Spectral asymptotics for large skew symmetric perturbations of the harmonic oscillator
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Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator Isabelle Gallagher Institut de Mathematiques de Jussieu Universite de Paris 7 Case 7012, 2 place Jussieu 75251 Paris Cedex 05, France Thierry Gallay Institut Fourier Universite de Grenoble I BP 74 38402 Saint-Martin-d'Heres, France Francis Nier IRMAR Universite de Rennes 1 Campus de Beaulieu 35042 Rennes, France Abstract Originally motivated by a stability problem in Fluid Mechanics, we study the spectral and pseudospectral properties of the differential operator H = ?∂2x + x2 + i?1f(x) on L2(R), where f is a real-valued function and > 0 a small parameter. We define ?() as the infimum of the real part of the spectrum of H, and ?()?1 as the supremum of the norm of the resolvent of H along the imaginary axis. Under appropriate conditions on f , we show that both quantities ?(), ?() go to infinity as ? 0, and we give precise estimates of the growth rate of ?(). We also provide an example where ?() ?() if is small. Our main results are established using variational “hypocoercive” methods, localization techniques and semiclassical subelliptic estimates.
- course ?
- lumer-phillips theorem
- then there exists
- sup ??r
- self- adjoint operator
- z?? ?
- operator then
- self-adjoint operators
Spectralasymptoticsforlargeskew-symmetricperturbationsof
theharmonicoscillator
IsabelleGallagherThierryGallay
InstitutdeMathematiquesdeJussieuInstitutFourier
UniversitedeParis7UniversitedeGrenobleI
Case7012,2placeJussieuBP74
75251ParisCedex05,France38402Saint-Martin-d’Heres,France
gallagher@math.jussieu.frThierry.Gallay@ujf-grenoble.fr
FrancisNier
RAMRIUniversitedeRennes1
CampusdeBeaulieu
35042Rennes,France
Francis.Nier@univ-rennes1.fr
Abstract
OriginallymotivatedbyastabilityprobleminFluidMechanics,westudythespectral
andpseudospectralpropertiesofthedierentialoperator
H
=
∂
x
2
+
x
2
+
i
1
f
(
x
)on
L
2
(
R
),where
f
isareal-valuedfunctionand
>
0asmallparameter.Wedene(
)asthe
inmumoftherealpartofthespectrumof
H
,and(
)
1
asthesupremumofthenormof
theresolventof
H
alongtheimaginaryaxis.Underappropriateconditionson
f
,weshow
thatbothquantities(
),(
)gotoinnityas
→
0,andwegivepreciseestimatesofthe
growthrateof(
).Wealsoprovideanexamplewhere(
)
(
)if
issmall.Ourmain
resultsareestablishedusingvariational“hypocoercive”methods,localizationtechniquesand
semiclassicalsubellipticestimates.
1Introduction
InmanyevolutionequationsarisinginMathematicalPhysics,oneencountersthesituationwhere
thegeneratoroftheevolutioncanbewrittenasasumofadissipativeandaconservativeoperator
whichdonotcommutewitheachother.Insuchacasetheconservativetermcanaectand
sometimesenhancethedissipativeeectsortheregularizingpropertiesofthewholesystem.For
instance,ifthesystemhasagloballyattractingequilibrium,therateofconvergencetowards
thissteadystatecanstronglydependonthenatureandthesizeoftheconservativeterms.
Typicalexamplesillustratingsuchaninterplaybetweendiusionandtransportarethekinetic
Fokker-Planckequation[13],andtheBoltzmannequation[5];seealso[25]foracomprehensive
studyofthesephenomenaatamoreabstractlevel.
Inthispaperwestudyasimplelinearsystemwhichtsintothisgeneralframework.Given
asmallparameter
>
0andasmooth,boundedfunction
f
:
R
→
R
,weconsiderthedierential
operator
iH
=
∂
x
2
+
x
2
+
f
(
x
)
,x
∈
R
,
(1.1)
1
actingontheHilbertspace
X
=
L
2
(
R
),withdomain
D
=
{
u
∈
H
2
(
R
);
x
2
u
∈
L
2
(
R
)
}
.
Clearly
H
isabounded,skew-symmetricperturbationoftheharmonicoscillator
H
∞
=
∂
x
2
+
x
2
.
Ourgoalistocomputethedecayrateintimeofthesolutionstotheevolutionequation
ud=
H
u,u
(0)=
u
0
∈
X.
(1.2)
tdAsweshallsee,thesolutionsto(1.2)decaytozeroatleastlike
e
t
as
t
→
+
∞
,buttheactual
convergenceratestronglydependsonthevalueof
andthedetailedpropertiesof
f
,duetothe
interactionbetweenthesymmetric(dissipative)andtheskew-symmetric(conservative)partof
thegenerator
H
.
OurinitialmotivationforthisstudyisaspecicprobleminFluidMechanicswhichwenow
brieydescribe.Asisexplainedin[6,7],toinvestigatethelong-timebehaviorofsolutionstothe
two-dimensionalNavier-Stokesequation,itisconvenienttousethethevorticityformulation.In
self-similarvariables,thesystemreads:
1ω∂+
u
r
ω
=
ω
+
x
r
ω
+
ω,x
∈
R
2
,t
0
,
(1.3)
2t∂where
ω
(
x,t
)
∈
R
isthevorticitydistributionand
u
(
x,t
)
∈
R
2
isthedivergence-freevelocityeld
obtainedfrom
ω
viatheBiot-Savartlaw.Equation(1.3)hasafamilyofstationarysolutions,
2called
Oseenvortices
,oftheform
ω
=
G
where
G
(
x
)=(4
)
1
e
|
x
|
/
4
and
∈
R
isafree
parameter(thecirculationReynoldsnumber).Itturnsoutthatthelinearizationof(1.3)at
G
hasthesameformas(1.2),inthesensethatthegeneratorcanbewrittenasadierence
L
,where
L
isaself-adjointoperatorintheweightedspace
L
2
(
R
2
,G
1
d
x
)andisa
skew-symmetricperturbation.Theanalogygoesevenfurtherifweconjugate
L
withthe
Gaussianweight
G
1
/
2
andifweneglectanonlocal,lower-ordertermintheperturbation.The
linearizedoperatorthenbecomes
21x||261H
e
=
+
+
f
˜(
x
)
∂
,x
∈
R
2
,
(1.4)
2where
∂
=
x
1
∂
2
x
2
∂
1
and
f
˜(
x
)=(2
|
x
|
2
)
1
(1
e
|
x
|
/
4
).Theoperator
H
in(1.1)isaone-
dimensionalanalogof
H
e
,andthelimit
→
0correspondstothefastrotationlimit
→∞
.
Remarkthat,inthisparticularexample,thefunction
f
˜hasauniquecriticalpointlocatedat
theorigin,anddecreasestozerolike
|
x
|
2
as
|
x
|→∞
.
Theaimofthispaperistostudythespectralandpseudospectralpropertiesofthelinear
operator
H
inthelimit
→
0.Besidesthespecicmotivationsexplainedabove,thisquestion
hasitsowninterestfromamathematicalpointofview,andturnsouttoberelativelycomplex.
Wehavetodealwithanon-self-adjointproblemof(almost)semiclassicaltypewhichexhibitsa
competitionbetweenvariousmicrolocalmodelsatdierentscales,dependingonthestructure
ofthecriticalpointsandthedecayrateatinnityofthefunction
f
.Inparticular,unlikeinthe
self-adjointcase,oreveninsomenon-self-adjointproblemssuchasthekineticFokker-Planck
operator(see[10,13,14]),thepseudospectralestimatesarenotmonotonewithrespecttothe
imaginarypartofthespectralparameter.Nevertheless,themodel(1.1)issimpleenoughso
thattheanalysiscanbepushedquitefar,andwebelievethatourresultsgiveagoodideaof
thephenomenathatcanbeexpectedtooccurinmoregeneralsituations.Wealsomentionthat
thespectraltheoryofnon-self-adjointoperators,especiallyinthesemiclassicallimit,isatopic
ofcurrentinterest[24,3,4,23].
Westartwithafewbasicobservationsconcerningtheoperator
H
.Asiswellknownthelim-
itingoperator
H
∞
=
∂
x
2
+
x
2
isself-adjointin
L
2
(
R
)withcompactresolvent,anditsspectrum
2
