsandtakesseveraldifferentforms
0 dependingonthesituation.Itisusedinhydrodynamics,nonlinearopticsandplasmaphysics
. toonlymentionafewareas.Incertainphysicalsituationsitmaybeappropriatetoincorporate
1
0 somekindofrandomnessintotheequation. Onepossibilityistoaddadrivingrandomforce
6 tothenobtainanequationoftheform(1.1). Seeforexample[12]andreferencesthereinfor
1 furtherdetails.
:
v StochasticSchrödingerequationshavereceivedmuchattentionfromamoretheoretical
i point of view during the last decades. Connected to the present article and without being
X
exhaustive,we mentionthe works[13, 14, 18] onIto problemsand[12, 14, 16, 30, 18] for
r
a theStratonovichsetting.
It is seldom possible to solve stochastic partial differentialequations exactly, and effi-
cientnumericalschemesarethereforeneeded.Forthetimeintegrationoftheabovestochastic
Schrödingerequations,wewillconsiderstochasticexponentialintegrators. Thesenumerical
methods are explicit and easy to implement, furthermore they offer good geometric prop-
erties. Exponential integrators are widely used and studied nowadays as witnessed by the
recentreview [23] for the time integrationof deterministicproblems. Applicationsof such
schemestothedeterministic(nonlinear)Schrödingerequationcanbefoundin,forexample,
Department of Mathematics and Mathematical Statistics, Umeå University, SE–901 87 Umeå, Sweden
∗
([email protected]).
†Department of Mathematics and Mathematical Statistics, Umeå University, SE–901 87 Umeå, Sweden
([email protected]). Department of Mathematics, University of Innsbruck, A–6020 Innsbruck, Austria
([email protected])
1
2 R.AntonandD.Cohen
[22,4,3,7,20,8,5]andreferencestherein. Thesenumericalmethodswererecentlyinvesti-
gatedforstochasticparabolicpartialdifferentialequationsin, forexample,[28,24,29]and
forthestochasticwaveequationsin[9,32,10,1].
We now review previous works on temporal discretisations of stochastic Schrödinger
equations. In [17] a Crank-Nicolson scheme is studied for the equation with nonlinearity
F(u). First order of convergence is obtained in the case of additive noise, and with mul-
tiplicative Ito noise the convergence rate is one half. Observe that this numerical scheme
is implicit. A stochastic Schrödinger equation with Stratonovich noise is considered in
[15], where, again, a Crank-Nicolson scheme is studied for the equation with nonlinearity
F(x,u)=l u2s u, with l = 1 and s >0. The authors prove convergence to the exact
| | ±
solution and mass preservation of the scheme. Further, in [26] a mass-preserving splitting
schemeforequation(1.1)withF(x,u)=V(x)uandG(u)=uisconsidered. Thenoiseisof
Stratonovichtype and first orderconvergenceis obtained. In [27],V(x) is replaced by u2
| |
and first orderconvergenceis againobtained. Still in the Stratonovichsetting, [25] derives
multi-symplecticschemesforstochasticSchrödingerequations. Wefinallymention[19,2],
inwhichthoroughnumericalsimulationsarepresentedforbothadditivenoiseandmultiplica-
tiveStratonovichnoise.
Inthepresentworkweshowthat
theexponentialintegratorappliedtothelinearstochasticSchrödingerequationwith
•
additivenoiseconvergesstronglywithorder1andsatisfiesexacttraceformulasfor
themass,theenergy,andforthemomentum;
theexponentialintegratorappliedtothestochasticSchrödingerequationwithamul-
•
tiplicativepotentialandadditivenoiseconvergeswithstrongorder1,buthasasmall
errorinthetraceformulasforthemassandenergy;
the exponential integrator applied to stochastic Schrödinger equations driven by
•
multiplicativeItonoisestronglyconvergeswithorder1/2.
Webegintheexpositionbyintroducingsomenotationsandusefulresultsthatwewilluse
inourproofs. Afterthatwewillfollowasimilarapproachasin[17]. Thatis,wewillbegin
byanalysingthenumericalmethodappliedtothelinearSchrödingerequationwithadditive
noise in Section 3. Then we study stochastic Schrödinger equations with a multiplicative
potential in Section 4 and finally we consider the stochastic Schrödinger equation with a
multiplicativepotentialandmultiplicativenoiseinSection5.Foreachoftheaboveproblems,
weanalysethespeedofconvergenceoftheexponentialmethods(inthestrongsense)andfor
additiveproblemswe show some trace formulas(such results couldbe interpretedas weak
error estimates). Various numerical experimentsaccompany the presentation and illustrate
the main properties of these exponential methods when applied to stochastic Schrödinger
equationsdrivenbyItonoise.
2. Notationsandsomeusefulresults. GiventwoseparableHilbertspacesH andH
1 2
with norms and respectively, we denote the space of bounded linear operators
fromH toHk·kbHy1L(Hk,·kHH2). WedenotebyL (H ,H )thesetofHilbert-Schmidtoperators
1 2 1 2 2 1 2
fromH toH withnorm
1 2
¥ 1/2
kF kL2(H1,H2):= k(cid:229)=1kF ekk2H2! ,
¥
where e isanyorthonormalbasisofH . Furthermore,forHilbertspacesH ,H ,and
H we{hakv}ek(=s1eetheproofofLemma2.1in[112])thatifS L (H ,H )andT L1(H2,H ),
3 2 1 2 2 3
thenTS L (H ,H )and ∈ ∈
2 1 3
∈
kTSkL2(H1,H3)≤kTkL(H2,H3)kSkL2(H1,H2). (2.1)
ExponentialintegratorsforstochasticSchrödingerequations 3
Also,weconsiderL2(Rd)thespaceofsquareintegrablefunctionsonRd withinnerproduct
(u,v)=Re uv¯dx. (2.2)
Rd
Z
For s R, we further denote the fractional Sobolev space of order s by Hs =Hs (Rd)
∈
withnorm s . Tomakethenotationscleanerandmorereadablewewillusethefollowing
k·k
shorternotations
L2=L2(Rd), L(H )=L(H ,H ), Ls =L (L2,Hs ).
1 1 1 2 2
Wenotethattheoperator iD ,appearinginthestochasticSchrödingerequation,isthegen-
eratorofasemigroupofis−ometriesofboundedlinearoperatorsS(t)=e itD . We willmake
−
useofthefollowingresult
LEMMA 2.1. (Seee.g.[26, Lemma 3.2])Considerany s 0 andS(t)=e−itD ,t 0,
s ≥ ≥
then,forw H ,onehas
∈
S(t)w s = w s ,
k k k k
and,forD w Hs andanyt 0,
∈ ≥
D (S(t)w) s = D w s , (S(t) I)w s t D w s .
k k k k k − k ≤ k k
Asaconsequenceoftheabovelemmaand(2.1),wehave,forF Ls +2andanyt 0,
∈ 2 ≥
(S(t) I)F Ls t F Ls +2.
k − k 2 ≤ k k 2
Wewillalsousethefollowingresult.
LEMMA 2.2. Foranys 0andt 0,wehave
≥ ≥
kS(t)−IkL(Hs +1,Hs )≤Ct1/2,
foraconstantC.
TheproofofthisresultisverysimilartotheproofofLemma2.1.
Finally,throughoutthepaper,C (andC ,C etc.) willdenotea genericconstantwhich
1 2
maychangefromlinetoline.
3. The linear Schrödinger equation with additive noise. In this section, we study
timediscretisationsofthelinearstochasticSchrödingerequation
idu D udt=dW in Rd (0,¥ ),
− × (3.1)
u(0)=u in Rd,
0
whereu isanF -measurablerandomvariableandthenoise W(t) isasquareintegrable
0 0 t 0
complex-valuedQ-Wienerprocesswithrespecttothefiltrati{on. In}t≥hissection,we haveno
restrictionsonthedimensiond.
GlobalexistenceanduniquenessinHs ,s 0,ofthesolutiontoequation(3.1)isguar-
anteed if u Hs a.s. and Q1/2 Ls . The≥proof follows the same line as the proof of
0 ∈ ∈ 2
Theorem7.4in[11]. Themildsolutionof(3.1)reads
t
u(t)=S(t)u i S(t r)dW(r),
0
− 0 −
Z
wherewerecallthenotationS(t)=e itD .
−
4 R.AntonandD.Cohen
We now consider the time integration of the above problem. This is done as follows.
LetT >0beafixedtimehorizonandN>0beaninteger. Wefirstdividetheinterval[0,T]
into subintervals 0=t <t <...<t <t =T of equal length k so that t =nk. An
0 1 N 1 N n
exponentialintegratorwithstepsizekis−nowderivedbyapproximatingtheabovestochastic
integral,inthemildsolution,attheleftendpoint.Wethusobtainanumericalapproximation
unoftheexactsolutionu(t )of(3.1):
n
un=S(k)un 1 iS(k)D Wn 1=S(t )u in(cid:229)−1 tj+1S(t t )dW(r), (3.2)
− − n 0 n j
− − j=0Ztj −
where D Wn 1 =Wn Wn 1 =W(t ) W(t ) denotes Wiener increments. We call this
− − n n 1
explicitnumericalme−thodanexponenti−alinteg−rator.
3.1. Errorestimates. Thissubsectionpresentsaresultontheerroroftheexponential
integrator(3.2)whenappliedtothelinearstochasticSchrödingerequation(3.1). Theseerror
estimatesaregiveninthenexttheorem.
THEOREM3.1. Lets 0,p N.Recallthatunisthenumericalapproximationgivenby
≥ ∈
theexponentialintegrator(3.2)oftheexactsolutionu(t )tothelinearstochasticSchrödinger
n
equationdrivenbyaQ-Wienerprocess(3.1). Assumethatu Hs a.s.,andQ1/2 Ls +2.
0∈ ∈ 2
ThenthereexistsaconstantCsuchthat
E n=m1,a..x.,Nkun−u(tn)k2s p ≤Ck2pkQ1/2k2Lp2s +2.
h i
Proof. Wehave
un u(t )=in(cid:229)−1 tj+1(S(t r) S(t t ))dW(r)
n n n j
− j=0Ztj − − −
tn
=i (S(t r) S(t [r/k]k))dW(r),
n n
0 − − −
Z
where[r/k] denotestheintegerpartofr/k. Thelast equalityis indeedcorrectsince ift
j
≤
r<t , then j r/k< j+1 so that [r/k]= j and [r/k]k= jk=t . Using Burkholder’s
j+1 j
≤
inequality[11,Lemma7.2],andLemma2.1,wehave
t
E max un u(tn) s2p E sup i (S(t r) S(t [r/k]k))dW(r) 2s p
n=1,...,Nk − k ≤ t [0,T]k Z0 − − − k
h i h∈ i
T p
CE (S([r/k]k r) I)Q1/2 2Ls dr
≤ 0 k − − k 2
(cid:20)(cid:18)Z (cid:19) (cid:21)
p
=CE N(cid:229) −1 tj+1 (S(tj r) I)Q1/2 2Ls dr
" j=0Ztj k − − k 2 ! #
p
CE N(cid:229) −1 tj+1 r t 2 Q1/2 2 dr
≤ " j=0Ztj | − j| k kL2s +2 ! #
Ck2p Q1/2 2p .
≤ k kLs +2
2
ExponentialintegratorsforstochasticSchrödingerequations 5
3.2. Atraceformulaforthemass. Underappropriateboundaryconditions,forexam-
ple periodic boundaryconditionsor homogeneousDirichlet boundaryconditions, the mass
(alsocalledL2-normordensity)
M(u):= u2dx
| |
Z
ofthedeterministiclinearSchrödingerequationi¶ u D u=0isaconservedquantity. Inthe
¶ t −
stochastic case, one immediately gets a trace formulafor the mass of the exact solution as
statedbelow.
PROPOSITION 3.2. AssumethattheinitialdataissuchthatE[M(u0)]isfiniteandthat
thecovarianceoperatorQ istrace-class. ThentheexactsolutionofthelinearSchrödinger
equationwithadditivenoise(3.1)satisfiesthetraceformulafortheexpectedmass
E[M(u(t))]=E[ u(t) 2 ]=E[M(u )]+tTr(Q) foralltimet.
k kL2 0
Proof.UsingLemma2.1andthefactthatthestochasticintegralsarenormallydistributed
withmean0,weget
t
E[ u(t) 2 ]=E S(t)u 2 + S(t)u , i S(t r)dW(r)
k kL2 k 0kL2 0 − 0 −
(cid:20) (cid:18) Z (cid:19)
t t
+ i S(t r)dW(r),S(t)u + S(t r)dW(r) 2
− 0 − 0 k 0 − kL2
(cid:18) Z (cid:19) Z (cid:21)
t
=E[ S(t)u 2 ]+ E[ S(t r)Q1/2 2 ]dr
k 0kL2 0 k − kL20
Z
=E[ u 2 ]+tTr(Q).
k 0kL2
REMARK 3.3. We would like to point out that, as in the case of the linear stochastic
waveequationtreatedin[9],analternativeproofoftheaboveresultcanbeobtainedusing
Ito’sformula(see, forexample[11,Theorem4.17]). Thisremarkisalsovalidfortheother
traceformulasgivenbelow.
Ourexponentialintegratordoesindeedsatisfythistraceformulaforthemassaswell,as
seeninthenextresult.
PROPOSITION 3.4. With the same assumptions as in Proposition 3.2, the stochastic
exponentialintegrator(3.2)satisfiesthefollowingtraceformulaforthemass
E[M(un)]=E[M(un 1)]+kTr(Q)
−
=E[M(u )]+t Tr(Q) forall t =nk.
0 n n
Proof. SimilarlytotheproofofProposition3.2,wehave
tn
E[M(un)]=E[M(un−1)]+E[kZtn 1S(k)dW(r)k2L2]
−
=E[M(un−1)]+kTr(Q).
Arecursionconcludestheproof.
WewouldliketoexaminethebehaviouroftheEuler-Maruyamascheme,thebackward
Euler-Maruyamascheme,andthemidpointrulewithrespecttothetraceformulaforthemass.
6 R.AntonandD.Cohen
Sincewewillconsiderperiodicdomainsandpseudospectraldiscretisationsinthenumerical
experimentspresentedbelow,wewillexaminethiscasefirst.
Apseudospectralspatialdiscretisationof(3.1)withanoisegivenbytherepresentation
W(x,t)= (cid:229) l 1/2b (t)e (x),
n n n
n Z
∈
where b n(t) n Z are i.i.d Brownian motions, l n are eigenvalues of Q, and en(x) n Z =
{√12p ei{nx}n∈Z}is∈anorthonormalbasisofL2(0,2p ),willleadtothefollowing{system}o∈fde-
coupledstochasticdifferentialequationsfortheFouriercoefficientsy oftheexactsolution
k
idy = k2y dt+l 1/2db .
k − k k k
Thismotivatesustoconsiderthescalartestproblem(withastandardBrownianmotionb and
realnumbersa,b)
idy=aydt+bdb (3.3)
andthequantityequivalenttothemassisthusthesecondmomentE[y2]. Wethushavethe
| |
traceformulafortheexactsolution
E[y(t)2]=E[y(0)2]+b2t foralltimes t.
| | | |
The following result states that the above classical numerical methods do not preserve the
trace formulafor thissimple test problem,andin particular, are notsuited whenappliedto
pseudospectraldiscretisationsoflinearstochasticSchrödingerequations.
PROPOSITION3.5. ConsiderapseudospectraldiscretisationofthestochasticSchrödinger
equation(3.1) with a trace-classnoise and periodic boundaryconditionsyielding to equa-
tionsoftheform(3.3). Wehavethefollowingresults:
1. TheEuler-Maruyamascheme
yn+1=yn ikayn ibD Wn
− −
producesasecondmomentthatgrowsexponentiallywithtime
E[yn 2] e(21ka2)tnE[y0 2] for tn=nk.
| | ≥ | |
2. ThebackwardEuler-Maruyamascheme
yn+1=yn ikayn+1 ibD Wn
− −
producesasecondmomentthatgrowsataslowerratethantheexactsolution
b2
E[yn 2] E[y0 2]+ forall n 0,
| | ≤ | | a2k ≥
hence lim E[yn 2]/t =0.
tn ¥ | | n
3. Themid→pointrule
(cid:0) (cid:1)
yn+1 yn yn+1+yn
i − a =bD Wn
k − 2
producesasecondmomentthatunderestimatethelineardriftoftheexactsolution
b2t
E[yn 2]=E[y0 2]+ n for t =nk.
| | | | 1+a2k2 n
2
ExponentialintegratorsforstochasticSchrödingerequations 7
Proof.Theproofofthispropositionisaneasyadaptationoftheresultspresentedin[31].
We start by looking at the behaviour of the Euler-Maruyama scheme and compute, using
propertiesoftheWienerincrements,
E[yn+1 2]=(1+a2k2)E[yn 2]+b2k (1+a2k2)E[yn 2] (1+a2k2)n+1E[y0 2]
| | | | ≥ | | ≥ | |
e(21ka2)tn+1E[y0 2].
≥ | |
ForthebackwardEuler-Maruyamaschemeweobtain,usingageometricseries,
1 b2k b2
E[yn+1 2]= E[yn 2]+b2k E[yn 2]+ E[y0 2]+ .
| | 1+a2k2 | | ≤ | | 1+a2k2 ≤ | | a2k
(cid:16) (cid:17)
Finally,forthemidpointrule,wehave
b2k b2t
E[yn+1 2]=E[yn 2]+ =E[y0 2]+ n+1 .
| | | | 1+a2k2 | | 1+a2k2
2 2
ForhomogeneousDirichletboundaryconditionsasimilarresultholds.
PROPOSITION 3.6. Consider the stochastic Schrödinger equation (3.1) with a trace-
classnoiseandhomogeneousDirichletboundaryconditions.Wehavethefollowingresults:
1. TheEuler-Maruyamascheme
un+1=un ikD un iD Wn
− −
producesanumericaltraceformulaforthemassthatgrowsexponentiallywithtime
E[M(un)] e(12kl1)tnE[M(u0)] for tn=nk,
≥
wherel denotesthesmallesteigenvalueoftheLaplacian.
1
2. ThebackwardEuler-Maruyamascheme
un+1=un ikD un+1 iD Wn
− −
producesa numericaltrace formula for the mass thatgrows ata slower rate than
theexactsolution
Tr(Q)
E[M(un)] E[M(u0)]+ , forall n 0,
≤ l 2k ≥
1
wherel isthesmallesteigenvalueoftheLaplacian.Thus lim E[M(un)]/t =0.
1 tn ¥ n
3. Themidpointrule[17] →
(cid:0) (cid:1)
un+1 un un+1+un
i − D =D Wn
k − 2
underestimatetheexpectedmass:
t
E[M(un)] E[M(u0)]+ n Tr(Q), forall n 0,
≤ 1+k2l1 ≥
4
wherel isthesmallesteigenvalueoftheLaplacian.
1
Proof.
8 R.AntonandD.Cohen
1. WestartbylookingatthebehaviouroftheEuler-Maruyamaschemeandcompute
E[M(un)]=E[M(un−1)]+k2E[kD un−1k2]+kTr(Q)≥E[M(un−1)]+k2l 12E[kun−1k2]+kTr(Q),
wherel denotesthesmallesteigenvalueoftheLaplacian.Hence,oneobtains
1
E[M(un)]≥(1+k2l 12)E[M(un−1)]≥(1+k2l 12)nE[M(u0)]≥e(21kl1)tnE[M(u0)].
2. ForthebackwardEuler-Maruyamaschemeweobtain
un+1+ikD un+1=un iD Wn.
−
Takingthenormandexpectationwehave
E[ un+1 2]+k2E[ D un+1 2]=E[ un 2]+E[ D Wn 2].
k k k k k k k k
Usingthattheeigenvaluesarepositiveandincreasing,andusingasimilarargument
asintheproofofProposition3.5,weget
1
E[ un+1 2] E[ un 2]+kTr(Q)
k k ≤ 1+k2l 2 k k
1
(cid:0) Tr(Q) (cid:1)
E[ u0 2]+ ,
≤ k k kl 2
1
wherel isthesmallesteigenvalueoftheLaplacian.
1
3. Let(l ,e )betheeigenpairsoftheLaplaceoperatorandlet(a ,f )betheeigenpairs
j j l l
ofthecovarianceoperatorQ. Writingu(t)=(cid:229) ¥ c (t)e wehavethattheexpected
j=1 j j
massisgivenby
¥
E[M(u(t))]= (cid:229) E[c (t)2].
j
| |
j=1
Themidpointrulethusbecomes
(cid:229)¥ 1+ikl j cn+1e = (cid:229)¥ 1 ikl j cne i(cid:229)¥ (cid:229)¥ a D b n(f ,e )e .
2 j j − 2 j j− l l l j j
j=1(cid:18) (cid:19) j=1(cid:18) (cid:19) j=1l=1
Thus,forevery j 1wehave
≥
1+ikl j cn+1= 1 ikl j cn i(cid:229)¥ a D b n(f ,e ).
2 j − 2 j− l l l j
(cid:18) (cid:19) (cid:18) (cid:19) l=1
Multiplyingbothsideswiththeconjugateandtakingtheexpectation,wehave
k2l 2 k2l 2 ¥
1+ j E[cn+1 2]= 1+ j E[cn 2]+(cid:229) a 2E[D b n 2](f ,e )2
4 ! | j | 4 ! | j| l=1| l| | l | | l j |
k2l 2 ¥
= 1+ j E[cn 2]+k(cid:229) a 2 (f ,e )2.
4 ! | j| l=1| l| | l j |
ExponentialintegratorsforstochasticSchrödingerequations 9
Usingthat l ¥ isapositiveandincreasingsequence,weget
{ j}j=1
¥
E[cn+1 2]=E[cn 2]+ k (cid:229) a 2 (f ,e )2
| j | | j| 1+k2l j l=1| l| | l j |
4
¥
=E[c0 2]+ tn+1 (cid:229) a 2 (f ,e )2
| j| 1+k2l j l=1| l| | l j |
4
¥
E[c0 2]+ tn+1 (cid:229) a 2 (f ,e )2.
≤ | j| 1+k24l1 l=1| l| | l j |
NowusingParseval’sidentity,wefinallyobtain
¥ ¥ ¥ ¥
E[M(un+1)]= (cid:229) E[cn+1 2] (cid:229) E[c0 2]+ tn+1 (cid:229) a 2(cid:229) (f ,e )2
j=0 | j | ≤ j=0 | j| 1+k24l1 l=1| l| j=0| l j |
¥ ¥
= (cid:229) E[c0 2]+ tn+1 (cid:229) a 2 f 2
j=0 | j| 1+k24l1 l=1| l| k ik
¥ ¥
= (cid:229) E[c0 2]+ tn+1 (cid:229) a 2
j=0 | j| 1+k24l1 l=1| l|
t
=E[M(u0)]+ n+1 Tr(Q).
1+k2l1
4
3.3. A trace formula for the energy. Again, under appropriate boundaryconditions,
forexampleperiodicboundaryconditionsorhomogeneousDirichletboundaryconditions,it
iswellknownthattheenergy
1
H(u(t)):= (cid:209) u2dx
2 | |
Z
of the deterministic linear Schrödinger equation i¶ u D u=0 remains constant along the
¶ t −
exactsolution,seeforexample[6]. Whenadditivenoiseisintroducedintotheproblem,we
getalineardriftintheexpectedvalueofthetotalenergyasstatedinthefollowingresult.
PROPOSITION 3.7. AssumethattheinitialdataissuchthatE[H(u0)]isfinite,andthat
(cid:209) Q1/2 is bounded. Then the exact solution of the linear Schrödinger equation with
L0
k k 2
additivenoise(3.1)satisfiesthetraceformulafortheexpectedenergy
t
E H(u(t)) =E H(u ) + Tr((cid:209) Q(cid:209) ) foralltime t.
0
2
(cid:2) (cid:3) (cid:2) (cid:3)
10 R.AntonandD.Cohen
Proof. Using Lemma 2.1 and the fact that the Ito integral is normallydistributed with
mean0,wehave
t
E[ (cid:209) u(t) 2 ]=E[ (cid:209) (S(t)u ) i (cid:209) S(t r)dW(r) 2 ]
k kL2 k 0 − 0 − kL2
Z
t
=E (cid:209) (S(t)u ) 2 + (cid:209) S(t r)dW(r) 2
k 0 kL2 k 0 − kL2
(cid:20) Z
t t
(cid:209) (S(t)u ),i (cid:209) S(t r)dW(r) i (cid:209) S(t r)dW(r),(cid:209) (S(t)u )
0 0
− 0 − − 0 −
(cid:18) Z (cid:19) (cid:18) Z (cid:19)(cid:21)
t
=E[ (cid:209) u 2 ]+ E[ (cid:209) (S(t r)Q1/2) 2 ]dr
k 0kL2 0 k − kL20
Z
=E[ (cid:209) u 2 ]+tTr((cid:209) Q(cid:209) ).
k 0kL2
Wenowshowthatourexponentialintegratorsatisfiestheverysameenergytraceformula
astheexactsolutiontothelinearstochasticSchrödingerequation(3.1).
PROPOSITION 3.8. With the same assumptions as in Proposition 3.7, the exponential
integrator(3.2)satisfiesthefollowingtraceformula
k
E H(un) =E H(un−1) + Tr((cid:209) Q(cid:209) )
2
(cid:2) (cid:3)=E(cid:2)H(u ) +(cid:3)tnTr((cid:209) Q(cid:209) ) forall t =nk.
0 n
2
(cid:2) (cid:3)
Proof. Similarlytotheproofofthepreviousproposition,wehave
1
E H(un) = E[ (cid:209) un 2 ]
2 k kL2
(cid:2) (cid:3) 1 tn
= E[ (cid:209) (S(k)un 1) i (cid:209) S(k)dW(r) 2 ]
2 k − − Ztn 1 kL2
1 1 tn−
= E[ (cid:209) un 1 2 ]+ E[ (cid:209) (S(k)Q1/2) 2 ]dr
2 k − kL2 2Ztn 1 k kL20
−
k
=E H(un−1) + Tr((cid:209) Q(cid:209) ).
2
(cid:2) (cid:3)
3.4. A formula for the momentum. In the deterministic case, one has an additional
conservedquantity,themomentum
p(u):=i (u(cid:209) u¯ u¯(cid:209) u)dx.
−
Z
Thenextresultinvestigatethebehaviourofthisquantityinthestochasticcase.
PROPOSITION 3.9. AssumethatE[p(u0)]<¥ andQ1/2∈L21. Thentheexactsolution
ofthelinearSchrödingerequationwithadditivenoise(3.1)exhibitsthefollowingformulafor
theexpectedmomentum
E[p(u(t))]=E[p(u )] 2tIm Q1/2,(cid:209) Q1/2 foralltime t.
0
− L0
2
D E