Near Sharp Strichartz estimates with loss in the presence of degenerate hyperbolic trapping

0.33 MB·English

by  Hans Christianson

NEAR SHARP STRICHARTZ ESTIMATES WITH LOSS IN THE
PRESENCE OF DEGENERATE HYPERBOLIC TRAPPING
2
1 HANSCHRISTIANSON
0
2
Abstract. We consider an n-dimensional spherically symmetric, asymptot-

n ically Euclidean manifold with two ends and a codimension 1 trapped set
a which isdegenerately hyperbolic. Byseparating variables and constructing a
J semiclassicalparametrixforatimescalepolynomiallybeyondEhrenfesttime,
2 we show that solutions to the linear Schro¨diner equation with initial condi-
1 tionslocalizedonasphericalharmonicsatisfyStrichartzestimateswithaloss
depending only on the dimension n and independent of the degeneracy. The
] Strichartzestimatesaresharpuptoanarbitraryβ>0loss. Thisisincontrast
P to [CW11], where it is shown that solutions satisfy a sharp local smoothing
A estimatewithlossdependingonlyonthedegeneracyofthetrappedset,inde-
pendent ofthedimension.
.
h
t
a
m
[ 1. Introduction

1 It is well known that there is an intricate interplay between the existence of
v
trapped geodesics, those which do not escape to spatialinfinity, and dispersive esti-
4
mates for the associatedquantum evolution. Trapping can occur in many different
6
4 ways, from a single trapped geodesic (see [Bur04,BZ04,Chr07,Chr11,Chr08b]),
2 to a thin fractal trapped set (see [NZ09,Chr09,Chr08a,Dat09]), to codimension 1
1. trapped sets in general relativity (see, for example, [BS03,BH08,DR09,MMTT10,
0 Luk10,TT11,LMar,WZ10] and the references therein), to elliptic trapped sets and
2 boundary value problems. Dispersive type estimates also come in many flavors,
1
but are all designed to express in some manner that the mass of a wave function
:
v tends to spreadout as the wavefunction evolves. Since the mass of wavefunctions
i tends to move along the geodesic flow, the presence of trapped geodesics suggests
X
some residual mass may not spread out, or may spread out more slowly than in
r
a the non-trapping case. In this paper, we concentrate on Strichartz estimates, and
exhibitaclassofmanifoldsforwhichweprovenear-sharpStrichartzestimateswith
a loss depending only on the dimension of the trapped set. This class of manifolds
has already been studied in [CW11], where a sharp local smoothing estimate is
obtained with a loss depending only on how flat the manifold is near the trapped
set. This presents an interesting dichotomy conjecture: “loss in local smoothing
depends only on the kind of trapping, while loss in Strichartz estimates depends
only on the dimension of trapping”.
The purpose of this paper is to study a very simple class of manifolds with a
hypersphere of trapped geodesics. If the dynamics near such a sphere are strictly
hyperbolic in the normal direction, then resolvent estimates are already obtained
in [WZ10] (see also [Chr08b,Chr08a]) which can be used to prove local smooth-
ing estimates with only a logarithmic loss. However, if the dynamics are only
weakly hyperbolic, resolventestimates and local smoothing estimates are obtained
1
2 CHRISTIANSON
in [CW11] with a sharp polynomial loss in both. We now turn our attention to
studying Strichartz estimates, which are mixed LpLq time-space estimates. The
typicalprocedureforprovingStrichartzestimatesistoconstructparametrices(ap-
proximatesolutions), whichencode how wavepacketsmovewith the geodesicflow.
For solutions of the Schr¨odinger equation, wave packets at higher frequency move
at a higher velocity, so the presence of trapping, or more precisely of conjugate
pointsmeansthatparametricescantypicallybe constructedonlyfortimeintervals
depending on the frequency of the wave packet. Then summing up many paramet-
rices to get an estimate on a fixed time scale leads to derivative loss in Strichartz
estimates.
However, if the trapped set is sufficiently thin and hyperbolic, we expect that
most of a wave packet still propagates away quickly, and a procedure developed
by Anantharaman [Ana08] allows one to exploit this to logarithmically extend the
timescale on which one can construct a parametrix leading to Strichartz estimates
with no loss [BGH10].
Forthe manifolds studiedinthispaper,the trappingis degeneratelyhyperbolic,
sowestillexpectsomemassofeachwavepacketto propagateaway,butata much
slower rate than the strictly hyperbolic case. As a consequence, we need to ex-
tend the parametrix polynomially in time to get sharp estimates. The techniques
in [Ana08,BGH10] will not work in this situation since the (h∞) estimate of de-
O
cayingcorrelationswill notcontrolthe exponential number of suchcorrelations. In
thispaper,wefailtoproveestimatesallthewaytothesharppolynomialtimescale,
but we are nevertheless able to extend the parametrix construction to the sharp
timescale up to an arbitrary β > 0 loss, which is expressed as a loss in deriva-
tive in the main theorem. Further, the technique of proof involves decomposing
the solution in terms of spherical harmonics in order to reduce the problem to a
1-dimensionalsemiclassicalparametrixconstruction. Lackingasquare-functiones-
timate forsphericalharmonics,the proofonly worksforinitialdatalocalizedalong
one spherical harmonic eigenspace. In this sense, the result shows more about the
natural semiclassical timescale, polynomially extended beyond Ehrenfest time, for
which we have good control of the rate of dispersion.
We begin by describing the geometry. We consider X = R Sn−1, equipped
x × θ
with the metric
g =dx2+A2(x)G ,
θ
where A ∞ is a smooth function, A > ǫ > 0 for some epsilon, and G is the
θ
metric on∈SnC−1. From this metric, we get the volume form
dVol=A(x)n−1dxdσ,
whereσ istheusualsurfacemeasureonthesphere. TheLaplace-Beltramioperator
acting on 0-forms is computed:
∆f =(∂x2+A−2∆Sn−1 +(n−1)A−1A′∂x)f,
where ∆Sn−1 is the (non-positive) Laplace-Beltramioperator on the sphere.
We study the case A(x) =(1+x2m)1/2m, m >2, in which case the manifold is
asymptotically Euclidean (with two ends), and has a trapped hypersphere at the
unique critical point x = 0. Since A(x) has a degenerate minimum at x = 0, the
trapped sphere is weakly normally hyperbolic in the sense that it is unstable and
isolated, but degenerate (see Figure 1).
STRICHARTZ ESTIMATES WITH LOSS 3
Our main theorem is the following, which expresses that a solution of the linear
homogeneous Schr¨odinger equation on this manifold satisfies Strichartz estimates
with loss depending only on the dimension n, up to an arbitrary β >0 loss.
Theorem 1. Suppose u solves
(D +∆)u=0,
t
(1.1)
(ut=0 =u0,
|
where u =H u is localized on the spherical harmonic subspace of order k. Then
0 k 0
for any T,β >0, there exists C >0 such that
T,β
(1.2) u 6C D (n−2)/pn+βu ,
Lp([0,T])Lq(M) T,β θ 0 L2(M)
k k kh i k
where
2 n n
+ = ,
p q 2
and 26q < if n=2.
∞
Remark 1.1. There are several important observations to make about Theorem
1. First, this theorem concerns endpoint Strichartz estimates. In dimension n>3,
if we take β < 1/n, the loss in derivative is then (n 2)/2n+ β < 1/2; that
−
is, the loss is always less than the loss following the argument of Burq-G´erard-
Tzvetkov [BGT04] (which gives a loss of 1/2 for endpoint estimates in n > 3).
Second, there is only a β >0 loss over the Euclidean (scale-invariant) estimates if
n=2, that is, if the trappedset is a single degenerateperiodic geodesic,so we can
get as close to the no-loss estimates as we like. We expect the β > 0 derivative
loss can actually be removed in all dimensions, but this is beyond our techniques.
Third,inalldimensions,the lossdepends only on the dimension of the trapped set.
It does not depend on m, the order of degeneracy of the trapping. This is in sharp
contrast to the local smoothing effect, which depends only on m, and not on the
dimension n (see [CW11] and below).
For dimensions n > 3, the estimate (1.2) is near sharp on natural semiclassical
time scales (see Corollary 3.3), in the sense that no better polynomial derivative
estimate can hold. In dimension n = 2, the same is true by comparing to the
scale-invariantcase.
Finally, since u is localized to a single spherical harmonic, the estimate in the
0
theorem can be rephrased, since
D (n−2)/pn+βu k (n−2)/pn+βu .
θ 0 L2(M) 0 L2(M)
kh i k ∼kh i k
Acknowledgements. TheauthorwouldliketothankNicolasBurq,KirilDatchev,
ColinGuillarmou,JeremyMarzuola,JasonMetcalfe,FabricePlanchon,andMichael
Taylorforhelpfulandstimulatingconversationsduringthepreparationofthisman-
uscript.
2. Reduction in dimension
Inthissectionweuseaseriesofknowntechniquesandestimatestoreducestudy
oftheSchr¨odinerequationonM tothestudyofasemiclassicalSchr¨odinerequation
on R with potential. The potential has a degenerate critical point, and we use a
technicalblow-upcalculus toconstructa sequenceofparametricesnearthe critical
point.
4 CHRISTIANSON
γ
Figure 1. A piece of the manifold X and the trapped sphere at
x=0.
Weobservethatwecanconjugate∆byanisometryofmetricspacesandseparate
variablesso that spectral analysisof ∆ is equivalent to a one-variablesemiclassical
problemwithpotential. Thatis,letT :L2(X,dVol) L2(X,dxdθ)betheisometry
→
given by
Tu(x,θ)=A(n−1)/2(x)u(x,θ).
Then ∆=T∆T−1 is essentially self-adjointon L2(X,dxdσ) for our choice of A. A
simple calculation gives
e
−∆f =(−∂x2−A−2(x)∆Sn−1 +V1(x))f,
where the potential
e
n 1 (n 1)(n 3)
V (x)= − A′′A−1+ − − (A′)2A−2.
1
2 4
Of course, conjugating the Laplacian by an L2 isometry does not necessarily
preserve Hs or Lq spaces.
Lemma 2.1. With the notation A(x)=(1+x2m)1/2m from above, for s>0,
Tu Hs(dxdσ) 6C u Hs(dVol),
k k k k
∆Sn−1 sTu L2(dxdσ) = ∆Sn−1 su L2(dVol),
kh− i k kh− i k
and for q >2,
u Lq(dVol) 6C Tu Lq(dxdσ).
k k k k
Proof. The result for 0 < s 6 1 follows from the L2 and H1 result, which follows
by observing that
(n 1)
∂ A(n−1)/2u=A(n−1)/2∂ u+ − A′A(n−3)/2u.
x x
2
But since A′ 6 CA, the L2(dxdσ) norm of ∂ Tu is bounded by the H1(dVol)
x
| |
norm of u.
The result for angular derivatives follows by commuting with A(n−1)/2(x).
STRICHARTZ ESTIMATES WITH LOSS 5
For the Lq result, we compute
u(x,θ)qA(n−1)(x)dxdσ = A(n−1)(1/q−1/2)Tu(x,θ)qdxdθ.
| | | |
Z Z
The function A(n−1)(1/q−1/2)(x) is bounded for q > 2, so the Lq inequality is true
as well. (cid:3)
As a consequence of this, to prove Theorem 1, it suffices to prove the following
Proposition, and apply Lemma 2.1 with v =Tu.
Proposition 2.2. Suppose v solves
(D +∆)v =0,
t
(2.1)
(v t=0 =v0,
|
e
where v = H v is localized on the spherical harmonic subspace of order k. Then
0 k 0
for any T >0 and β >0, there exists C >0 such that
T,β
(2.2) v 6C D (n−2)/pn+βv ,
Lp([0,T])Lq T,β θ 0 L2
k k kh i k
where
2 n n
+ = ,
p q 2
and 26q < if n=2.
∞
Wenowseparatevariablesbyprojectingontothekthsphericalharmoniceigenspace.
Thatis,let bethektheigenspaceofsphericalharmonics,sothatv implies
k k
H ∈H
−∆Sn−1v =λ2kv,
where
λ2 =k(k+n 2).
k −
Let H :L2(Sn−1) be the projector.
k k
→H
Sincev isassumedtosatisfyv =H v forsomekandtheconjugatedLaplacian
0 0 k 0
preserves spherical harmonic eigenspaces, we have also v = H v. Motivated by
k
spectral theory, we compute:
( ∆ λ2)v =P v,
k
− −
where
∂2 e
(2.3) P v =P H v =( +k(k+n 2)A−2(x)+V (x) λ2)v.
k k k −∂x2 − 1 −
Setting h = (k(k+n 2))−1/2 and rescaling, we have the one-dimensional semi-
−
classical operator
d2
P(z,h)ψ(x)=( h2 +V(x) z)ψ(x),
− dx2 −
where the potential is
V(x)=A−2(x)+h2V (x)
1
and the spectral parameter is z =h2λ2.
For our case where A(x) = (1 + x2m)1/2m, the subpotential h2V is seen to
1
be lower order in both the semiclassical and scattering sense. Furthermore, the
principal potential A−2(x) is even, smooth, decays like x−2 at and has a
±∞
unique degenerate maximum of the form 1 x2m at x=0.
−
6 CHRISTIANSON
3. Endpoint Strichartz estimates
Before proceeding to the endpoint Strichartz estimates, let us briefly recall the
localsmoothingestimateswhichwilleventuallyallowustogluetogetherStrichartz
estimates on semiclassical timescales.
3.1. Local smoothing estimates. Inthis subsection,werecallthe localsmooth-
ing estimates from [CW11], as well as the dual versions which we will use in this
paper.
Theorem 2 ( [CW11]). Let V(x) = A−2(x)+h2V (x) as above. Then for any
1
T >0, there exists a constant C =C >0 such that
T
T
k|x|m−1hxi−m−1−3/2eit(−∂x2+h−2V)u0k2L2dt6Chku0k2L2,
Z0
and
T
khxi−3/2eit(−∂x2+h−2V)u0k2L2dt6Ch1/(m+1)ku0k2L2.
Z0
The dual versions of these estimates are given in the following Corollary.
Corollary 3.1. Let V(x)=A−2(x)+h2V (x) as above. Then for any T >0, there
1
exists a constant C =C >0 such that
T
2
T
(cid:13)(cid:13)Z0 |x|m−1hxi−m−1−3/2e−it(−∂x2+h−2V)fdt(cid:13)(cid:13)L2 6Chkfk2L2TL2,
(cid:13) (cid:13)
and (cid:13) (cid:13)
(cid:13) 2 (cid:13)
T
(cid:13)(cid:13)Z0 hxi−3/2e−it(−∂x2+h−2V)fdt(cid:13)(cid:13)L2 6Ch(1−m)/(m+1)kfk2L2.
(cid:13) (cid:13)
Thepurp(cid:13)oseoftheseresultsistodemons(cid:13)tratethatthereisperfect1/2derivative
(cid:13) (cid:13)
localsmoothingawayfromx=0,orlocalsmoothingwitheitheralossinderivative
or with a vanishing multiplier at x=0.
3.2. TheendpointStrichartzestimates. TheendpointStrichartzestimatesare
the L2L2⋆1 estimates, where 2⋆ is the Strichartz dual:
T
n n
1+ = ,
2⋆ 2
which implies 2⋆ =2n/(n 2) for n>3, and 2⋆ = if n=2.
We wantto estimate v i−nL2⋆(M), whichwedou∞sing the followingestimatedue
to Sogge [Sog86]:
Theorem 3 ( [Sog86]). Let (M,g) be a d-dimensional compact Riemannian man-
ifold without boundary, and let ∆ be the Laplace-Beltrami operator on M. If ϕ
j
−
are the eigenfunctions,
∆ϕ =λ2ϕ
− j j j
with 0=λ 6λ 6 the eigenvalues, then
1 2
···
ϕ 6Cλ(d−1)/2(d+1) ϕ .
k jkL2(d+1)/(d−1) j k jkL2
1Throughout this manuscript, we use the notation LpLq = Lp([0,T])Lq(M) to denote the
T
localintime,globalinspaceStrichartznorm.
STRICHARTZ ESTIMATES WITH LOSS 7
In particular, for the situation at hand,
kvkL2⋆(R)L2⋆(Sn−1) =kHkvkL2⋆(R)L2⋆(Sn−1)
6Ck(n−2)/2nkHkvkL2⋆(R)L2(Sn−1).
Now let Λ be an index set for the kth harmonic subspace and write
k k
H
v(t,x,θ)= v (t,x)H (θ),
lk lk
lX∈Λk
whereH aretheorthonormalsphericalharmonicsin . Nowifp>2,26q 62∗
lk k
H
(q < if n=2), we have
∞
kvkLpTLq(M) 6Ck(n−2)/2nkvkLpTLq(R)L2(Sn−1)
1/2
6Ck(n−2)/2n v 2 ,
(cid:13)(cid:13)(cid:13) lX∈Λk| lk| ! (cid:13)(cid:13)(cid:13)LpLq(R)
(cid:13) (cid:13) T
byPlancherel’stheorem. We furtherestim(cid:13)ate usingMinko(cid:13)wski’sinequalityrepeat-
(cid:13) (cid:13)
edly:
1/2 q/2 2/q p/2 1/p
T
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) lX∈Λk|vlk|2! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)LpTLq(R) =Z0 TZ Xl |vlk|2! 2/dqxp/2  1/dpt
6C v qdx dt
lk
Z0 " l (cid:18)Z | | (cid:19) # 
X
 p 1/p
1/2
T
=C v 2 dt
Z0  l k lkkLq!  
X
  p/2 2/p 1/2
T
=C v 2 dt
Z0 l k lkkLq!  
X
 
 2/p 1/2 
T
6C v p dt
 l Z0 k lkkLq ! 
X
 1/2 
=C v 2 .
k lkkLpTLq!
lX∈Λk
All told then we have
1/2
(3.1) kvkLpTLq(M) 6Ck(n−2)/2n kvlkk2LpTLq! ,
lX∈Λk
where v (t,x)= v(t,x, ),H .
lk h · lkiL2(Sn−1)
Using (2.3), we see that v satisfies the equation (D +P )v = 0, which is
lk t k lk
a 1-dimensional Schr¨odinger equation with potential. We want to estimate v in
lk
the L2L2⋆(R) norm when n > 3, and in LpLq Strichartz duals with 2 6 q <
T T ∞
8 CHRISTIANSON
in dimension n = 2. However, since we are now looking at a solution to a one
dimensionalSchr¨odingerequation,2and2⋆ arenotStrichartzdualsin1dimension.
The Strichartz dual p to 2⋆ =2n/(n 2) (n>3) in one dimension satisfies
−
2 1 1
+ = ,
p 2⋆ 2
or
p=2n.
WethereforefirstuseHo¨lder’sinequalityint,withweightsnandn/(n 1)respec-
−
tively to get
T
v 2 = v 2 dt
k lkkL2TL2⋆ Z0 k lkkL2⋆
6T(n−1)/n v 2 .
k lkkL2TnL2⋆
In dimension n = 2, we use p > 2, 2 6 q < and the same weights in Ho¨lder’s
∞
inequality to get
T
v 2 = v 2 dt
k lkkLpTLq Z0 k lkkL2⋆
6T1/p v 2 .
k lkkL2TnL2⋆
We have the following proposition.
Proposition 3.2. Suppose v solves
lk
(D +P )v =0,
t k lk
(vlk|t=0 =vl0k,
where v0 Hs for some s > 0. Then for any T > 0, there exists a constant
lk ∈
C =C >0 such that
T
kvlkkL2TnL2⋆ 6Ckhkiβvl0kkL2.
Thatis,eventhoughv solvesaSchr¨odinerequationwithadegeneratepotential
lk
barrier, v nevertheless satisfies Strichartz estimates with an arbitrary β >0 loss.
lk
As a consequence, we have the following estimate on natural semiclassical time
scales.
Corollary 3.3. Suppose v solves (2.1) with initial data v = v0 H . Then for
0 lk lk
ǫ>0 sufficiently small and T =ǫk−2/(m+1), v satisfies the Strichartz estimate
kvkL2TL2⋆ 6Ckhkiη+βv0kL2,
where
1
η = (m(n 2) n).
2n(m+1) − −
Moreover, if v is a zonal spherical harmonic, this estimate is near-sharp, in the
lk
sense that no polynomial derivative improvement is true for every β >0.
Remark 3.4. This corollary shows that on natural semiclassical time scales the
Strichartzestimatesareimproved. Indeed,indimensionn=2,thereisasmoothing
effect. The proof of the near-sharpness of this estimate is in Section 5.1.
STRICHARTZ ESTIMATES WITH LOSS 9
3.3. Proof of Proposition 2.2 and Corollary 3.3. Assuming Proposition 3.2,
we have from (3.1) (in dimension n>3):
v 2 6Ck(n−2)/n v 2
k kL2TL2⋆(M) k lkkL2TL2⋆(R)
lX∈Λk
2
6Ck(n−2)/nT(n−1)/n k βv0
h i lk L2(R)
lX∈Λk(cid:13) (cid:13)
(cid:13) (cid:13)
6CT(n−1)/n k (n−2)/2n+(cid:13)βv 2 (cid:13) ,
kh i 0kL2(M)
by orthonormality, which is Proposition 2.2, and hence also Theorem 1. A similar
computation using (3.1) holds when n=2, and 26q < .
∞
For Corollary 3.3, the sum is over only one term, and T k−2/(m+1). Then in
∼
this case,
v 2 6Ck−2(n−1)/n(m+1)k(n−2)/n v 2
k kL2TL2⋆(M) k lkkL2⋆(R)
6(1+|k|)n(m1+1)(m(n−2)−n)khkiβvl0kk2L2(M)
6C k η+βv 2 ,
kh i 0kL2(M)
where η is as in Corollary 3.3. A similar computation holds for q < in the case
∞
n=2.
(cid:3)
4. The parametrix
ItremainstoproveProposition3.2. Forthatpurpose,inthissectionweconstruct
a parametrix for the separated Schr¨odinger equation:
(D +( ∂2+λ2A−2(x)+V (x)))u=0,
t − x k 1
(ut=0 =u0.
|
We rescale h2 =λ−2 to get
k
(D ( ∂2+h−2A−2(x)+V (x)))u=0,
t− − x 1
(ut=0 =u0.
|
Let v(t,x)=u(ht,x), so that
(hD ( h2∂2+A−2(x)+h2V (x)))v =0,
t− − x 1
(v t=0 =u0.
|
Fortherestofthissection,weconsidertheone-dimensionalsemiclassicalSchr¨odinger
equation with barrier potential:
(hD +( h2∂2+V(x)))v =0,
(4.1) t − x
(v t=0 =v0.
|
The potential V(x) = A−2(x) + h2V (x) decays at x = , is even, and the
1
| | ∞
principal part A−2(x) has a degenerate maximum at x = 0 with no other critical
points. Denote P = h2∂2+V(x).
− x
Let us give a brief summary of the steps involved in the construction. We
will use a WKB type approximation, although, since we are in dimension 1, we
do not need a particularly good approximation. The first step is to approximate
10 CHRISTIANSON
the solution away from the critical point at (x,ξ) = (0,0). Since this is a non-
trapping region, standard techniques can be used to construct a parametrix and
prove Strichartz estimates on a timescale t h−1 for the semiclassical problem, or
∼
on a fixed timescale for the classical problem. A similar construction applies for
energies away from the trapped set.
The remaining regions can be divided into an h-dependent strongly trapped re-
gionanda“transitionregion”,wherewavepacketspropagate,butnotatauniform
rate. By restricting attention to a sufficiently small h-dependent neighbourhoodof
(0,0), we can extend a semiclassical parametrix to a timescale t h(1−m)/(1+m),
∼
whichisaclassicaltimescaleofh2/(m+1). Wedividethetransitionregionintoalog-
arithmicnumberofh-dependentregionsonwhichasimilarparametrixconstruction
works. Summingoveralloftheseregionsgivesaparametrixconstructionandcorre-
spondingStrichartzestimatesinacompactregioninphasespacewithalogarithmic
loss due to the number of summands in the transition region. These constructions
and Strichartz estimates hold for a frequency dependent timescale h2/(m+1)+β,
∼
β > 0, or with a β > 0 loss in derivative on timescale h2/(m+1). We then use
∼
the local smoothing estimate from [CW11] to glue estimates on h−2/(m+1) time
∼
intervals to get the Strichartz estimates with a β >0 loss overall.
4.1. WKB expansion. We make the following WKB ansatz:
v =h−1/2 eiϕ(t,x,ξ)/he−iyξB(t,x,ξ)u (y)dydξ,
0
Z
and compute
(hD )2v = eiϕ(t,x,ξ)/h((ϕ )2B ihϕ B 2ihϕ B h2B )u (y)dxdξ,
x x xx x x xx 0
− − −
Z
and
hD v = eiϕ(t,x,ξ)/h(ϕ B ihB )u (y)dydξ.
t t t 0
−
Z
In order to approximately solve the semiclassical Schr¨odinger equation for v, we
use the WKB analysis. We begin by trying to construct ϕ so that
ϕ +(ϕ )2+V(x)=0,
t x
(ϕt=0 =xξ.
|
Givensuchϕ, we solve the transportequations for the amplitude using a semiclas-
sical expansion:
B = hjB (t,x,ξ),
j
j>0
X
and
ihB ihϕ B 2ihϕ B h2B =0.
t xx x x xx
− − − −
This amounts to solving:
B 2ϕ B ϕ B =0,
0,t x 0,x xx 0
(4.2) − − −
( iBj,t iϕxxBj 2iϕxBj,x Bj−1,xx =0, j >1.
− − − −