Quantum Ergodicity for a Class of Mixed Systems

1.1 MB·English

by  Jeffrey Galkowski

QUANTUM ERGODICITY FOR A CLASS OF MIXED SYSTEMS
JEFFREYGALKOWSKI
Abstract. WeconsidersetsofquasimodesfortheDirichletLaplacianonadomainwithboundary
2
wherethegeodesicflowexhibitsmixeddynamicalbehavior. Weassumethatthebilliardflowhas
1
aninvariantergodiccomponent,U,andstudydefectmeasures,µ,ofpositivedensitysequencesof
0
almostorthogonalquasimodes. Wedemonstratethatthesemeasureshaveµ| = cµ | whereµ
2 U LU L
istheLiouvillemeasure. Inordertodothis, weadaptquantumergodicityresultstothecaseof
t
c quasimodes. Finally,usingBunimovich’smushroombilliards[2],wegiveanexamplewhereour
O resultsapply.

7

]
h
p
ntroduction
- 1. I
h
t
a Thedistributionofeigenfunctionsoverphasespaceinthesemiclassicallimitisanimportant
m objectofstudyinthetheoryofquantumchaos. Thefundamentalresultisaquantumergodicity
[ theorem of Shnirelman [14], Zelditch [15], and Colin de Verdie`re [4] which states that, for clas-

sically ergodic systems, high energy eigenfunctions distribute uniformly in phase space. Since
3
v wewillstudydomainswithboundary,wenotethatZelditch-Zworski[16]generalizedquantum
8 ergodicitytothatcase.
6
9 Some progress has been made toward understanding semiclassical limits of eigenfunctions
2 for systems with dynamical behavior that is not completely ergodic. Marklof and O’Keefe, [8],
.
9 examineseparatedphasespacesforcertainmaps. Morerecently,MarklofandRudnick[9]have
0 madefurtherstridestowardsanunderstandingofquantumbehaviorformixedphasespace. In
2
particular, they prove that, for rational polygons, the eigenfunctions of the Dirichlet Laplacian
1
: equidistributeinconfigurationspace. Weuse[9]asinspirationforfurtherworkonsemiclassical
v
limitsforsystemswithmixeddynamicalbehavior.
i
X
Weexaminesystemswhosephasespaceshaveinvariantsubsets,U,suchthattheflowrestricted
ar to U is ergodic and the Liouville measure of U is positive. In particular, let (M,g) be a smooth
Riemannian manifold of dimension d with a piecewise smooth boundary, ∂M. Then M ⊂ M(cid:101),
whereM(cid:101)isamanifoldwithoutboundarytowhichgextendssmoothly,and∂M = (cid:83)J N where
j=1 j
N are smooth embedded hypersurfaces in M(cid:101). Define ∂oM ⊂ ∂M to be the open set of all points
j
wheretheboundaryissmooth. Thenthecomplement∂M\∂oMhasmeasurezero.
WeconsiderP(h) = −h2∆+VwithDirichletboundaryconditionsandletp(x,ξ)beitsprincipal
symbol. Sincep(x,ξ)issmoothuptotheboundary,wecanextenditsmoothlytoT∗M(cid:101). Weassume
that
(1.1) forE ∈ [a,b], dp|p−1(E) (cid:44) 0
1
2 J.GALKOWSKI
igure
F 1. WeshowtwobilliardstrajectoriesinaBunimovichmushroom[2]with
/
semicircular hat of radius 1 and centered base of width 1 2 and height 1. On the
left, we have a trajectory in the ergodic portion of phase space. On the right, we
haveoneintheintegrableportion.
and
(1.2) x ∈ ∂oM,V(x) = E ⇒ dV (cid:60) N∗∂oM.
x
sothatp−1(E)andT∗ Mintersecttransversally.
∂oM
Wethenwrite
p−1(E)∩T∗ M = Ω+(cid:116)Ω−(cid:116)Ω0,
∂oM E E E
where (x,ξ) lies in Ω+ if the vector H x ∈ TM(cid:101) points outside of M, in Ω− if it points inside M,
E p E
and Ω0 if it is tangent to ∂M. The set Ω0 contains the glancing covectors and, under (1.2), has
E E
measurezeroinp−1(E)∩T∗ M.
∂oM
We define the broken Hamiltonian flow as follows. (see, for example, [5, Appendix A]). We
denote this flow by ϕ := exp(H t). Assume without loss of generality that t > 0. We consider
t p
exp(tH )(x,ξ) defined on T∗M(cid:101), and let t be the first nonnegative time when exp(tH )(x,ξ) hits
p 0 p
theboundary. Ifthishappensatanon-smoothpointoftheboundary,orifexp(t H )(x,ξ) ∈ Ω0,
0 p E
then the flow cannot be extended past t = t . Otherwise, exp(t H )(x,ξ) ∈ Ω+ and there exists
0 0 p E
unique(x ,ξ ) ∈ Ω−suchthatthenaturalprojectionsofexp(t H )(x,ξ)and(x ,ξ )ontoT∗∂Mare
0 0 E 0 p 0 0
the same. We then define ϕ inductively, by putting ϕ (x,ξ) = exp(tH )(x,ξ) for 0 < t < t and
t t p 0
ϕt(x,ξ) = ϕt−t0(x0,ξ0)fort > t0.DefineforT > 0,theset
B ⊂ T∗M∩p−1([a,b])
T
to be the closed set of all (x,ξ) such that ϕ (x,ξ) intersects a glancing point for some t ∈ [−T,T].
t
Then,asshownin[16,Lemma1],B hasmeasurezeroinp−1(E).
T
QUANTUMERGODICITYFORACLASSOFMIXEDSYSTEMS 3
igure
F 2. We show two high energy eigenfunctions for the laplacian in a Buni-
movich mushroom [2]. The left hand eigenfunction spreads uniformly through
theergodicprotionofphasespace,andtherighthandeigenfunctionconcentrates
intheintegrableportion. (ImagescourtesyofA.BarnettandT.Betcke[1]).
The ease with which quasimodes can be constructed in examples motivates us to generalize
existing quantum ergodicity results to that setting. Although this is a natural generalization
of quantum ergodicity results, we are not aware of any reference in the literature. We present
this generalization in section 2. In order to do this, we make the following definition (with the
obviousanalogintheclassicalsettingundertherescalingh = λ−1).
Definition1. ApositivedensitysetofquasimodesforPon[a,b], 0 ≤ a < bisacollection{(ψ ,E ),j =
j j
1,...}satisfying,
(cid:107)ψ (cid:107) =
(1) 1
j L2
(2) (cid:107)(P−E )ψ (cid:107) = o(h2d+1)
j j L2
(3) |(cid:104)ψ ,ψ (cid:105)| = o(h2d) j (cid:44) k
j k
(4) |{E ∈ [a,b]}| ≥ ch−d, c > 0.
j
Remark: Noticethatthesetofalleigenfunctionsisapositivedensitysetofquasimodes. Hence,
allofourresultswillapplyinthiscase.
Underassumptions(1.1),(1.2),and
(1.3) ϕ isergodiconp−1(E), E ∈ [a,b]
t
4 J.GALKOWSKI
we have the following analog of quantum ergodicity for a positive density set of quasimodes
forP. Forasemiclassicalpseudodiffentialoperatoroforderm,wewriteB ∈ Ψm,anddenoteits
h
semiclassicalsymbol,σ (B)(see,forexample,[17,Section14.2]fordefinitions).
h
Theorem1. Supposethat(M,g)isacompactmanifoldwithapiecewisesmoothboundaryand(1.3)holds.
Let {(ψ ,E ),j = 1,...} be a positive density set of quasimodes of the Dirichlet realization of P on [a,b].
j j
Thenthereexistafamilyofsubsets
Λ(h) ⊂ {E |a ≤ E ≤ b}
j j
offulldensitysuchthatforallB ∈ Ψ0(Mo)withsymbol,σ (B),compactlysupportedawayfrom∂M,
h h
(cid:90)
(cid:104)Bψ ,ψ (cid:105) → − σ (B)dµ
j j h Ej
p−1(Ej)
forE ∈ Λ(h).
j
In section 3, we specialize to the case P = −h2∆ and pass from semiclassical to the classical
setting. For a classical pseudodifferential operator of order m, we write B ∈ Ψm and denote its
classicalsymbol,σ(B)(see,forexample,[6,Chapter18]fordefinitions).
Theorem 2. Suppose that (M,g) is a compact manifold with a piecewise smooth boundary and (1.3)
holds. Let{(ψ ,E ),j = 1,...}beapositivedensitysetofquasimodesoftheDirichletrealizationof−∆on
j j
[0,1]. Then, thereisafulldensitysubsequenceψ suchthatforallclassicalpseudodifferentialoperators
j
k
B ∈ Ψ0(Mo)withsymbolcompactlysupportedawayfrom∂M,
(cid:90)
(cid:104)Bψ ,ψ (cid:105) → − σ(B)dµ .
j j L
k k S∗M
In section 4, we continue to restrict our attention to the classical setting. But, instead of the
ergodicassumption(1.3),wemakethemixeddynamicalhypothesis
(1.4) ∃U ⊂ S∗M, µ (U) > 0, µ (∂U\U) = 0
L L
(1.5) ϕ isergodiconU, ϕ (U) = U.
t t
Inthiscase,wehavethefollowingresult
Theorem3. Supposethat(M,g)isacompactmanifoldwithapiecewisesmoothboundaryandthat(1.4),
(1.5)hold. Let{(ψ ,E ),j = 1,...}beapositivedensitysetofquasimodesoftheDirichletrealizationof−∆
j j
on [0,1]. Then there is a full density subsequence of ψ such that if any further subsequence ψ has a
j j
k
defectmeasure,µ,µ| ≡ cµ | .
U LU
ψ µ
Here,adefectmeasureforasequence isthelimitofthesemiclassicalmeasures associated
j j
ψ
to . (see,forexample,[17,Chapter5]fordetails)Intheboundarylesscase,Rivie´re[12]recently
j
provedatheoremonaccumulationpointsofsemiclassicalmeasureswhichgivessimilarresults
forseparatedphasespaces. Healsoprovidesexamplesofseparatedphasespacesforthegeodesic
flowonboundarylessmanifolds.
Remark: Hypotheses(1.4)includethecasewhereUisclosedwithpositivemeasureboundary.
QUANTUMERGODICITYFORACLASSOFMIXEDSYSTEMS 5
Finally, in section 5, we apply our results to the Dirichlet Laplacian for mushroom billiards.
Figure 1 shows two billiards trajectories, one in the ergodic region and one in the integrable re-
gion. Percival[13]madeconjecturesoneigenfunctionsthatwerenumericallyverifiedbyBarnett
andBetckein[1]inthecaseofmushroombilliards. Ourresultsproviderigorousproofsforsome
partsoftheseconjectures. (Foracompletedescriptionofthebilliardmaponmushroomssee[2].)
cknowledgemnts
A . The author would like to thank Maciej Zworski for valuable discussion,
Ste´phane Nonnenmacher and Zee´v Rudnick for pointing him toward mushroom billiards and
[9], to Semyon Dyatlov for advice on boundary value problem quantum ergodicity, and to
Alex Barnett for allowing him to use the images in Figure 2. The author is grateful to the
Erwin Schro¨dinger Institue for support during the Summer School on Quantum Chaos and to
the National Science Foundation for support under the National Science Foundation Graduate
Research Fellowship Grant No. DGE 1106400, and for support during the Erwin Schro¨dinger
Institute’sSummerSchoolonQuantumChaos.
uantum rgodicityfor uasimodes
2. Q E Q
Inthissection,wedemonstratehowtoadaptquantumergodicityresultsforeigenfunctionsto
positivedensitysetsofquasimodes. Wemakethefollowingdefinition.
Definition2. LetE ,j = 1,...bealltheeigenvaluesofPwithmultiplicity. Acompletesetofquasimodes
j
forPisacollection{ψ ,j = 1,...}satisfying
j
(cid:107)ψ (cid:107) = 1, (cid:107)(P−E )ψ (cid:107) = o(h2d+1), |(cid:104)ψ ,ψ (cid:105)| = o(h2d), j (cid:44) k.
j L2 j j L2 j k
We will need the following lemmas relating various spectral quantities for operators to their
correspondingexpressionsinvolvingcompletesetsofquasimodes.
Lemma 1. Let {u ,k = 1,...} be the eigenfunctions of P corresponding to E and let {ψ ,j = 1,...} be a
k k j
completesetofquasimodesforP. Then,
(cid:88)
u = c ψ +o(hd).
j jk k
|Ej−Ek|<Chd+1
Proof. First,observethat,nearanenergylevelE,thereexists(c (h),c (h))with|c (h)−c (h)| ≥ chd+1
1 2 1 2
suchthat((E−c (h),E−c (h))∪(E+c (h),E+c (h)))∩{E ,j = 1,...} = ∅–ifthiswerefalse,then
2 1 1 2 j
thespectrumofPwouldviolatetheWeyllaw.
Now,letΛ = {ψ |E ∈ (E−c (h),E+c (h))}andΠbethespectralprojectiononto[E−c (h),E+
j j 1 1 1
c (h)]. BytheWeyllaw,|Λ| ≤ Ch−d. Therefore,by[7,Proposition32.4],
1
Π Λ = |Λ|,
dim ( span( ))
since
o(h2d)+o(h2d+1)O(h−d−1) = o(hd).
Π = |Λ| Λ= Π
But,rank . Hence,span range andtheresultfollowsfromthealmostorthogonality
ψ (cid:3)
of .
j
6 J.GALKOWSKI
Lemma 2. Let A be a Hilbert-Schmidt operator and {ψ ,j = 1,...} be a complete set of quasimodes for P
j
anda < b. Then,
(cid:88)
(2.1) hd (cid:107)Aψj(cid:107)2 = hd(cid:107)Π[a,b]A(cid:107)2HS+o(1)
a≤Ej≤b
andifAisoftraceclass,
(cid:88)
(2.2) hd (cid:104)Aψj,ψj(cid:105) = hdTrΠ[a,b]A+o(1).
a≤Ej≤b
Proof. First,let(c ) = (cid:104)u ,ψ (cid:105). Then,byLemma1,
jk k j
(cid:88) (cid:88)
u = c ψ +o(hd) ψ = c u ,
j jk k j jk k
k k
whereallsumsaretakenoverksuchthat|E −E | < Chd+1. Observethat
k j
(cid:88) (cid:88)
uj = cjkψk+o(hd) = cjkckk(cid:48)uk(cid:48) +o(hd).
k k,k(cid:48)
Hence,bytheorthogonalityofu ,
j
(cid:88)
cjkckk(cid:48) = δkj+o(hd).
k
Now,
(cid:88) (cid:88) (cid:88)
hd (cid:104)Aψj,Aψj(cid:105) = hd cjkcjk(cid:48)(cid:104)Auk,Au(cid:48)k(cid:105) = hd (δkk(cid:48) +o(hd))(cid:104)Auk,Auk(cid:48)(cid:105)
a≤Ej≤b j,k,k(cid:48) k,k(cid:48)
(cid:88)
= hd(cid:107)Π[a,b]A(cid:107)2HS+hd o(hd)(cid:104)Auk,Auk(cid:48)(cid:105) = hd(cid:107)Π[a,b]A(cid:107)2HS+o(1)
k,k(cid:48)
wherethelastequalityfollowsfromthefactthatthereareatmostO(h−d)termsineachsum.
(cid:3)
Ananalgousargumentshowsthat(2.2)holds.
ToproveTheorem1,weneedthefollowinglemmasimilarto[5,LemmaA.2]
Lemma3. Letχ ∈ C∞(Mo)with0 ≤ χ ≤ 1.Thenfora(cid:48) < a < b < b(cid:48),
c
(cid:90) (cid:90)
(cid:88)
(2πh)d (1−χ)|ψj|2dVol ≤ (1−χ)dµσ+o(1)ash → 0.
Ej∈[a,b] M T∗M∩p−1([a(cid:48),b(cid:48)])
Proof. Thisfollowsdirectlyfrom[5,LemmaA.2]andLemma2onceweobservethat
(cid:90)
(cid:88) (cid:88)
(2πh)d (1−χ)|ψj|2dVol = (2πh)d (cid:104)(1−χ)ψj,ψj(cid:105) = (2πh)dTrΠ[a,b](1−χ)+o(1).
Ej∈[a,b] M Ej∈[a,b]
(cid:3)
QUANTUMERGODICITYFORACLASSOFMIXEDSYSTEMS 7
The following lemma allows us to take arbitrary collections of orthogonal quasimodes and
completethem. Hence,weonlyhavetoproveresultsforcompletesetsofquasimodesandthey
followforpositivedensitysets.
Lemma4. Let{(ψ ,E ),j = 1,...,J}beasetofalmostorthogonalquasimodeswith
j j
(cid:107)ψ (cid:107) = 1, (cid:107)(P−E )ψ (cid:107) = o(h2d+1), |(cid:104)ψ ,ψ (cid:105)| = o(h2d), j (cid:44) k.
j L2 j j L2 j k
Then,thereexists{ϕ ,k = 1,...K}suchthattheset{ϕ ,k = 1,...,K}∪{ψ ,j = 1,...,J}isacompletesetof
k k j
quasimodesforP.
Proof. Let u be eigenfunctions of P corresponding to E . Select an energy E. Then, by the
j j
proof of Lemma 1, there are gaps at distance c (h) of size hd+1 in the spectrum of P near E. Let
1
Λ = {u : |E −E| ≤ c (h)}. LetΛ(cid:48) := {ψ : |E −E| = o(hd+1)}. Then,forψ ∈ Λ(cid:48),wehavethat
k j 1 j j j
(cid:88)
ψ = c u .
j jk k
u ∈Λ
k
Now, define N := |Λ| and M = |Λ(cid:48)| letting b = (c ,c ,...,c ), j = 1,..M. Let {e ,...,e } be an
j j1 j2 jN 1 m
orthonormal basis for span ({b : j = 1,...,M}). Apply the Gram-Schmidt process to obtain an
j
orthonormalbasis{e1,...,eM,vM+1,...,vN}ofRN where
v = (v ,...,v ).
k k1 kN
Then,letting
(cid:88)
ϕ = v u , M+1 ≤ k ≤ N,
k kj j
j
{ψ ,...,ψ ,ϕ ,...,ϕ } Λ
1 M M+1 N is an almost orthonormal basis for span . Repeating this process for
(cid:3)
eachcluster,weobtainacompletesetofquasimodes.
Wealsoneedthefollowingrestatementofresultsin[3,Section4.3]thatisfoundin[5,Lemma
A1].
Lemma 5. Fix T > 0. Assume that A ∈ Ψcomp(Mo) is supported away from the boundary of M and
WF (A) ⊂ p−1([a,b])\B .Then,foreachχ ∈ C∞(Mo)andforeacht ∈ [−T,T],theoperatorχe−itP/hAisa
h T c
Fourierintegraloperatorsuppostedawayfrom∂Mandassociatedtotherestrictionofϕ toaneighborhood
t
of WFh(A)∩ϕ−t1(suppχ), plus an OL2→L2(h∞) remainder. The following version of Egorov’s Theorem
holds:
χeitP/hAe−itP/h = At,χ+O(h∞)L2(M)→L2(M),
whereAT,χ ∈ Ψcomp(Mo)issupportedawayfrom∂Mandσh(AT,χ) = χ(a◦ϕt).
Now,weproveTheorem1,following[5,AppendixA]
Proof. Let (ψ ,E ) be a positive density set of quasimodes. By Lemma 4, and the fact that our
j j
originalquasimodeshavepositivedensity,wemayassumewithoutlossthat(ψ ,E )isacomplete
j j
setofquasimodes(seedefinition2).
8 J.GALKOWSKI
Wefirstshowthat
(cid:88) (cid:12)(cid:12) (cid:90) (cid:12)(cid:12)
limsuphd (cid:12)(cid:12)(cid:104)Bψ ,ψ (cid:105)−− σ (B)dµ (cid:12)(cid:12) = 0.
h→0 Ej∈[a,b](cid:12)(cid:12) j j p−1(Ej) h Ej(cid:12)(cid:12)
Then, by a standard diagonal argument that can be found, for example, in [17, Theorem 15.5],
weextractthesetΛ(h).
Take a(cid:48),b(cid:48) such that a(cid:48) < a < b < b(cid:48) and the assumptions (1.2) and (1.1) hold for E ∈ [a(cid:48),b(cid:48)].
(If they do not hold when E (cid:60) [a,b], then we need to take a(cid:48),b(cid:48) getting close to a and b. e.g. Let
a(cid:48) = a−1/T andb(cid:48) = b+1/T.ThenestimatetheextracontributionsbytheWeyllaw.) Now,fixa
T > 0andchooseχ ∈ C∞(Mo)with0 ≤ χ ≤ 1and
T c T
(cid:90)
(1−χT)2dµσ ≤ T−1.
T∗M∩p−1([a(cid:48)−1,b(cid:48)−1])
Letψ ∈ C∞(a(cid:48)−1,b(cid:48)+1)have
c
(cid:90) (cid:90)
ψ(E) χ dµ = σ (B)dµ , E ∈ [a(cid:48),b(cid:48)].
T E h E
p−1(E) p−1(E)
Then, by Lemma 3, it is enough to show that the conclusion holds for B − ψ(P(h))χ , whose
T
symbolintegratesto0onp−1(E).Therefore,weassume,withoutloss,that
(cid:90)
σ (B)dµ = 0, E ∈ [a(cid:48),b(cid:48)].
h E
p−1(E)
Byellipticestimates, wemayassume thatWF (B) ⊂ p−1((a(cid:48),b(cid:48))). Morespecifically, B ∈ Ψcomp.
h
Thus,wecanwriteB = B(cid:48)T +B(cid:48)T(cid:48),whereWFh(B(cid:48)T)∩BT = ∅and(cid:107)σh(B(cid:48)T(cid:48))(cid:107)L2(p−1[a(cid:48),b(cid:48)]) ≤ T−1.
Then,byHo¨lder’sinequality,Lemma2and[5,Lemma2.2],wehavethat
 1/2  1/2
hd (cid:88) |(cid:104)B(cid:48)T(cid:48)ψj,ψj(cid:105)| ≤ Chd (cid:88) (cid:107)B(cid:48)T(cid:48)ψj(cid:107)2 = Chd (cid:88) (cid:107)B(cid:48)T(cid:48)uj(cid:107)2 +o(1)
Ej∈[a,b] Ej∈[a,b] Ej∈[a,b]
≤ C(cid:107)σh(B(cid:48)T(cid:48))(cid:107)L2(p−1[a(cid:48),b(cid:48)])+o(1).
(cid:48)(cid:48)
H(cid:48)ence,thecontributionofBT goesto0inthelimitlimT→∞limsuph→0 andwemayreplaceBby
B .
T
Now,byDuhamel’sformulaandtheunitarityofeitP/h,
(cid:90)
eitP/hψ = eitEj/hψ + i tei(t−s)P/ho(h2d+1) = eitEj/hψ +o (h2d).
j j j T
h
0
So,defining
(cid:90)
T
(cid:104)A(cid:105) := 1 eitP/hAe−itP/hdt,
T
T
0
QUANTUMERGODICITYFORACLASSOFMIXEDSYSTEMS 9
andusingLemma2andHo¨lder’sinequality,wehave
(cid:88) (cid:88) (cid:88)
hd |(cid:104)B(cid:48)ψ ,ψ (cid:105)| = hd |(cid:104)B(cid:48)e−itEj/hψ ,e−itEj/hψ (cid:105)| = hd |(cid:104)(cid:104)B(cid:48)(cid:105) ψ ,ψ (cid:105)|+o (h2d)
T j j T j j T T j j T
Ej∈[a,b] Ej∈[a,b] Ej∈[a,b]
 1/2  1/2
≤ hd (cid:88) ((cid:107)(cid:104)B(cid:48)T(cid:105)Tψj(cid:107)+oT(h2d))2 = hd (cid:88) (cid:107)(cid:104)B(cid:48)T(cid:105)Tuj(cid:107)2 +oT(1)
Ej∈[a,b] Ej∈[a,b]
Fromthispointforward,theproofisidenticaltothatin[5,TheoremA.2]. ByLemma5,(cid:104)B(cid:48)(cid:105) χ
T T T
is, up to an O(h∞)L2→L2 remainder, a pseudodifferential operator in Ψcomp compactly supported
insideMo andwithprincipalsymbol
(cid:90)
χ T
σ ((cid:104)B(cid:48)(cid:105) χ ) = T σ (B(cid:48))◦ϕ dt.
h T T T T h T t
0
Now,allthatremainstoshowis
(cid:88)
lim limsuphd (cid:107)(cid:104)B(cid:48)(cid:105) u (cid:107)2 = 0.
T→∞ h→0 Ej∈[a,b] T T j
Since,byLemma3,
(cid:88)
limsup(2πh)d (cid:107)(1−χ )u (cid:107)2 ≤ T−1,
T j L2
h→0 Ej∈[a,b]
wecanreplace(cid:104)B(cid:48)(cid:105) by(cid:104)B(cid:48)(cid:105)χ .Thus,by[5,Lemma2.2],itremainstoshowthat
T T T T
(2.3) Tl→im∞(cid:107)σh((cid:104)B(cid:48)T(cid:105)TχT)(cid:107)L2(p−1([a(cid:48),b(cid:48)])) ≤ Tl→im∞(cid:107)(cid:104)σh(B(cid:48)T)(cid:105)T(cid:107)L2(p−1([a(cid:48),b(cid:48)])) = 0.
Todothis,write
(cid:107)(cid:104)σh(B(cid:48)T)(cid:105)T(cid:107)L2(p−1(E)) ≤ (cid:107)(cid:104)σh(B)(cid:105)T(cid:107)L2(p−1(E))+(cid:107)(cid:104)σh(B(cid:48)T(cid:48))(cid:105)T(cid:107)L2(p−1(E)).
Thefirsttermontherightgoesto0whenT → ∞bythevonNeumannergodictheoremandthe
secondtermisboundedby(cid:107)σh(B(cid:48)T(cid:48))(cid:107)L2(p−1(E)) andhencealsogoesto0. (cid:3)
Remark: Notice that (2.3) is the only step in which the ergodicity of the flow is used. This will
beimportantwhenweadapttheresulttoergodicinvariantsubsetsofphasespace.
rom emiclassicalto lassical uantum rgodicity
3. F S C Q E
For completeness and to present the proof in the quasimode case, we will now pass from
Theorem1withP = −h2∆toTheorem2.
Proof. Leth2λ2j = Ejandλ = h−1,χ ∈ C∞χ(ξ) ≡ 0for|ξ| ≤ 1andχ ≡ 1for|ξ| ≥ 2,andχ(cid:15) = χ(ξ/(cid:15)).
Let Aˆ be a classical pseudodifferential operator of order 0 on M with symbol compactly
supportedawayfrom∂M. Definea := σ(Aˆ). Then,a ishomogeneousdegree0onT∗M. Hence
0 0
a0(x,D)χ(D) = a0(x,hD)χ(hD/h). Now,defineA(cid:15) ∈ Ψ0(Mo)by
A(cid:15) := a(x,hD)χ(cid:15)(hD).
10 J.GALKOWSKI
Theorem1gives,for0 < a < b,
(cid:88) (cid:12)(cid:12) (cid:90) (cid:12)(cid:12)
hdhλj∈[a,b](cid:12)(cid:12)(cid:12)(cid:12)(cid:104)A(cid:15)ψj,ψj(cid:105)−−p−1(Ej)σh(A(cid:15))dµEj(cid:12)(cid:12)(cid:12)(cid:12) → 0, h = λ−1 → 0.
But,sinceσh(A(cid:15)) = a0(x,ξ)χ(cid:15)(ξ)anda0 ishomogeneousdegree0,wehave
(cid:90) (cid:90)
− σh(A(cid:15))dµEj = − σ(Aˆ)dµL+O((cid:15)).
p−1(Ej) S∗M
Hence,weneedtoshow,
(cid:88)
limlimsuphd (cid:104)(Aˆ −A(cid:15))ψj,ψj(cid:105) = 0.
(cid:15)→0 h→0 hλj∈[a,b]
ByHo¨lder’sinequality,Lemma2,and[5,Lemma2.2],
 1/2  1/2
hd (cid:88) |(cid:104)(Aˆ −A(cid:15))ψj,ψj(cid:105)| ≤ Chd (cid:88) (cid:107)(Aˆ −A(cid:15))ψj(cid:107)2 = Chd (cid:88) (cid:107)(Aˆ −A(cid:15))uj(cid:107)2 +o(1)
hλj∈[a,b] hλj∈[a,b] hλj∈[a,b]
≤ C(cid:107)σh(Aˆ −A(cid:15))(cid:107)L2(p−1([a,b]))+o(1).
But,lim(cid:15)→0(cid:107)σh(Aˆ −A(cid:15))(cid:107)L2(p−1([a,b])) = 0.
δ >
Thus,forany 0,wehave
(cid:12) (cid:90) (cid:12)
(cid:88) (cid:12) (cid:12)
limsuphd (cid:12)(cid:12)(cid:12)(cid:104)Aˆψj,ψj(cid:105)−− σ(Aˆ)dµL(cid:12)(cid:12)(cid:12) = 0.
h→0 hλj∈[δ,1] S∗M
Allthatremainsistoshowthat
(cid:12) (cid:90) (cid:12)
(cid:88) (cid:12) (cid:12)
δli→m0limh→s0uphdhλj∈[0,δ)(cid:12)(cid:12)(cid:12)(cid:104)Aˆψj,ψj(cid:105)−−S∗Mσ(Aˆ)dµL(cid:12)(cid:12)(cid:12) = 0.
(cid:82)
But,lettinga¯ = − σ(Aˆ)dµ ,andapplyingtheWeyllaw,andHo¨lder’sinequality,
0 S∗M L
 1/2
hd (cid:88) |(cid:104)Aˆψj,ψj(cid:105)−a¯0| ≤ Chd (cid:88) (cid:107)(Aˆ −a¯0)ψj(cid:107)2 = O(δd)(cid:107)(Aˆ −a¯0)(cid:107)L2→L2 +Oδ(h).
hλj∈[0,δ) hλj∈[0,δ)
Onceagain,toobtainthefulldensitysubsequence,weemployastandarddiagonalargument.
(cid:3)
uantum rgodicityfor ubsets
4. Q E S
Now, we prove Theorem 3 using the fact that only step (2.3) uses the ergodicity of the flow.
Forcompleteness,weincludethefollowingelementarylemma.
Lemma 6. Let U satisfy (1.4), let µ be a finite measure on S∗M and suppose that for all a ∈ C∞(S∗M)
2 (cid:82) (cid:82)
compactly supported away from ∂M with adµ = 0, we have adµ = 0. Then, µ | ≡ cµ | for
U L U 2 2U LU
somec ≥ 0.