A Probabilistic Approach to Semi-classical Approximations

Journal of Functional Analysis  FU2845 journal of functional analysis 137, 243280 (1996) article no. 0046

A Probabilistic Approach to Semi-classical Approximations Gerard Ben Arous* Laboratoire de Mathematiques de l'Ecole Normale Superieure, 45 rue d 'Ulm, 75230 Paris Cedex 05, France

and Fabienne Castell Laboratoire de Modelisation Stochastique et Statistique, URA 743, Universite Paris-Sud (Ba^t. 425), 91405Orsay Cedex, France Received March 1995

We study in this paper the semi-classical expansion of the Schrodinger equation, using a probabilistic approach based on the Wiener measure. Using almost-analytic extensions, we exhibit a probabilistic ansatz for the wave function. We show that this ansatz approximates very well the wave function in the semi-classical regime, and gives the semi-classical expansion under mild hypothesis on the potential at infinity, and no analyticity conditions. In this paper, the study takes place before the caustics.  1996 Academic Press, Inc.

1. Introduction We give in this paper a new probabilistic approach to the semi-classical approximation of the Schrodinger equation, i.e. the behavior when the Planck's constant  tends to zero, of the solution 8(t, x) of

{

i

8  2 + 28&V8=0 t 2

8(0, x)=f (x) e (i) s(x)

where f, s, V are smooth functions. This is indeed an old question, and with no claim of completeness, we can quote the following references in the physics literature, starting from * E-mail address: benarousdmi.ens.fr. E-mail address: castellgyptis.univ-mrs.fr.

243 0022-123696 18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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BEN AROUS AND CASTELL

Van Vleck ('28) [23], Feynman ('45) [10], Voros [24], and two recent surveys in the books of Schulman [20] and Gutzwiller [13]. In the mathematical literature, one can distinguish two different approaches. The first one, for which a good source could be the books of Guillemin 6 Sternberg [12], and of Robert [19], is purely analytical. This approach is very complete and efficient, but very far from the physical picture given by the path integral formalism. The second one tries to stay closer to this physical intuition and is probabilistic in nature. One can distinguish there two different lines of attack: the first one is to build rigorously a Feynman integration, and has been pursued by Albeverio 6 Hoegh-Krohn [3], Elworthy 6 Truman [9], Kallianpur, Kannan 6 Karandikar [15], and more recently by Albeverio 6 Brzezniak [2]. The intrinsic limitation of the scope of these results seems to be on the class of potentials V that can be handled, i.e. V should be a quadratic form, plus the Fourier transform of a signed measure. This very global hypothesis is necessary for the rigorous Feynmann integral formalism developed by Albeverio 6 Hoegh-Krohn. The second line is to extend analytically the FeynmanKac formula for solutions of the heat equation. The general opinion about this very appealing strategy is resumed by Berezin 6 Shubin [7] when they say that ``it runs into practically insurmontable difficulties''. We want here to show that this strategy based on Wiener measure, can indeed be implemented for general smooth potentials. It must be noticed that if one is ready to assume very strong analyticity hypothesis on the potentials, this strategy has been successfully used by Doss [8], and by Azencott 6 Doss [5]. Our trick to avoid these analyticity assumptions is to use the almost-analytic machinery introduced by Melin 6 Sjostrand [18], to complexify the FeynmanKac formula. This almost-analytic extension of the FeynmanKac formula does not give an exact probabilistic representation of the wave function, but it enables us to give a probabilistic ansatz which, although it does not solve the Schrodinger equation, gives a very good approximation of its solution; and to which the available stationary phase results on Wiener space proved in [6], can be applied to get semi-classical expansions in any Sobolev norm, or uniform norm. The error between the ansatz and the true solution is of order O(  ) in 2 L -norm with no assumption on V, except smoothness and essential selfadjointness of the operator & 22 2+V. To get estimates in better norms is a purely analytical problem. We propose some results in H 1-norm, and uniform norm, which are certainly not optimal. But it should be noticed that our ansatz gives the semi-classical approximation in a given norm as soon as the problem is ``semi-classically localizable'' in this norm, which is a necessary condition for the expansion to be valid (see the discussion of Section 4.4).

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SEMI-CLASSICAL APPROXIMATIONS

Our discussion is for this paper, limited to the short time problem, i.e. before caustics. We expect to extend this approach after caustics in a forecoming paper.

2. A Probabilistic Ansatz Using Almost-Analytic Extensions The following assumptions will be made throughout the paper. H1. f, V, s are C  from R d to R. H2. f has compact support K in R d. H3. The classical motion defined by V is complete. Assumption H3 is not essential. It simply avoids to consider explosion times for the classical trajectories. But it is easy to see that all our results will be true if we remove H3, and if we limit the study before these explosion times. 2.1. Description of the Probabilistic Ansatz For each t>0, and each x # R d, let us consider the classical mechanics system

{

# s +{V(# s )=0, # 0 =x

\st (C xt )

#* t +{s(# t )=0

where {V and {s stand for the gradient of the functions V and s, and # is a continuous path from [0, t] to R d. Proposition 2.1. Let V be a smooth potential from R d to R, such that the classical motion associated to V is complete. Let K be a compact set in R d. K t =[x # R d, (C xt ) has a solution ending at time t in K] is a compact set of R d. Moreover, there exists T K >0 such that for all t0, such that \x # K, \y # R d,

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BEN AROUS AND CASTELL

|D (:, ;) f (x+iy)| C | y | k |D (:, ;)V (x+iy )| C | y | k |D (:, ;) s~ (x+iy )| C | y | k where D (:, ;) = |:| + | ;| (z :1 1 } } } z :d d z 1; 1 } } } z d; d ), z k =x k +iy k . The reader is referred to [22] for the definition, existence and elementary properties of almost-analytic continuations. We just want to underline that almost-analytic continuations are not unique. Two almost-analytic continuations differ one from the other by a smooth function, which is flat on the real axis (i.e. in x), but whose behavior in y near  is free. For this reason, it can be assumed that f , s~ , V are null for | y | r, for some r>0. Moreover, since f has compact support, f can be taken with compact support in C d. From now on, the same notations will be used for f, s, V and their almost-analytic continuations. Moreover, = 2 will denote . Using these almost-analytic extensions, we guess that the function 9 (t, x) defined (whenever it is possible) by

_

\

9 (t, x)=E f (# xt(t)+- i=B t ) exp &

H(t, x, =B) =2

+&

(2)

where H is defined by

|

H(t, x, =B)=&is(# xt(t)+- i=B t )+i

t

V(# xt(s)+- i=B s ) ds

0

1 + -i

|

t

#* xt(s) $(=B s )& 0

i 2

|

t

|#* xt(s)| 2 ds

(3)

0

should be a good approximation of the solution to Schrodinger equation. Before explaining to what extent this assertion is true, we would like to tell where this ansatz comes from. Under analyticity assumptions for s, f, V, and additional assumptions for which we refer the reader to [8], H. Doss has proved in [8] that

_

2

E ( fe is= )(x+- i=B t ) exp

\

&i =2

|

t

V(x+- i=B s ) ds o

+&

(4)

is a solution of the Schrodinger equation. Extending analytically the CameronMartin formula, R. Azencott 6 H. Doss ([5]) have shown that (4) is equal to (2), which is the natural expression to obtain semi-classical approximations.

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SEMI-CLASSICAL APPROXIMATIONS

247

In our context, where the analyticity assumptions are removed, the analytical extension of CameronMartin formula is no longer valid, so that (2) and (4) are no longer equal. Our goal being here to obtain semi-classical expansions, we have chosen to work directly on (2). 2.2. Equation Satisfied by the Probabilistic Ansatz Proposition 2.2. 1. _T r >0 such that 9 can be defined on [0, T r[_R d, and is C 1,  on this space (that is C 1 in t, and C  in x). For all t
i

V 9 = 2 2 + 29& 2 9=E[Z(t, x, =, =B) exp &(1= ) H(t, x, =B) ] t 2 =

(5)

where (a) (b)

H is defined by (3), and Z(t, x, =, =B)=

:

= 2jZ j (t, x, =B).

(6)

j=&1, 0, 1

Z j (t, } , |) has compact support in K rt , and satisfies \k # N, _C>0, such that \t
a.s

(7)

x # Rd

where &|& t denotes the uniform norm in C([0, t], R d ). 3. When the functions f, s, V, are analytic on a strip around the real axis, then _r 0 >0 such that Z j (t, x, |) 1 &|& t r 0 =0 a.e. 2

Remark 2.3. Using property (7), one can therefore expect E(Ze &H= ) to be O(= k ) for all k, i.e. the remainder term to be ``small''. This is actually the case, and will be proved in Section 4. Proof of Proposition 2.2. Proof of Proposition 2.2.1. We first prove that  is well defined, at least for small times. Let K be the (compact) support of f in C d tR 2d. Then K is a subset of [x+iy, x # K, | y | r]. Let us define K r =[x # R d, _x~ # K, |x&x~ | r] K r is a compact set in R d, and we associate to K r the time T r, and the sets K rt and O rt , in the same way as in Proposition 2.1, the time T K and the sets K t and O t were associated to K. Whenever # xt(t)  K r, 9 (t, x)=0.

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248

BEN AROUS AND CASTELL

Therefore, the support of 9 (t, } ) is included in K rt . Moreover, for all t
}

\

f (# xt(t)+- i=B t ) exp + & f & K exp

\

i i s(# xt(t)+- i=B t )& 2 2 = =

1 t &Im s& K exp 2 2 = =

+ \

sup

|

t

V(# xt(s)+- i=B s ) ds

0

|Im V(x+- iy)|

| y | - 2r r x #  st K s

+}

+

where & f & K =sup z # K | f (z)|. But, when t<+,  st K rs is compact. Therefore,

\

f (# xt(t)+- i=B t ) exp &

1 1 s(# xt(t)+- i=B t )+ 2 i= 2 i=

|

t

V(# xt(s)+- i=B s ) ds 0

+

is bounded, and expression (2) makes sense for all t
_

\

G(t, q, p)=E F(.(q, p)+- i=B) exp & &

1 2i= 2

|

t

|.* s(q, p)| 2 ds 0

1 - i=

|

t

.* s(q, p) $B s 0

+&

(8)

where for all continuous path | from [0, t] to C d, F(|)=f(| t ) exp

\

i i s(| t )& 2 =2 =

|

t

+

V(| s ) ds . 0

(9)

Then 9 (t, x)=G(t, x, #* xt(0)). The differentiability of G in t is an easy consequence of Ito^ formula. The differentiability of G in (q, p) follows from the

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249

SEMI-CLASSICAL APPROXIMATIONS

differentiability of , and its derivatives in (q, p), and from the smoothness of the coefficients f, s, V. Proposition 2.1 enables us to conclude that before the caustics, 9 is C 1, . Proof of Proposition 2.2.2. Before computing i(9 t)+(= 22) 29& (V= 2 )9, we introduce some notations. v \z # C d, |z| 2c = |Re z| 2 & |Im z| 2 +2i(Re z, Im z), v Wt =[continuous paths from [0, t] to R d ], v Wt(C)=[continuous paths from [0, t] to C d ]. Wt and Wt(C) are endowed with the uniform convergence topology, denoted by & } & t . Denote for every function F : Wt(C)  R Frechet differentiable (with derivative dF ), for all | # Wt(C), and all h # Wt , D r F(|) h=dF(|) } h,

D i F(|) h=dF(|) } (ih)

DF= 12 (D r F&iD i F ),

DF= 12 (D r F+iD i F ).

Finally, let 3 be a bounded domain in R n, and ( g s(%), st) a path in Wt(C), depending smoothly on % # 3. We define for all k # N and all p>1, the space

{

D p,g k(3)= F: Wt(C) [ R, k-times Frechet differentiable, such that : sup &|d (l )F( g(%)+- i=B)| HS & L p <+ lk % # 3

=

where | } | HS denotes the HilbertSchmidt norm. We have then the following lemma: Lemma 2.4. Let ( g s(%), st) a path in Wt(C), depending smoothly on % # 3, and 2-times continuously differentiable in s. Assume that F is in D p,g k(3) for some k2, and some p>1. Let us define 1 G(t, %)=E F(g(%)+- i=B) exp & - i=

_

\

|

t

g* s(%) $B s &

0

1 2i= 2

|

t

|g* s(%)| 2c ds

0

+& .

Then G is (k&1)-differentiable in 3, and \j # [1, ...n], \% # 3, G (t, %)=E % j

_\

Dr F }

\

_exp &

g r0 g i g r g r0 g i g i0 +D i F } 0 +(1+i ) DF } & & + % j % j % j % j % j % j 1

-i =

|

t

g* s $B s & 0

1 2i= 2

|

t

0

| g* s | 2c

\ ds +&

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++

(11)

250

BEN AROUS AND CASTELL

where 1. the derivatives of F are evaluated at g(%)+- i=B; 2. g and its derivatives are evaluated at %; 3. g l0 % j (l=r, i ) denotes the constant path in R d : s [ g l0 % j . The reader is referred to Appendix 2 for the proof of Lemma 2.4. From Lemma 2.4, we get Lemma 2.5. i

V G = 2 V H 9 = 2 + 29& 2 9=i + 2 q G& 2 G+E Z exp & 2 t 2 = t 2 = =

_

\ +&

(12)

where Z satisfies (6) and (7), and G is defined by (8). Proof of Lemma 2.5. i

A simple computation yields

V 9 = 2 + 29& 2 9 t 2 = =i

G = 2 V = 2 G G #* xt(0) + 2 q G& 2 G+ 2 x #* xt(0)+i } t 2 = 2 p p t

+= 2 Trace

\

=2  2G #* xt(0)  2G #* xt(0) #* xt(0) + Trace q p x 2 p 2 x x

+

\

\

t

++

where  2G  2G = q p q i p j

\

+

 2G  2G = p 2 p i p j

\ + G G = , p \ p +

, i, j

d

 2G , q 2i i=1

2 q G= : x t

x, i t

#* (0) #* (0) = x x j

\

j

+

d

, i, j

, i, j

j

2 x t

 #* (0) . x 2i i=1

2 x #* xt(0)= :

In order to compute Gp, let us apply Lemma 2.4 with n=2d, %=(q, p), F defined by (9), and the real path g(%)=.(q, p). It is clear that F is in all the spaces D .p, k(O rt ) (remember that f (. u +- i=B u ), s(. u +- i=B u ), V(. u +- i=B u ) are null whenever |=B u | - 2 r). Since . 0(q, p)=q, (11) leads to 1 G (t, x, #* xt(0))=E exp & 2 H(t, x, =B) M j (t, x, =, =B) p j =

_ \

+

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&

251

SEMI-CLASSICAL APPROXIMATIONS

where H is defined by (3) and M j =(1+i)

_

s . t  f . t f } + & } + z p j i= 2 z p j

\

|

t 0

V . s } ds z p j

+& .

In the preceding expression, s, f and their derivatives are evaluated at # xt(t)+- i=B t ; the derivates of V at # xt(s)+- i=B s ; and the derivatives of . at (x, #* xt(0)). Moreover, the notation ( fz ) } r stands for  di=1 ( fz i ) r i . Thus, the almost-analyticity of f, s, V enables us to conclude that #* x(0) #* x(0) =2 G = 2 2 2 x #* xt(0)+i t 2 x #* xt(0)+i t =E exp &(1= ) HZ p 2 t 2 t

\

+ _

\

+&

with Z satisfying (6) and (7). The same kind of arguments holds for the terms involving  2Gq p, and  2Gp 2. They also have the form E[exp(&(1= 2 ) H ) Z], with Z more complicated than previously, but involving only  -derivatives of f, s, V. This completes the proof of Lemma 2.5. K It remains now to compute i(Gt)+(= 22) 2 q G&(V= 2 ) G. It is done in Lemma 2.6. i

V H G = 2 + 2 q G& 2 G=E Z exp & 2 t 2 = =

_

\ +&

where Z satisfies (6) and (7). Proof of Lemma 2.6. To prove Lemma 2.6, we introduce the process (X s(q, p, z, y)) st , defined by

{

X 1s =. s (q, p) X 2s =.* s(q, p) X 3s =z+. s(q, p)&q+- i=B s X 4s =y&

i =2

|

s 0

V(X 3u ) du&

1

| - i=

s 0

X 2u $B u &

1 2i= 2

|

s

|X 2u | 2 du

0

(X s ) is an homogeneous diffusion process with value in R d_R d_C d_C, and with initial condition (q, p, z, y). For all suitable function r: R d_R d_C d_C  R, define 6 s r(q, p, z, y)=E[r(X s(q, p, z, y ))].

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252

BEN AROUS AND CASTELL

For r(q, p, z, y)=f (z) exp(+(i= 2 ) s(z)) exp( y ), we get 6 t r(q, p, z, y).u t(q, p, z) e y

(13)

G(t, q, p) exp( y)=6 t r(q, p, q, y)

(14)

G(t, q, p)=u t(q, p, q).

(15)

so that

Ito^ formula (applied in R 4d+2 ) gives us the Meyer decomposition of the process (6 t&s r(X s ), st). But, from the Markov property, (6 t&s r(X s ), st) is a martingale. Writing that its bounded variation part is null for s=0, yields &

6 t r 6 t r 6 t r 1 1 6 t r 6 t r + } p&{V(q) } + } p+ } V(z)& 2 | p| 2 t q p z y i= 2 2i=

\

+

+

6 t r 6 t r 1 1 i= 2 i= 2 } p+ } & 2 V (z)+ 2 | p| 2 + 2 z 6 t r& 2 z 6 t r z y i= 2i= 2 2

\

+= 22 z, z 6 t r+ +i

+

| p| 2  26 t r | p| 2  26 t r | p| 2  26 t r  26 t r & }p & 2 + 2 2i= 2 y 2 2i= y 2 = y y y z

 26 t r 2 6t r  26 t r } p&i } p& } p=0 y z y z y z

(16)

where 2 z = di=1  2z 2i , 2 z = di=1  2z 2i , 2 z, z = di=1  2z i z i ,  2y z= ( 2y z i ) i , and so on... Equation (13) implies that 6 t ry =0, and 6 t ry=6 t r. (16) reads then u t 1 u t u t } p&{V(q) } + V(z) u t & + t q p i= 2 +(1+i)

i= 2 u t i= 2 } p+ 2 z u t & 2 z u t += 22 z, z u t =0. z 2 2

(17)

Using (15), it follows that i

V G = 2 + 2 q G& 2 G t 2 = =i

u t = 2 u t } p&i {V(q) } + 2 u += 22 q, z u t += 22 q, z u t q p 2 q t

+(&1+i )

u t } p+= 22 z u t +(1+i ) = 22 z, z u t . z

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(18)

253

SEMI-CLASSICAL APPROXIMATIONS

But u t(q, p, z)=E[F(z+.(q, p)&q+- i=B) 1 exp & - i=

\

|

t

.* s(q, p) $B s & 0

1 2i= 2

|

t

|.* s(q, p)| 2 ds 0

+& .

Apply Lemma 2.4 with n=4d, %=(q, p, z) # R d_R d_C d, and g(%)= z+.(q, p)&q. Since g r0 p j =g i0 p j =0, g r0 q j =g i0 q j =0, the terms in (18) involving u t p, u t q, 2 q u t , 2 q, z u t , 2 q, z u t , have the form 2 E[exp &1= H Z] with Z satisfying properties (6) and (7) of Proposition 2.2. r From g 0 z rj =g i0 z ij =e j , and g r0 z ij =g i0 z rj =0, we obtain 1 u t =E DF } e j exp & z j - i=

_

\

This ends the proof of Lemma 2.6.

|

t

.* s $B s & 0

1 2i= 2

|

t

|.* s | 2 ds 0

+& .

K

Proof of Proposition 2.2.3. When the functions f, s, V are analytic on a strip around the real axis, their almost-analytic continuations can be chosen analytic around the real axis. In this case, assertion 3 is satisfied, since Z j is a function of the  -derivatives of f, V, s. This ends the proof of Proposition 2.2. K

3. Semi-classical Expansion of the Probabilistic Ansatz The aim of this section is to obtain the semi-classical expansion of 9. Since 9 has the form E[Z exp(&H= 2 )], this will be done by the stationary phase method in the Wiener space. For this reason, section 3.1 is devoted to the study of the critical points of the phase function. 3.1. Critical Points of Phase Function We denote by Ht the space of continuous paths from [0, t] to R d, starting from 0, absolutely continuous with respect to Lebesgue measure on [0, t], and whose derivative is square integrable. Ht is an Hilbert space, with respect to inner product ( h, k) Ht = t0 h4 s } k4 s ds. Let us then consider the phase function r F x, t : Ht  C

h [ &is(# xt(t)+- ih t )+i +

1

| -i

t

0

#* xt(s) h4 s ds&

| i 2

t 0

|

V(# xt(s)+- ih s ) ds t

|#* xt(s)| 2 ds+ 0

1 2

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|

t

|h4 s | 2 ds 0

(19)

254

BEN AROUS AND CASTELL

r The superscript r in F x, recalls the dependence of s and V on r. Therefore, t x, r iF t is just the classical action defined by

S xt : Ht  R h [ s(x+h t )&

|

t

V(x+h s ) ds+

0

x t

1 2

|

t

(20)

|h4 s | 2 ds

0

x t

taken on the complex path # +- ih. When S is considered as an operator acting on real paths, it is a well-known fact that its critical points are the classical mechanics trajectories, and that before the caustics, S xt has a unique critical point, which is a non degenerate minimum of S xt . The question is now to prove that this remains true, when the paths are allowed to visit a neighborhood of the real axis in the complex domain. This is the object of Proposition 3.7. Let K be the compact support of f in R d. Let us fix t0 such that 1. t
|

+i - i +

1 -i

|

t

V(l s ) } k s ds+i - i 0

t

#* xt(s) k4 s ds+

0

|

|

t

 V(l s ) } k s ds 0

t

h4 s k4 s ds 0

=&i - i (&i ) s(l t ) } k t + +i - i

|

1 -i

#* xt(t) k t +h4 t k t

t

(&i ) V(l s ) } k s ds& 0

1

| -i

t

# xt(s) k s ds&

0

=&i - i [l4 t +(&i ) s(l t )] } k t +i - i

|

t

[l s +(&i ) V(l s )] k s ds.

0

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|

t

0

h s k s ds

255

SEMI-CLASSICAL APPROXIMATIONS

r Thus, h is a critical point of the phase function F x, if and only if l= t x # t +- i h is solution to

l s +(&i ) V(l s )=0,

(21)

\st

l 0 =x

(22)

l4 t +(&i ) s(l t )=0.

(23)

From the almost analyticity of V and s, it follows that \z # R d,  V(z)=  s(z)=0, and that V(z)={V(z), s(z)={s(z). It is then equivalent to say that h#0 is a critical point, and that # xt satisfies (C xt ). Let us look at the degeneracy of 0 as a critical point. A straightforward computation leads to r x D 2F x, t (0)(h, h)=s"(# t (t))(h t , h t )&

|

t 0

V"(# xt(s))(h s , h s ) ds+

|

t

|h4 s | 2 ds 0

=D 2S xt(# xt &x)(h, h)

(24)

where the operator S xt is the classical action, defined by (20). But it is a well-known fact that before the caustics (i.e. t
(25)

where the operator A xt is defined on Ht by A xt(h, k)=s"(# xt(t))(h t , k t )&

|

t

V"(# xt(s))(h s , k s ) ds

(26)

0

and satisfies for some constant C (depending on x and t), |A xt(h, k)| C &h& t &k& t

(27)

A xt defines thus a continuous quadratic form on the space C0([0, t]) of continuous paths starting from 0, endowed with uniform convergence. It results then from a result of L. Gross [11], that A xt is a trace operator on Ht , so that there is a basis of Ht formed by eigenfunctions of A xt . The nondegeneracy of # xt means therefore that for all t0 such that \h # Ht ,

r x 2 |D 2F x, t (0)(h, h)| : t &h& Ht .

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Moreover, the function x [ : xt is lower semi-continuous. Indeed, : xt &1 is the lowest eigenvalue of A xt , i.e. : xt &1=inf(A xt(h, h), h # Ht , &h& Ht =1)

(29)

where the extremum is reached, since A xt is a trace operator. Thus, \r>0, x [ : xt reaches its minimum value on the compact set K rt . We have then proved the following assertion: \r>0 such that t0, such that \x # K rt ,

D 2S xt(# xt &x)(h, h):(r)&h& 2Ht .

\h # Ht ,

(30)

Note that the lower semi-continuity of x [ : xt implies that lim inf :(r)Min : xt >0. r0

x # Kt

r We are now going to estimate D 2F x, t (0) on a critical point of the phase x, r function F t . Using integration by parts, we can rewrite for all h in Ht

r x 4 D 2F x, t (0)(h, h)=(s"(# t (t)) h t +h t , h t )&

|

t 0

(h s +V"(# xt(s)) h s , h s ) ds.

(31)

r When h is a critical point of F x, t , each of these terms can be evaluated in view of expressions (23) and (21). Using a Taylor expansion of (&i ) s(# xt(t)+- i h t ) around # xt(t) to rewrite (23), we obtain

0=- i (h4 t +s"(# xt(t)) h t )

|

+i

|

1

1

+

(1&u)  3s(# xt(t)+- i uh t )(h t , h t ) du

0

(1&u)  2 s(# xt(t)+- i uh t )(h t , h t ) du

0

&i - i

|

1

 s(# xt(t)+- i uh t )(h t ) du&i  s(# xt(t)+- i h t ).

0

Thus,

{

|h4 t +s"(# xt(t)) h t |  C(r)+ sup

(| 3s(x+- i y)|

| y | - 2r x # Kr

=

+| 2  s(x+- i y)| ) |h t | 2

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SEMI-CLASSICAL APPROXIMATIONS

257

where the constant C(r) comes from the last two terms, and from the almost-analyticity of s. We deduce then that \r>0 such that t0 such that \x # K rt , \h # Ht critical point of F x, t , |h4 t +s"(# xt(t)) h t | C(r)|h t | 2.

(32)

Note that C(r) remains bounded when r  0 (C(r) ww sup x # K |s (3)(x)| ). r0 We rewrite expression (21) in the same way to obtain that \r>0 such that t0 such that \x # K rt , \h # Ht critical point r of F x, t , \st, |h s +V"(# xt(s)) h s | C(r)|h s | 2.

(33)

Again, the constant C(r) remains bounded when r  0. Comparing (32), (33), and (31), it follows that there exists C 1 , C 2 >0 r such that \r # ]0, 1] such that t
(34)

From (30) and (34), it follows that _:>0, _c>0 such that \r # ]0, 1] r such that &h& Ht {0, such that t
(35)

We need then the following lemma to conclude. Lemma 3.8. _C >0 such that \r # ]0, 1] such that t0, there is a critical point r h of F x, such that &h& Ht {0. From (35) and Lemma 3.8, it results then t that :CCr, for all r # ]0, 1]. Therefore, when r is sufficiently small, h=0 is the unique critical point r K of F x, t . r such that Proof of Lemma 3.8. Let h be a critical point of F x, t &h& Ht {0. By integration by parts, it follows from (21) and (23) that

i &h& 2Ht =&[#* xt(t)+(&i ) s(# xt(t)+- i h t )] - i h t

|

t

[# xt(s)+(&i ) V(# xt(s)+- i h s )] - i h s ds.

+

0

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BEN AROUS AND CASTELL

Using (C xt ) and a Taylor expansion of (&i ) s(# xt(t)+- i h t ) around # xt(t), it follows that |#* xt(t)+(&i ) s(# xt(t)+- i h t )|  

| |

1

|(&i ) 2 s(# xt(t)+- i uh t )| |h t | du

0 1 0

|(&i ) 2 s(# xt(t)+- i uh t )| 1 u |h t | - 2r |h t | du

|

C 1(r)

1

0

1 u |h t | - 2r d(u |h t | )

C 2(r) r with C 2(r) bounded when r  0. In the same way, it is easy to prove that \st, |# xt(s)+(&i ) V(# xt(s)+- i h s )| C 3(r) r,

with C 3(r) bounded.

Therefore,

\

&h& 2Ht C 4(r) r |h t | +

|

t 0

+

|h s | ds 2C 4(r) r &h& t C (r) &h& Ht

so that &h& Ht C (r) r. In order to apply stationary phase method, we have now to look at the r global minima of the real part of F x, t . It is done in exactly the same way x, r as for the critical points of F t , so we omit the proof of the third assertion of Proposition 3.7. K 3.2. Asymptotic Expansion of 9 We are now able to give the asymptotic expansion when =  0, of 9. The result is the following Proposition 3.9. x # R d,

Let t
9 (t, x)=exp

\

i S(t, x) =2

+

N

: ; k(t, x) k=0

= 2k +O(= 2(N+1) ) (2k)!

(37)

where 1. the support of ; k(t, } ) is included in K t . 2.

; 0(t, x)=f (# xt(t)) det(D 2S xt(# xt &x)) &12

where S xt is defined by (20).

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(38)

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SEMI-CLASSICAL APPROXIMATIONS

3. S(t, x)=S xt(# xt &x)=s(# xt(t))& t0 V(# xt(s)) ds+ 12  t0 |#* xt(s)| 2 ds 4. the remaining term O(= 2(N+1) ) can be understood in any Sobolev norm, or in C -norm. Remark 3.10. It should be noted that although we have given only the expression of the first term ; 0 , there is an explicit way of computing all the other terms (see the remark following Lemma 3.12). Remark 3.11. The usual expression for ; 0 is ; 0(t, x)=

}

# xt (t) x

}

12

f (# xt(t)).

(39)

To see that (38) and (39) coincide is classical (see for instance [1]). Proof of Proposition 3.9. It would be sufficient to apply Theorem 7 of [6] to obtain Proposition 3.9. In our context, the proof of this theorem is much easier, so that we give it for the sake of completeness. The main fact is the following lemma, whose proof is given in Appendix 3. Lemma 3.12. Let t
_

\

J =(s, x)=E L(s, x, =B) exp &

1 H(s, x, =B) =2

+& .

Then, J =(s, x)=exp

\

i S(s, x) =2

+

N

= 2p ; p(s, x)+O(= 2(N+1) ) (2p)! p=0 :

where v The O(= 2(N+1) ) is uniform in (s, x) # [0, t]_R d. v ; 0(s, x)=L(s, x, 0) det &12(D 2S xs(# xs &x)). v ; p(s, } ) has compact support in K rs .

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(40)

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BEN AROUS AND CASTELL

Moreover, if there exists r 0 >0 such that \st, \|, &|& s r 0 , L(s, x, |)#0, then J =(s, x) is exponentially small. Remark 3.13. ; p(s, x) can be expressed as x

; p(s, x)=E[Z p(s, x) exp &(12) A s (B, B) ], where Z p(s, x) is an element of the (6p)-th Wiener chaos, and A xs is defined by (26). It is then possible to compute the expression of ; p in the following way. Let ( f xs(i )) denote an orthonormal basis (in Hs ) of eigenfunctions of A xs (eigenvalues (: xs(i ))), and let (! xs(i )) denote the corresponding basis of the first Wiener chaos. If for all J=( j 1 , ..., j m ), m

HJ (! xs )= ` k=1

1 - jk !

H jk (! xs(k))

(where H j is the j-th Hermite polynomial), then (H J (! xs )) J=( j 1 , ..., j m);  j i 6p form a basis of the 6p-th Wiener chaos. Therefore, one can write Z p(s, x)=  J, |J | 6p c J (s, x) H J (! xs ). The expression of ; p is then ; p(s, x)=det &12(D 2S xs(# xs &x))

c J (&1) m H J (0)

: J=( j 1 , ..., j m ) | J | 6p

m

_` k=1



|: xs ( j k )| . 1+: xs( j k )

To get the result of Proposition 3.9 in uniform norm, it is sufficient to apply Lemma 3.12 with L(s, x, |)=f (# xs(s)+- i | s ), which satisfies trivially Assumptions 13. To get the result in C 1-norm, let us consider the first derivatives of 9. As in Section 2.2, we write that 9 (s, x)=G(s, x, #* xs(0)). It follows that G G #* x(0) 9 (s, x)= (s, x, #* xs(0))+ (s, x, #* xs(0)) } s . x q p x Applying Lemma 2.4, it appears that 9 x(s, x) can again be expressed as H = 2 j E L j (s, x, =B) exp & 2 = j=&1, 0, 1 :

_

\ +&

where v For j=&1, 0, 1, L j (s, x, |) satisfies Assumptions 13 of Lemma 3.12. v For j=&1, 1, \k # N, _C such that \x # K rs , &L j (s, x, |)&C &|& ks .

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SEMI-CLASSICAL APPROXIMATIONS

261

Lemma 3.12 yields then the result of Proposition 3.9 in C 1-norm. The same kind of arguments holds for the other derivatives. The assertion concerning the Sobolev norms is then straightforward, since everything happens inside the compact K rt .

4. Using the Probabilistic Ansatz to Get Semi-classical Estimates on the Wave Function We prove in this section that our probabilistic ansatz is a good approximation of the wave function 8 solution of the Schrodinger equation, in various norms. We begin by the L 2-norm, for which the hypothesis needed on V is minimal (we just require the Schrodinger operator to be essentially self-adjoint). To get better norms, we have to strengthen our hypothesis on V. We treat the case of H 1-norm, and uniform norm, but as already said in the introduction, we do not claim here for optimality in our hypothesis on the potential. 4.1. L 2-Estimates We will assume in this section that H4. The operator A.&( 22) 2+V defined on C  c (that is, the set of smooth functions with compact support in R d ) is essentially self-adjoint. Under this assumption, the Schrodinger equation has a unique solution 8(t, x) in L 2(R d, dx). Proposition 4.14. Let t0 such that sup &8(s, } )&9 (s, } )& L 2 ( R d, dx) C= k. st

Therefore, the semi-classical expansion given in Proposition 3.9 is also valid for 8 in L 2-norm. If the functions V, s, f are analytic on a strip around the real axis, the L 2-error between 9 and 8 is exponentially small. Proof of Proposition 4.14. Before proving Proposition 4.14, we are going to demonstrate the following lemma, which will be useful in the sequel. Lemma 4.15. such that

Let E s(x) and G s(x) be two functions from R +_R d to C,

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BEN AROUS AND CASTELL

v \s, E s and G s are in L 2(R d, dx).

v

{

i

E s (x)=AE s(x)+G s(x) s

E 0(x)#0.

Let {=sup[t, sup st &E s & L 2 ( R d ) <]. Then, for all t<{,  sup &E s & L 2 ( R d ) 2

t

|

&G u & L 2 ( R d ) du

0

st

We adopt here the convention that sup < =0. Proof of Lemma 4.15. i &E s & 2L 2 (R d ) =i &E 0 & 2L 2 (R d ) + =

|

s 0



i

E u , Eu u

|



s

i 0

 &E u & 2L 2 (R d ) du u du&

L2( R d )

|

s

E u

E , i u 

du

u

0

L2( R d )

where the minus sign comes from the complex conjugation. Thus, i &E s & 2L 2 (R d ) =

|

=

|

s 0

( AE u +G u , E u ) L 2 (R d ) du&

s

0

( G u , E u ) L 2 (R d ) du&

|

|

s

( E u , AE u +G u ) L 2 (R d ) du

0

s 0

( E u , G u ) L 2 (R d ) du

since A is a symmetric operator. Therefore, for all st,  &E s & 2L 2 (R d ) 2

|

s 0

&G u & L 2 (R d ) &E u & L 2 (R d ) du.

Taking the supremum over s, it follows that (sup &E s & L 2 (R d ) ) 2 2(sup &E s & L 2 (R d ) ) st

st

|

t 0

&G u & L 2 (R d ) du.

For t<{,  sup st &E s & L 2 (R d ) 2  t0 &G u & L 2 (R d ) du, and the proof of Lemma 4.15 is complete. K We return now to the proof of Proposition 4.14. Define for all st the error function E s : Rd  C x [ 9 (s, x)&8(s, x).

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263

SEMI-CLASSICAL APPROXIMATIONS

When t
st

where G s is the right-hand term in (5), that is

_

\

G s (x)=E Z(s, x, =, =B) exp &

1 H(s, x, =B) =2

+&

with Z satisfying (6) and (7). We recall that G s has compact support in K rs , so that

\ + meas . K \ +

12

sup &G s & L 2 (R d ) meas . K rs st

sup sup |G s(x)| st x # K r s

st

12

r s

st

C(t, =, r)

sup

:

=2j

sup

2

|E [Z j exp &H= ]|

j=&1, 0, 1 st, x # K rs &R=H= 2

|E [exp

]|

st, x # K rs

where we have used the estimates (7) on the coefficients Z j . Therefore, for all t
st

We estimate now &G s & L 2 (R d ) using Lemma 3.12. G s(x)=exp

\

i S(s, x) =2

+

N&j

:

: : kj (s, x) = 2(k+j ) +O(= 2(N+1) ).

j=&1, 0, 1 k=0

The coefficients : kj are defined by : kj (s, x)=

1 E [% (2k) (s, x, 0)] j (2k)!

where % (2k) (s, x, =) is the (2k)-th derivative in = of j % j (s, x, =)#Z j (s, x, =B) exp(&D 2H(s, x, =B)(B } B)). Using the estimates of Z j given in Proposition 2.2, it follows that \ j, \k, : kj (s, x)=0. Furthermore, when f, V, s are analytic in a strip around the real axis, Z(s, x, |) 1 &|& s r 0 =0; so it follows from Lemma 3.12 that the L 2-error between 8 and 9 is exponentially small. K

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264

BEN AROUS AND CASTELL

4.2. H 1-Estimates We consider here the space H 1(R d, dx) of functions f # L 2(R d, dx), which are absolutely continuous with respect to Lebesgue measure, and whose derivative is in L 2(R d, dx). Instead of H4, we assume here the stronger condition H5. V&C 0 for some constant C 0 . Under H5, H4 is automatically satisfied (see Theorem 1.1, Chapter 3 in [7]) and the wave function is (as a function of x), in H 1 (see Proposition 1.1, Chapter 3 in [7]). Proposition 4.16. Let t0 such that &8(t, } )&9 (t, } )& H 1 (R d, dx) C= k. Therefore, the semi-classical expansion given in Proposition 3.9 is also valid for 8 in H 1-norm. Here again, if the functions V, s, f are analytic on a strip around the real axis, the H 1-error between 9 and 8 is exponentially small. Proof of Proposition 4.16. Proposition 4.14.

We use the same notations as in the proof of

2 2 &{E t & 2L 2 = 2 2 =&

| |{E (x)| t

2

dx

2 ( 2E t , E t ) L 2 2

=( (A&V ) E t , E t ) L 2 ( AE t , E t ) L 2 +C 0 &E t & 2L 2 (C 0 +&AE t & L 2 )&E t & L 2 . But, 9 (t, } ) and 8(t, } ) are in the domain of the Schrodinger operator, so that &AE t & L 2 <. Proposition 4.16 follows then from Proposition 4.14. K 4.3. Sobolev and C -Estimates The space in consideration is C (R d, C), i.e. the space of functions f: R d  C, which are infinitely differentiable. This space is endowed with the uniform norms of f and its derivatives. We will assume that H6. The derivatives of V of order greater than 2 are bounded, and the wave function 8 is C  in x with derivatives in L 2(R d, dx).

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265

SEMI-CLASSICAL APPROXIMATIONS

Under H6, H4 is automatically satisfied (see Theorem 1.1, Chapter 3 in [7]). Proposition 4.17. Let t0 such that sup &8(s, } )&9 (s, } )& H l (R d, dx) C= k. st

Therefore, the semi-classical expansion given in Proposition 3.9 is valid for 8 in any Sobolev norm, and hence in C -norm. Here again, if the functions V, s, f are analytic on a strip around the real axis, the H l-error between 9 and 8 is exponentially small. Proof of Proposition 4.17. notations.

During the proof, we will use the following

v If M # N d, M=(m 1 , ..., m d ), |M| = di=1 m i . v If M, L # N d, M  L.(m 1 +l 1 , ..., m d +l d ). v If M, L # N d, we say that LM iff \ j # [1, ..., d ], l j m j . In this case, we will denote M  L.(m 1 &l 1 , ..., m d &l d ). Moreover, we will say that L
v If m, l # N, lm, and f # C (R d, C), & f & l, m . m&l

:

:

M, |M | =m L, |L| =l LM

l1

V x 1

V x d

"\ + \ + }}}

ld

D M L f

"

L2

m

& f & m . : & f & l, m l=0 m

_ f _m. : & f &l . l=0

Hence _ f _ 0 =& f & 0 =& f & L 2 , d

& f &1 = :  i=1

f x i

" "

d

+: L2

i=1

V f x i

" "

, L2

so that  & f & H 1 _ f 1_ for 1. \m # N,  m & f & H m _ f _ m for 1. We are going to prove that all m # N, _C such that \1, \st  sup _E u _ m C sup _G u _ m . us

us

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(41)

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BEN AROUS AND CASTELL

This is done by induction on m. For m=0, (41) reduces to  sup &E u & L 2 C sup &G u & L 2 us

us

which has already been proved under assumption H4. Let us assume that (41) is true for all nm&1. Let { m .sup[t, sup _E s _ m <]. st

We claim that \m1, _C such that \st, s<{ m , \1,  sup &E u & m C sup &G u & m +C 2 sup _E u _ m&1 . us

us

(42)

us

Thus, the induction hypothesis and (42) allow one to say that \st, s<{ m , \1,  sup _E u _ m C sup _G u _ m . us

us

But, exactly in the same way as in the proof of Proposition 4.14, one can see that \t
|

s

&E u & l+1, m du.

(43)

0

us

Indeed, we derive from (43) that \lm&1,  &E s & l, m C sup &G u & m +C 2 sup _E u _ m&1 +C us

us

|

s

&E u & m, m du 0

with &E u & m, m =

: M, |M| =m

V x 1

"\ +

m1

}}}

V x d

md

\ + E"

But, i

 s

m1

V x 1

md

V x d

\_ & _ & E + E V V = _x & } } } _x & \i s + }}}

s

m1

1

md

.

u

s

d

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L2

(44)

267

SEMI-CLASSICAL APPROXIMATIONS

=A

V x 1

m1

V x d

md

V x 1

m1

V x d

\_ & _ & + __ & _ & V V + _x & } } } _x & G }}}

Es +

m1

}}}

md

&

; & 22 E s

md

s

1

d

where [B; A] denotes the commutator of the operators A and B. Now, for all f, g in C (R d, C), [ f, &2] g=2 f g+2 { f } {g. Hence, using H6, V x 1

m1

V x d

"__ & _ & }}}

md

& "

; & 22 E s

C 2 &E s & m&1, m&1 +C &E s & m&1, m . L2

Lemma 4.15 then yields that \st, s{ m ,  &E s & m, m C sup &G u & m, m +C 2 sup _E u _ m&1 +C us

us

|

s

&E u & m&1, m du. 0

Using (43) for l=m&1, it follows then that  &E s & m, m C sup &G u & m +C 2 sup _E u _ m&1 +C us

us

|

s

&E u & m, m du. 0

And Gronwall lemma leads to  sup &E u & m, m C sup &G u & m +C 2 sup _E u _ m&1 us

us

us

which together with (44), yields (42). It remains now to show (43). We prove it first for all couples (0, m). We have thus to estimate quantities such as  m &D M E s & L 2 , for |M| =m. i

E  (D M E s )=D M i s s s

\

+

=A(D M E s )+[D M ; V] E s +D M G s =A(D M E s )+ : : M(J ) D MJ VD J E s +D M G s . J
For | J |  |M| &2 and 1,  m &D MJ VD J E s & L 2 C 2 _E s _ m&2 . For | J | =m&1,  m &D M J VD J E s & L 2  &E s & 1, m . We obtain then from Lemma 4.15, that \st, s<{ m , \1,  &E s & 0, m C sup &G u & 0, m +C 2 sup _E u _ m&2 +C us

us

|

that is (43) for (0, m).

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s

&E u & 1, m du 0

268

BEN AROUS AND CASTELL

Assume now (43) is true for (l&1, m) and let us prove it for (l, m). We have now to estimate quantities such as  m&l &(Vx 1 ) l1 } } } (Vx d ) ld D M L E s & L 2 for |L| =l and |M| =m.  s

l1

V x 1

ld

V x d

\_ & _ & D  V V = _x & } } } _x & i s D V V = _x & } } } _x & \A(D

i

}}}

l1

ML

Es

+

ld

ML

Es

d

1

l1

ld

M L

Es)

d

1

+ V V V V =A }}} D E + }}} \_x & _x & + __x & _x & ; & 2& D V V + : : (J) }}} V D E _x & _x & D V V + }}} _x & _x & D G . +

: M L( J ) D M (L  J )V D J E s +D MLG s

:

J
l1

ld

l1

ld

M L

2

ML

s

1

d

1

l1

d

ld

M  (L  J )

J

M L

s

1

J
l1

d

ld

ML

s

1

d

For | J | m&l&2, and 1, V x 1

l1

V x d

ld

V x d

ld

V x 1

l1

"_ & _ &

 m&l

}}}

D M(L  J )V

D J Es

"

C _E s _ m&2 . L2

For | J | m&l&1,  m&l

V x 1

l1

"_ & _ & }}}

D M(L  J )V

D J Es

"

C &E s & l+1, m . L2

Moreover,  m&l

V x d

ld

"__ & _ & ; & 2& D }}}

2

ML

Es

"

L2

2

C &E s & l&1, m&1 +C &E s & l&1, m . By Lemma 4.15, we obtain therefore that for all st, s{ m , \1,  &E s & l, m C sup &G u & l, m +C 2 sup _E u _ m&1 us

+C

us

|

s

&E u & l+1, m du+C 0

|

s

&E u & l&1, m du. 0

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Es

269

SEMI-CLASSICAL APPROXIMATIONS

The induction hypothesis implies then that \st, s<{ m , \1,  &E s & l, m C sup &G u & m +C 2 sup _E u _ m&1 us

+C

us

|

s

&E u & l+1, m du+C

0

|

s

&E u & l, m du. 0

And Gronwall lemma leads to (43). It follows from (41) that  m+1 sup &E s & H m C sup _G s _ m st

(45)

st

with m

k

_G s _ m = : :  k&l k=0 l=0

: K, |K | =k L, |L| =l, LK

"

V l 1 V l d KL }}} D Gs x 1 x d

"

L2

But &(V l 1x 1 ) } } } (V l dx d ) D KLG s & L 2 C

m 12(K rs ) sup &D KLG s(x)&, r

x # Ks

since G s has compact support in K rs . Therefore, _G s _ m P() sup &G s & C m

(46)

st

where P() is a polynomial of degree lower than m. As in the proof of Proposition 4.14, we estimate &G s & C m using Lemma 3.12. We obtain that \k # N, _C such that sup &G s & C m C k

(47)

st

(45), (46) and (47) imply the result of Proposition 4.17. K 4.4. Localization It is worth to notice that our probabilistic ansatz 9 sees only the values of V on a compact set of R d, so that it is unsensitive to a truncation of V. This truncated potential trivially satisfies the assumptions H6, so that our probabilistic ansatz is an O(  ) approximation in C -norm to the wave function associated to this truncated potential. This remark leads us to the definition:

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270

BEN AROUS AND CASTELL

Definition 4.18. We will say that (s, V ) is semi-classically localizable in norm & } &, if for any initial condition f compactly supported, for any T, there is a compact subset K T of R d (depending on (s, V )), and a truncation / T (with support in K T ), such that &8 V (t, } )&8 /T V(t, } )&=O(  ), for t0, & f &C & f & C k

(48)

In this case, our strategy can be applied to obtain the semi-classical expansion of 8. On the other side, to be semi-classically localizable in norm & } & is a necessary condition for the semi-classical expansion to be true in norm & } & (note that the semi-classical expansion (37) depends only on the values of V on a compact set). Therefore, for the norms satisfying (48), we have (s, V ) semi-classically localizable in norm & } &  The semi-classical expansion (37) is valid for 8 in norm & }&  &8&9 &=O(  ) To determine the class of potentials and initial conditions which are semi-classically localizable in a given norm, is a purely analytical problem. In this section, we have given some answers to this question, but we are aware that these results are certainly not optimal.

APPENDIX 1:

Proof of Proposition 2.1

Let X H denote the Hamiltonian vector field associated to V, X H : R d_R d  R d_R d (q, p) [ ( p, &{V (q)). Let , s(q, p) denote the Hamiltonian flow, that is the flow of diffeomorphisms of R 2d associated to X H . To solve (C xt ) is equivalent to look for some initial speed p such that , t(x, p) is in L0 .[(q, &{s(q)), q # R d ]. If for all s0, Ls ., &s(L0 ), we have then K t =[x # R d, _p such that (x, p) # Lt and ? b , t(x, p) # K ]

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271

SEMI-CLASSICAL APPROXIMATIONS

where ? is the first projection in R 2d : R 2d  R d. (q, p) [ q. From the diffeomorphism property of , t , it is not difficult to see that K t is a compact subset of R d. Let us consider T K .sup[t0,

v ? t =? | Lt is non singular on Kt ., t ? &1 0 K v ? &1 t (?Kt )=Kt ].

Since L0 is projectable, a continuity argument shows that T K >0. T K represents the time of appearance of caustics. For t
so

that

# xt(t)

is

implicitly

defined

by

F(t, x, y )=? b , &t( y, &{s( y ))&x =? t b , &t( y, &{s( y ))&x. Now (Fy )(t, x, y )=T , &t ( y, &{s( y )) ? t b T ( y, &{s( y )) , t is invertible for all t
APPENDIX 2: Proof of Lemma 2.4 M rs(%). s0 g* ru(%) $B u is a Gaussian process with 0 mean and joint quadratic variation A r(%, %$, s)=

|

s

( g* ru(%), g* ru(%$)) du. 0

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272

BEN AROUS AND CASTELL

It follows from the regularity of g that there exists a modification of (M rs(%)) which is C  in % # 3, and that \: # N n, D :% M rs(%)=

|

s

D :% g* ru $B u 0

(see for instance Theorem 3.1.2, or exercise 3.1.6 of [16]). The same holds for M is(%). s0 g* iu $B u . Therefore, 1 Z t(%).F(g(%)+- i=B) exp & - i=

\

|

t

g* s(%) $B s &

0

1 2i= 2

|

t 0

| g* s(%)| 2c ds

+

is differentiable in %, and \ j # N m, 1 Z (%)=L% j (F)(%, g(%)+- i=B) exp & % j - i=

\

|

t

g* s(%) $B s &

0

1 2i=2

|

t

| g* s(%)| 2c ds

0

+

where \| # Wt(C), L % j (F)(%, |)=D r F(|)

g r g i 1 (%)+D i F(|) (%)& 2 F(|) % j % j i=

|

g* s (%) $| s . % j

t 0

Remark.  t0 (g* s % j )(%) $| s is defined by integration by parts, since g* s % j is continuously differentiable. To invert expectation and differentiation, it is sufficient to prove \: # N n, |:| =2, sup E[ |D :% Z t(%)| ]<.

(49)

%#3

But, it follows from a computation of the second derivatives and from the hypothesis on F, that _C>0 such that \: # N n, |:| =2, n

\

M t (%)

exp &Re

n

j, k=1 n

+ : j, k=1

g* s

0

j

s

j=1

+ :

t

" \ \ - i= ++ | % $B " M (%) g* g* "exp \&Re \ - i= ++ | % $B | % $B " M (%)  g* "exp \&Re \ - i= ++ | % % $B " +

E[ |D :% Z t (%)| ]C 1+ :

t

t

Lq

t

s

s

s

t

0

j

t

2

0

j

s

0

k

s

s

with 1r+1q=1. This implies easily (49).

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k

Lq

Lq

273

SEMI-CLASSICAL APPROXIMATIONS

Therefore, G is differentiable in 3, and \% # 3, \j # [1, ..., n], G 1 =I j +J j exp & 2 % j 2i=

\

|

t

+

| g* s | 2c ds , 0

with g r

g i

1 exp & - i=

1

t

t

| g* | ds _\ % \ +& % + 2i= 1 1 1 g* g* J =E F } & $B & | g* ds exp & | _ \ - i= % i= % + \ - i= | g* $B +& I j =E

Dr F }

+D i F }

j

j

t

t

s

j

s

0

Consider l: Wt  C, |[

|

t 0

2

j

% j

g* s $B s &

2

0

|

t

s

s

0

2 s c

0

s

s

0

j

K : Wt  W t* , c

and

g* s

|

\

| [ exp &

$| s

1 - i=

|

t

+

g* s $| s l 0

,c where W * is the dual space of W ct . Then, K=K r +iK i, with K r, K i # D H t  (see [25] p. 51 for the definition of this space). If $ denotes the dual operator of D defined in corollary of Proposition 1.9 in [25] ($ : D H   D  ), then

\

$K(|)=exp &

1

- i= |

t

g* s $| s 0

+_

|

t

&

0

g* % j

$| s &

1

- i= |

t

g* s

0

% j

&

g* s ds .

Thus, a Malliavin integration by parts leads to 1 E[( (D r +D i ) F( g(%)+- i=B), K(B)) ] J j =& - 2i =

&1+i -2

_

E (D r +D i ) F }

g

\%

j

&

g 0 % j

1 exp & - i=

+ \

|

t

0

g* s $B s

+& .

This proves the identity of the lemma. Moreover, since G% j has the same form as G, the lemma follows by induction on k.

APPENDIX 3: Proof of Lemma 3.12 We recall that forall s, Ws is the space C([0, s], R d ) endowed with the uniform norm & } & s (associated distance d s ). Let us write J = (s, x)=J 1, = (s, x)+J 2, =(s, x)

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274

BEN AROUS AND CASTELL

with J 1, = (s, x)#E[L(s, x, =B) exp((&1= 2 ) H(s, x, =B)) 1 &=B&s < \ ]. J 2, = (s, x)#E[L(s, x, =B) exp((&1= 2 ) H(s, x, =B)) 1 &=B&s  \ ]. We are going to show that J 2, = is exponentially small, and that J 1, = gives the asymptotic expansion (40). r Treatment of J 1, = . H(s, x, 0)=&iS(s, x), DH(s, x, 0)=DF x, s (0)=0, x, r since 0 is critical point of F s . A Taylor expansion of = [ H(s, x, =B) yields then 2

J 1, = (s, x)=e (i= ) S(s, x)E[L(s, x, =B) e &A(x, s, =) 1 &=B&s < \ ]

(50)

where A(s, x, =)= 10 (1&v) D 2H(s, x, =vB)(B, B) dv, is C  in =. Let us write now the Taylor expansion of %(s, x, =)=L(s, x, =B) e &A(s, x, =) around ==0. We obtain N

=k Z k (s, x)+= N+1R(s, x, =) k ! k=0

%(s, x, =)= :

(51)

with x

Z k (s, x)=Z k (s, x) e &A(s, x, 0) =Z k (s, x) e (&12) As (B, B) where Z k (s, x) is a sum of terms such as C(s, x)(B, ..., B), with C(s, x) any deterministic multilinear functional of order lower than 6k on the Wiener space Ws . Hence, Z k (s, x) is an element of the (6k)-th Wiener chaos. The coefficients of C(s, x) involve the derivatives of L(s, x, } ) at 0, and the derivatives of order greater than 2 of H(s, x, } ) at 0. Using Assumption 2, it is easily checked that for all st, C(s, } ) has compact support in K rs , so that the same is true for Z k (s, } ), and for R(s, } , =) by (51). Moreover, assumption 3 yields that \p0, sup sup E[ |Z k (s, x)| p ]<

(52)

st x # K rs

It follows then from (50) and (51) that N

2

N =k =k E(Z k (s, x))+ : E(Z k (s, x) 1 &=B&s  \ ) k! k! k=0 k=0

e &(ie ) S(s } x)J 1, = (s, x)= :

+= N+1E(R(s, x, =) 1 &=B&s < \ ).

(53)

The first sum gives the semi-classical expansion (40), if we put ; k (s, x)= E(Z k (s, x)). We have indeed already prove that ; k (s, } ) has compact

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275

SEMI-CLASSICAL APPROXIMATIONS

support in K rs . Moreover, by symmetry of Brownian motion, ; k (s, x)=0 for odd k. Let us then compute ; 0 . x

x

; 0(s, x)=E(Z 0(s, x) e (&12) As (B, B) )=L(s, x, 0) E(e (&12) As (B, B) ) A xs(B, B) is an element of the second Wiener chaos. Let us write its decomposition A xs(B, B)=: : xs(k) ! 2k(s, x) k

where (! k (s, x)) k1 are i.i.d N(0, 1) random variables, and (: xs(k)) k1 are the eigenvalues of the trace-class operator A xs (in increasing order). Since t0. Therefore 

x

E(e (&12) As (B, B) )= ` (1+: xs(k)) &12 =det &12(D 2S xs(# xs &x)) k=1

since D 2S xs(# xs &x)=A xs +Id. Note that this product is finite by the trace property. For the computation of the other terms, the reader is referred to [6]. We are now going to prove that the second sum in (53) is exponentially small. sup |E[Z k (s, x) 1 &=B&s  \ ]| st r x # Ks x

 sup |E[Z k (s, x) e (&12) As (B, B) 1 &=B&s  \ ]| st r x # Ks x

 sup E[ |Z k (s, x)| p1 ] 1p1 sup E[e (&p2 2) As (B, B) ] 1p2 st r x # Ks

st r x # Ks

_P[&=B& t  \] 1p3

(54)

with 1p 1 +1p 2 +1p 3 =1, p i >1. The first term in the right-hand side is finite using (52). Furthermore,

\

P[&=B& t  \] 1p3 exp &

\2 . 2p 3 t= 2

+

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276

BEN AROUS AND CASTELL

Thus, our goal is achieved as soon as we have found p 2 >1 such that the second term in the right-hand side of (54) is finite. From the lower semicontinuity of (s, x) [ : xs(1), it follows that Min Minr : xs(1)>&1 st x # K s

so that one can found p 2 >1, such that 1 Min Minr : xs(1)>& >&1. st x # K s p2 It implies that \st, \x # K rs ,

_ \

E exp &

p2 x A (B, B) 2 s

+&



=` k=1

1 - 1+ p 2 : xs(k)

<

Fatou's lemma yields that the function (s, x) [ E[exp(&( p 22) A xs(B, B))] is lower semi-continuous, and reaches therefore its maximum value on any compact set. Hence,

_ \

sup E exp & st r x # Ks

p2 x A (B, B) 2 s

+& <

It remains now to treat E[R(s, x, =) 1 &=B&s < \ ], R(s, x, =)=

|

1

0

(1&v) N  N+1 %(s, x, =v) dv N! =

with ( N+1=) %(s, x, =v)=M(s, x, =vB) e &A(s, x, =v) and \p>1, sup st, x # K rs sup v # [0, 1] E[ |M(s, x, =vB)| p ]<. Let us choose q such that 1
|

r

r

x # Ks

x # Ks

_ sup E[e &q Re A(s, x, =v)1 &eB&s < \ ] 1q dv st r x # Ks

C sup

sup E[e &q Re A(x, s, =v)1 &=B&s < \ ] 1q

st v # [0, 1] r x # Ks

But A(s, x, =v)= 12 A xs(B, B)+v  10 (1&u) 22 D 3H(s, x, =uvB)(=B, B, B) du, so that |Re A(s, x, =v)& 12 A xs(B, B)| 1 &=B&s < \ C\ &B& 2s

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277

SEMI-CLASSICAL APPROXIMATIONS

where the constant C can be chosen uniform in (s, x, v), st, x # K rs , v # [0, 1]. Therefore, sup

sup E [exp(&q Re A(s, x, =v)) 1 &=B&s < \ ]

st v # [0, 1] r x # Ks



sup

x

E [e &(qp2 p2 &q)(Re A(s, x, =v)&(12) Ax (B, B)) ] ( p2 &q)p2

st, v # [0, 1] r x # Ks x

_ sup E [e &( p2 2) As (B, B) ] qp2 st x # K rs 2

CE [e C\ &B&t ]<

as soon as

\<

1 . 2Ct

We have thus proved that _\ 0 >0, such that \\ \ 0 J 1, = (s, x)=exp

\

i S(s, x) =2

+

N

: ; k (s, x) k=0

=k +O(= N+1 ) k!

(55)

where the O(= N+1 ) is uniform in (s, x). Treatment of J 2, = . From now on, we fix \ such as (55) holds. Here are some notations we use in the sequel v When st,

4 s : Ws  R |[

{

1 2

&|& 2Hs +

if | # Hs otherwise

is the rate function for the large deviations of Brownian motion. 4 s is lower semi-continuous, and the level sets of 4 s are compact. v When k>0, C sk =[| # Ws , 4 s (|)k], v When k>0, and $>0, C sk($ )=[| # Ws , d s (|, C ks )<$ ]. As being a good rate function, C sk & [&|& s  \] is a compact subset of Ws . Thus the lower semi-continuous function | # Ws [ Re H(s, x, w)+4 s (|) reaches its minimum value m(s, x) on C sk & [&|& s  \]. Since s0. The function (s, x) [ m(s, x) being lower semicontinuous, _m>0 such that \st, \x # K rs , \| # C ks & [&|& s  \], Re H(s, x, w)+4 s (|)m.

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BEN AROUS AND CASTELL

Let us fix m~ such that 00 and ' 0 < \ such that \$<$ 0 , \', ' 0 <'< \, \st, \x # K rs , \| # C sk ($ ), &|& s  \&', Re H(s, x, |)+4 s (|)m~. For such ($, '), let us write Re H(s, x, =B)

_ \ +1 & = Re H(s, x, =B) E exp & 1 1 _ \ = Re H(s, x, =B) +E exp & _ \ +1 1 =

|J 2, = (s, x)| E exp &

2

&=B&s  \

2

&=B&s > \&'

&=B&s  \

2

k

=B # C s ($ )

&

k

ds(=B, C s )$

&.

(56)

Let us treat the second term in the right-hand side of (56) Re H(s, x, =B)=Im s(# xs (s)+- i=B s )& +

-2 2

|

|

s

ImV(# xs(u)+- i=B u ) du 0

s 0

#* xs (u) $(=B u ).

Therefore, _A>0 such that \st, \x # K rs ,

_ \

E exp &

Re H(s, x, =B) 1 &=B&s  \ 1 ds (=B, C ks )$ =2

+

-2 A E exp & 2 = 2=

&

s

#* (u) $B 1 \+ _ \ + & -2 A exp \= + E _exp \& = | #* (u) $B +& P[d (=B, C )$ ] A 1 exp \= + exp \= | |#* (u)| du+ P[d (=B, C )$ ] A exp \= + P[=B  C ($ )] exp

|

x s

u

x s

u

0

12

s

2

k

ds (=B, C s )$

k s

s

12

0

s

2

2

2

x s

2

t

k t

12

0

k t

12

where the constant A does not depend on (s, x). C kt($ ) is an open set of Wt , an the large deviations for the Brownian motion yield (for = sufficiently small), 1 P[=B  C kt($ )]exp & 2 Inf k 4 t (|) = |  Ct

\ k exp & \ =+

+

2

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SEMI-CLASSICAL APPROXIMATIONS

for $ sufficiently small. It is then sufficient to take k>A to obtain the exponential decay. It remains now to treat the first term in the right-hand side of (56). Let us choose m, such that 0
{

if | # C sk ($ ) otherwise.

Re H(s, x, |) m &4 s (|)

and &|& s > \&'

Since C ks ($ ) & [|, &|& s > \&'] is an open set of Ws , | # Ws [ H 1(s, x, |) is lower semi-continuous \st, \x # R d. Moreover, Re H(s, x, =B) E exp & 1 =B # C ks ($ ) 1 &=B&s > \&' =2

_ \

+

_ \

E exp &

H 1(s, x, =B) =2

&

+&

Using corollary 2.9 in [21] (which can be rendered uniform in (s, x)), we obtain that \st, \s # K rs , for = sufficiently small H 1(s, x, =B) E exp & =2

_ \

m

+& exp \&= + 2

The proof of Lemma 3.12 is thus complete.

References 1. S. Albeverio, A. M. Boutet de Monvel Berthier, and Z. Brzezniak, Stationary phase in infinite dimensions by finite dimensional approximations: Applications to the Schrodinger equation, Potential Anal. 4, No. 5 (1995), 469502. 2. S. Albeverio and Z. Brzezniak, Finite dimensional approximations approach to oscillatory integrals in infinite dimensions, J. Funct. Anal. 113 (1993), 117244. 3. S. Albeverio and R. Hoegh-Krohn, ``Mathematical Theory of Feynman Path Integrals,'' Lecture Notes in Math., Vol. 523, Springer-Verlag, BerlinNew York, 1976. 4. V. I. Arnold, ``Mathematical Methods of Classical Mechanics,'' Graduate Texts in Mathematics, Springer-Verlag, New York, 1978. 5. R. Azencott and H. Doss, L'equation de Schrodinger quand  tend vers 0. Une approche probabiliste, in ``2 eme rencontre franco-allemande entre physiciens et mathematiciens, CIRM, Marseille, 1983,'' Lecture Notes in Mathematics, Vol. 1109, pp. 117, Springer-Verlag, New YorkBerlin, 1985. 6. G. Ben Arous, Methode de Laplace et de la phase stationnaire sur l'espace de Wiener, Stochastics 25 (1988), 125153. 7. F. A. Berezin and M. A. Shubin, ``The Schrodinger Equation,'' Mathematics and Its Applications, Kluwer, Dordrecht, 1991.

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