A Semi-classical Trace Formula for Schrödinger Operators in the Case of a Critical Energy Level

A semi-classical trace formula for Schrodinger operators in the case

of a critical energy level

KHUAT-DUY David

CEREMADE, Universite Paris-Dauphine, Place de Lattre de Tassigny,

75775 Paris Cedex 16, France.

e-mail : dkd@ceremade.dauphine.fr

Abstract Let b H =?  h 2 2
+ V(x) be a Schrodinger operator onIR

n, with smooth potential

V(x)!+1

asjxj!+1. The spectrum of b

His discrete, and one can study the asymptotic of the smoothed

spectral density (E;h) = X k '  E k( h)?E  h  ; as h!0. Here, fE k( h)g k 2IN is the spectrum of b H and ^'2C 1 0 (

IR). We investigate the case

whereEis a critical value of the symbolH of b

H and, extending the work of Brummelhuis, Paul

and Uribe in [3], we prove the existence of a full asymptotic expansion for  in terms of p



h

and lnhand compute the leading coecient. We consider some new Weyl-type estimates for

the counting function : N Ec;(

h) = #fk2IN = jE k(

h)?Ejhg.

Contents

1 Introduction and main statements.

1

2 A result concerning the classical dynamic.

7

3 Microlocal decomposition and Proof of the main statements.

12

4 Proof of Theorem 3.2

19

4.1 The phase function

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

19 4.2 The asymptotic expansion

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

24 4.3 Calculation of the leading term

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

30

5 The case with an observable

33

6 An example for the case



= 0

34

A Determination of the density dened on



36

B A theorem for the singularity in

t

= 0

.

38

1 Introduction and main statements.

Consider a Schrodinger operator

b

H

=? 

h

2 2
+

V

(

x

) on

IR

n (

n

1) with

V

2

C

1(

IR

n) and lim jxj!+1

V

(

x

) = +

1. Then, from well known results, the spectrum of b

H

consists in a discrete set of real eigenvalues :

E

0(

h

) 

E

1(

h

)




where we count each eigenvalue

according to its (nite) multiplicity.

It is still an open problem to nd, in all the cases, an asymptotic expansion, as 

h

!0, of these

eigenvalues in terms of the classical dynamic in phase space

IR

2n generated by the corresponding

Hamiltonian :

H

(

x;

) =



2

=

2 +

V

(

x

). This problem is only solved in the case of an integrable

classical system with the WKB method which leads to the EBK (or Bohr-Sommerfeld) quantization. For more generic Hamiltonians, the semi-classical trace formula gives us one of the most important tool in this direction. There has been many mathematical publications on the subject which lead, many years after Gutzwiller has introduced it [15], to a rigorous version of this formula [19, 14, 4, 18, 21, 20, 7, 8]. Let us now describe in detail this result.

For every

'

in the Schwartz space S(

IR

), we will note ^

'

for the Fourier transform of

'

^

'

(

t

) =Z +1 ?1

e

?it

'

(



)

d

and 

'

for the inverse Fourier transform, so that 

'

(

t

) = (2



)?1

'

^( ?

t

).

Consider a strictly positif real number



and dene the smoothed spectral density (

E;



h

) = X Ek (h)2[E?;E+]

'



E

k(

h

)?

E



h

 for any ^

'

2

C

1 0 (

IR

) and any

E

2

IR

.

Consider the following two assumptions

H1

E

is a regular value of the classical Hamiltonian

H

,

We recall that



t is a clean ow on E if

- PE =f(

t;x;

)2

IR

E

= 

t(

x;

) = (

x;

)g is a smooth submanifold of

IR

E and

- 8(

t;x;

)2PE,

T

t;x;PE =f(

t;x;

)

= t:

(

x;

) +

d

t(

x;

)

:

(

x;

) = (

x;

)g

where (

x;

) is the innitesimal generator of the ow at (

x;

).

Then the semiclassical trace formula says that if (H1) and (H2) are satised, one can nd a sequence of distributionf

d

l;kgsuch that, as 

h

!0,

(

E;



h

) =XL l=1

e

i  hS l

h

1?dimY l 2 0 @ X k2IN

< d

l;k

;

' >

^ 

h

k 1 A+ O(h 1 ) where - (

Y

1

;

;Y

L) are the connected components of the periodic orbit manifoldPE such that

Y

l\supp ^

'

6=; and

-

S

l is the action of any arbitrary periodic orbit of

Y

l.

The leading distribution

d

l;0 of each connected component can be eectively calculated and

related to an intrinsic density dened on

Y

l. If, for example, we assume that the periodic points of

non-zero period are of zero (Liouville) measure on E, we obtain

(

E;



h

) = ^

'

(0)LVol(₍₂

_

_)nE)

h

?n+1+ O(

h

?n+2)

:

where LVol(E) stands for the Liouville measure of E.

Remark 1.1

In the case of strictly positive Liouville measure of the periodic points of E, we

have : (

E;



h

) =



E;h 0 (^

'

)

h

?n+1+ O(

h

?n+2) (1) where



E;h

0 (^

'

) is a density which is oscillating in 

h

and supported by the periods (non necessarily

discrete) of the periodic orbits (including the zero period). If we assume, for example, that E is

fully periodic with a constant period equal to

T

, then (cf [8])



E;h 0 (^

'

) = LVol( E) (2



)n X k2ZZ

e

? i  hk (S+TE)?ik=2

'

^(

kT

)

where

S

and



are the common action and Maslov index of E. In [22] was nd an explicit

expression for



E;h

0 (^

'

) in the general case, and it is proven that formula (1) remains valid even if

the Hamiltonian ow is not clean (with a worse error term) : cf Th. 1.4 of [22].

The proof of all these results is based on the theory of FIO (Fourier Integral Operators) where (

E;



h

) can be seen \asymptoticaly" as the composition of two such FIO. The assumptions (H1) and (H2) are just the conditions that allow us to apply the general theorem for the clean composition of two semi-classical FIO (cf [18]). In this paper, as in [3], we wish to eliminate the hypothesis (H1) which will lead to dealing with degenerate phase functions in the oscillatory integrals that we will meet. The theory of oscillatory integrals with degenerate phase functions has an important background in the mathematical literature. A general review may be found in the book of Arnold, Gusein-Zade and Varchenko [1] which shows the complexity of the problem. In order to deal with simple degeneracies, we will make the following assumptions on  which is the set of critical points of

H

and which is simply, in our case,

 =f(

x;

0) 2

IR

2n

= dV

(

x

) = 0 g

;

H3

 is a compact submanifold (without boundary) of

IR

2n,

H4

H

has a non degenerate normal hessian on , that is

d

2

H

(

x;

0) restricted to

T

x;0

IR

2n

=T

x;0

is non degenerate for all (

x;

0)2,

H5

the multiplicity of the eigenvalues of the normal hessian of

H

is independent of (

x;

0)2.

Remark 1.2

These are the same assumptions as in [3] and, because the connected components of  are contained in level sets of

H

and since we are interested in estimates that are local in energy, we will also assume, with no loss of generality, that  is connected and included in Ec =

H

?1(

E

c)

for a certain value,

E

c, of

H

. We will suppose moreover that Ec is also connected. Because our

proofs use only micro-local technics and not global properties, we only need to sum up, in the case of many connected components of  or Ec, all the contributions coming from these connected

components. We remark that in this general case, the assumptions (H3), (H4) and (H5) need only to be satised for the part of  contained in a small neighborhood of Ec

In [3] was shown, under the assumptions (H3), (H4) and (H5), that (

E

c

;



h

) has an asymptotic

expansion in terms ofp



h

and ln

h

, but it was assumed that the support of ^

'

does not contain any period of the linearised ow 1 on  (see appendix B).

As we will see later, we will have to include the contribution of the periods of the linearised ow in the asymptotic expansion of (

E

c

;



h

) in order to eliminate the hypothesis on the support of

^

'

. When (H1) is not assumed, the hypothesis (H2) can be not true (except for very special cases like the top of a double well in one dimension), and we have to weaken (H2) in the following way :

H2bis

the Hamiltonian ow



t is a clean ow 2 on 

Ec n. Let (



1(

x

) 2

;

;

r(

x

) 2

;

?



r +1(

x

) 2

;

;

?



r +(

x

) 2

;

0

;

;

0) (



i(

x

)

>

0)

be the eigenvalues of

d

2

V

(

x

) for all (

x;

0)

2 and dene the codimension of  in

IR

2n by

N

:= codim = 2

n

?dim =

n

+

r

+

:

The assumptions (H3), (H4) and (H5) tell us that

N

,

r

and



are independent of the point (

x;

0)2.

We can now state our main result :

Theorem 1.3

Let

'

be a test function such that

'

^2

C

1

0 (

IR

). Assume, for simplicity, that the

periodic points on Ec

n of non zero period are of zero (Liouville)-measure. Then, under the

assumptions (H2bis), (H3), (H4) and (H5), we have, as 

h

!0,

1_{We say that} T is a period of the linearised ow at z 2  if there exist a vector u 2 T z IR2 n nT z so that dT(z):u=u. 2_i.e.  P E c n= f(t;x;)2IR( Ec n) =  t( x;) = (x;)gis a submanifold ofIR1+2 n  8(t;x;p)2P Ecn ; Tt;x;P Ecn= f(t;x;)2T( t;x;)( IR E c) = t:(x;) +dt(x;):(x;) = (x;)g

1. if



1 and if



and

N

?



are odd, then

N

is even and (

E

c

;



h

)

h

?n+1 2 4 N=2?2 X j=0

c

j

h

j+ 

h

N=2?1 1 X j=0 X l=0;1

c

j;l

h

j(?ln

h

) l 3 5

;

(2)

with leading coecients

a) if

N

4 :

c

0 =

LV ol

(Ec) (2



)n

'

^(0) (3) b) if

N

= 2 :

c

0;1 = ? 1



1 ^

'

(0) and (4)

c

0;0= Z

'

(



)

f

(



)

d:

(5)

where

f

is analytic in the region jIm



j

<

1

2 and

f

(



) =

o

(

j



j) at innity.

2. if



1 and if



or

N

?



is even, then

N

3 and

(

E

c

;



h

)

h

?n+1 2 4

LV ol

(E c) (2



)n

'

^(0) + +1 X j=1

c

j

h

j=2 3 5 (6) 3. if



= 0, then  = Ec and (

E

c

;



h

)

h

N= 2?n 1 X j=0

c

j

h

j (7)

with leading coecient

c

0 = (2



) N=2?n2n?N

e

?iN=4 Z 

<

₍

_t

1 ?

i

0)n ?N=2 QN ?n i=1 sin( i (x) 2 (

t

?

i

0))

;

'

(

t

)

> dx:

(8)

Remark 1.4

1. Formula (41) gives a more explicit expression for

f

.

2. The distribution introduced in formula (8) is studied in Theorem 3.1.11 of [16] and can be dened as

<

₍

_t

1 ?

i

0)n ?N=2 QN ?n i=1 sin( i (x) 2 (

t

?

i

0))

;

'

(

t

)

>

= lim_ !0+ Z IR ₍

t

?

i

)n

'

(

t

) ?N=2 QN ?n i=1 sin( i (x) 2 (

t

?

i

))

dt:

3. In the case of strictly positive measure of the periodic points, the theorem is the same except for the leading coecients of (3) and (6) where they have to be replaced as in the remark (1.1).

4. The only possibility for the case 1)b) is n=1 and dim = 0. This corresponds to the top(s) of a double-well (or multiple-well in the case of many connected components of ).

5. The proposition 2.1 is a crucial point in the demonstration of this theorem because it tells us that any periodic point of Ec of period smaller than a given constant stands outside a

suciently small neighborhood of , this neighborhood depending only on that constant. This is wrong for a general Hamiltonian as is shown in the remark 2.2 ; that's why we had to restrict ourselves to Schrodinger-type operators.

6. The contribution coming from the periods of the linearised ow appears as a regularization of the Duistermaat-Guillemin density which has been dened in [11], [14],[3] and is studied in appendix A. This regularization appears explicitly in (8), but it is also present in the other cases as a contribution to lower orders. They are nevertheless dierent because they make use of all the support of ^

'

and not only the period of the periodic orbits. This is because every real can be seen as the period of a point on .

In the third case of this theorem, we can express the leading coecient as the action of a density on

'

in the following way:

Theorem 1.5

With the hypothesis of Theorem 1.3 and in the case where



= 0, we have (

E

c

;



h

)

h

N= 2?n 1 X j=0

c

j

h

j (9)

with leading coecient

1) if

d >

0 :

c

0 = (2



)?d=2 ?(

d=

2) Z IR 2 4 Z  X k2IN N?n (



?



k(

x

)) d=2?1 +

dx

3 5

'

(



)

d

(10) 2) if

d

= 0 :

c

0 = X k2IN N?n

'

(



k) (11)

where

d

is the dimension of,



k(

x

) =



1(

x

)(

k

1+ 12)+


+



N ?n(

x

)(

k

N?n+ 12) if

k

= (

k

1

;

;k

N ?n) and (



?



) d=2?1

+ is the function equal to (max(0

;

?



))d= 2?1.

Remark 1.6

1. The sum of (10) is necessarily a nite sum.

2. In this theorem, the hypothesis (H2bis) is trivially satised because  = Ec.

3. The case

d

= 0 corresponds to the case where

E

c is the energy of the bottom of a potential

well. This case was studied more precisely in [24].

Using these results, one would like to obtain some Weyl-type estimates for the counting function

N

Ec;(

h

) := #

f

k

2

IN =

j

E

k(

h

)?

E

cj

< 



h

g

by letting

'

equal to the characteristic function :



[?;+]. In the case where

N

= 2, it is proved

in [3] that this is possible because the linearised ow does not have any non-zero periods so that we can apply Theorem 1.3 of [3]. If this could be done in the other cases, we would obtain for example, in the case where



= 0 and with some easy calculations

N

Ec;(

h

) = (2





h

)?d=2 ?(

d=

2 + 1) Z  X k0 (



?



k(

x

)) d=2 +

dx

+

o

(

h

?d=2 ) (12)

Unfortunately, the Tauberian lemma of [3] does not apply in this case and we didn't achieve in generalizing it ; so that the formula (12) is only a conjecture. The fact that this result is true in the

case where

d

= 0 is a simple corollary of the results of [24]. In the last part of this paper, we will study the example of an integrable Hamiltonian satisfying the hypothesis of this conjecture with

d

= 1 and we will prove, by studying directly the eigenvalues of the associated quantum problem, that (12) is veried. We would like also to remark that a density similar to the one dened on  in (12) has already been introduced in the context of a Schrodinger operator with a magnetic eld (cf [5, 25]).

The paper is organised as follows : In section 2 of this paper, we will prove a particular result on the classical dynamic which will prove that the points (

t;x;

) 2 PE

c where (

t;x;

)

2 supp

^

'

(E c

n) can not accumulate on . In section 3, we use the fundamental theorem of this paper

(proved in section 4), to prove the results announced in this introduction. Section 5 is devoted to the case with an observable.

Acknowledgments

: I wish to thank T. Paul for suggesting the problem treated in this paper and for many helpful discussions. I thank E. Sere for useful conversations concerning the classical dynamics part of this paper. Finally, I thank R. Brummelhuis, S. Dozias, O. Fernandez and P. Leboeuf for helpful conversations.

2 A result concerning the classical dynamic.

Let

H

be a classical Hamiltonian of the form

H

(

x;p

) =

p

2

=

2 +

V

(

x

) where we assume that

V

2

C

4(

IR

n) and let

E

cbe a critical value of

H

.

We will assume moreover that :

 , the set of critical points of

H

, is a compact submanifold of

IR

2n contained in Ec =

H

?1(

E

c), 

d

2

H

(

x;

0) jTIR2n=T

 is non degenerate for all (

x;

0) 2.

In the following, when we will talk about periodic orbits, we will only consider periodic orbits which are not xed points.

In this section, we show that there exists a neighborhood

U

of  in

IR

2n such that every

periodic orbit on Ec with a period smaller than a given constant will never meet

U

. Of course,

U

depends on this constant. This result will allow us, in the following parts, to isolate, on the one hand, the periodic orbit of Ec

n which will be studied as in the regular trace formula, and, on

the other hand, the singularity coming from .

Proposition 2.1

Let

T >

0.

There exists

U

T, a neighborhood of in

IR

2nsuch that for every periodic orbit

E

c with period

T

2]0

;T

], we have :

\

U

T =;

Remark 2.2

This proposition is no more true in the case of a general Hamiltonian. Indeed, if we choose :

H

(

x;p

) =

p

2 1

=

2 ?

p

2 2

=

2+

x

2 1

=

2 ?

x

2 2

=

2 in a neighborhood of (0

;

0), then  = f(0

;

0)gand

all the points (

x

1 =

;x

2 =

;p

1 = 0

;p

2 = 0), where

 >

0 belong to

H

?1(0) and are periodic with

period equal to 2



.

We will now be concerned, until the end of this part, with the proof of this proposition. Let (

x

0

;

0)

2 . Because  is compact, it is sucient to prove that we can nd a

neigh-bourhood of (

x

0

;

0) in

IR

2n which verify the proposition. We will rst prove the following lemma

:

Lemma 2.3

There exists , a neighborhood of (

x

0

;

0) in

IR

2n such that every periodic orbit of

Ec will not stay forever in .

Proof of the lemma: We can assume that

E

c= 0 and

x

0= 0. We can moreover assume that there

exists

U

, a neighborhood of (

x

0 = 0) in

IR

n_{such that} \(

U



IR

n) =f(

x

0

;x

00

;

0) 2

IR

s

IR

n ?s 

IR

n

=x

0 =

f

(

x

00) g\(

U



IR

n)

where dim =

n

?

s

,

f

(0) = 0,

df

(0) = 0 and

d

2

V

(0

;

0) =diag( ?

a

2 1

;

;

?

a

2 r

;a

2 r+1

;

;a

2 s

;

0

;

;

0)

with

a

i

>

0. This can be easily obtain by some translation and rotation in the

x

-space.

Because a canonical change of coordinate will not aect the periodic orbits, we construct now a symplectomorphism dened on a neighborhood of (0

;

0) in order to make  at. Let's consider:



:

U

?!



(

U

) = ~

U

(

x

0

;x

00) 7!(

x

0 ?

f

(

x

00)

;x

00)

It is easy to see that



is a dieomorphism and that

d

(

x

) =

Id

?

df

(

x

00)

0

Id

!

does not depend on

x

0.

Let's write now:

:

U



IR

n?!

U

~ 

IR

n

(

x;p

)7!(



(

x

)

;

[t

d

(

x

)] ?1

:p

) = (

q;

)

:

We can notice that is a dieomorphism and that

(

dq

)

:

(

x;p

) =

< dq;d

(

x;p

)

:

(

x;p

)

>

= ([t

d

(

x

)]?1

:p

)

:

(

d

(

x

)

:x

) =

p:x:

We can thus deduce that is an (exact) symplectomorphism.

If we proceed to this change of coordinate, we get the following new Hamiltonian ~

H

(where we will also write (

x;p

) for the new coordinates) :

~

H

(

x;p

) =

H

( ?1(

x;p

)) = 12

< R

(

x

00)

p;p >

+~

V

(

x

0

;x

00) with ~

V

(

x

0

;x

00) =

V

(

x

0+

f

(

x

00)

;x

00) and

R

(

x

00 ) =

Id

+

df

(

x

00)

:

t

df

(

x

00) ?

df

(

x

00) ?t

df

(

x

00)

Id

!

:

Because

df

(0) = 0 and with a neighborhood ~

U

of 0 that may have been reduced, we can assume

that ₁

2

< R

(

x

00)

p;p >



p

2

4 8(

x;p

)2

U

~ 

IR

n

:

In particular,

R

(

x

00) is a positive denite matrix on ~

U

and so,

8(

x;p

)2

U

~ 

IR

n,

d

H

~(

x;p

) = 0 , 8 > < > :

p

= 0

@

x0

V

(

x

0+

f

(

x

00)

;x

00) = 0 1 2

< @

x 00

R

(

x

00)

p;p >

+

@

x0

V:@

x00

f

+

@

x00

V

= 0 , (

p

= 0

dV

(

x

0+

f

(

x

00)

;x

00) = 0 , (

p

= 0

x

0= 0

:

In these coordinates, we have \(~

U



IR

n) =f(0

;x

00

;

0)

2

U

~ 

IR

ng.

One can write: ~

H

(

x;p

_{) = 12}

< R

(

x

00 )

p;p >

₊₁₂

< d

2 x0

V

~(0

;x

00 )

x

0

;x

0

>

+

L

(

x

) (13) where

L

(

x

) = ~

V

(

x

0

;x

00 )? 1 2

< d

2 x0

V

~(0

;x

00 )

x

0

;x

0

> :

As ~

V

(0

;

0) = 0 (because

E

c = 0),

d

V

~(0

;x

00) = 0 for all (0

;x

00)

2

U

~ and

d

2

x0

V

~(0

;

0) =

d

2

x0

V

(0

;

0) is

non degenerate, we obtain that j

L

(

x

)j=O(jj

x

0 jj 3) uniformly in

x

00. Moreover, jj

@

x0

L

(

x

)jj=jj

@

x0

V

~(

x

0

;x

00) ?

d

2 x0

V

~(0

;x

00)

:x

0 jj=O(jj

x

0 jj 2) and jj

@

x 00

L

(

x

) jj=jj

@

x 00

V

~(

x

0

;x

00) ? 1 2

< @

x00

d

2 x0

V

~(0

;x

00)

x

0

;x

0

>

jj=O(jj

x

0 jj) because

V

2

C

4(

IR

n) Let's consider

A

= sup_i =1;


;r

a

2 i

;

B

= inf_i =1;


;s 1 2

a

2 i and

x

0 = (

x

1

;

;x

r

;x

r +1

;

;x

s) = (

x

0 ?

;x

0 +)

:

We thus have, for all (

x;p

)2

H

~ ?1(0) \(~

U



IR

n),

A

jj

x

0 ? jj 2 =

A

r X i=1

x

2 i  r X i=1

a

2 i

x

2 i = ~

H

(

x;p

) +Xr i=1

a

2 i

x

2 i = 12

< R

(

x

00 )

p;p >

₊₁₂

< Q

(

x

00 )

x

0

;x

0

>

+

L

(

x

) 

p

2 4 +12

< Q

(

x

00)

x

0

;x

0

>

+

L

(

x

) where

Q

(

x

00 ) =

d

2 x0

V

~(0

;x

00 ) + 0 B B B B @ 2

a

2 1 ... 2

a

2 r 0 1 C C C C A

:

We notice that

Q

(0) =diag(

a

2 1

;

;a

2

s) and so, with ~

U

that may have been reduced, we get

1 2

< Q

(

x

00 )

x

0

;x

0

>



B

2jj

x

0 jj 2

:

Asj

L

(

x

)j=O(jj

x

0 jj

3) and with ~

U

eventually again reduced, we also have j

L

(

x

)j

B

4jj

x

0 jj 2

:

We can nally conclude that there exists a constant

C

, strictly positive, such that for all (

x;p

)2

H

~ ?1(0) \(~

U



IR

n), jj

x

0 ? jj 2 

C

(jj

p

jj 2+ jj

x

0 jj 2)

:

(14)

We choose = ~

U



IR

n. This is not the nal , because we may reduce it (implicitly) from

line to line.

Let's nish the proof of the lemma by assuming that there exists

= (

x

(

t

)

;p

(

t

))t2[0;T

], a periodic

orbit, which stay forever in ~

H

?1(0) \.

Thus, there exists

t

0 such that:

- 1 2 jj

x

0 ?(

t

) jj 2 is maximum at

t

0 and - jj

x

0 ?(

t

0) jj

2

>

0 (or else we would stay on ).

This implies that

d

2

dt

2(12 jj

x

0 ?(

t

) jj 2)(

t

0) = jj

x

_ 0 ?(

t

0) jj 2+

x

0 ?(

t

0)

:

x

 0 ?(

t

0) 0

:

(15)

Let's show now that

x

0 ?(

t

0)

:

x

 0

?(

t

0)

>

0, which will contradict (15) and will end the proof of the

lemma. If we note

p

= ((

p

1

;

;p

r)

;

(

p

r +1

;

;p

n)) = (

p

0 ?

;

p

~)

R

(

x

00 )

:p

=

R

1

R

2

R

3

R

4 !

p

0 ? ~

p

!

d

2 x0

V

~(0

;x

00 )

:x

0 =

V

1

V

2

V

3

V

4 !

x

0 ?

x

0 + !

;

we have _

x

0 ?(

t

) =

@

p 0 ? ~

H

(

x;p

) =

R

1

:p

0 ?+

R

2

:

p

~ =

p

0 ?+ (

R

1 ?

Id

)

:p

0 ?+

R

2

:

p;

~ and thus 

x

0 ?(

t

0) = _

p

0 ?(

t

0) + (

R

1 ?

Id

)

:

p

_ 0 ?+

@

x 00

R

1

:

x

_ 00

:p

0 ?+

R

2

:

p

_~+

@

x 00

R

2

:

x

_ 00

:

p:

~ But _

p

0 ?(

t

0) = ?

@

x 0 ? ~

H

=?

V

1

:x

0 ? ?

V

2

:x

0 ++

@

x 0 ?

L;

(16) and as

V

1(0) =diag( ?

a

2 1

;

;

?

a

2 r),

V

2(0) = 0 and jj

@

x 0 ?

L

(

x

) jj=O(jj

x

0 jj 2), we obtain

x

0 ?(

t

0)

:

p

_ 0 ?(

t

0) 



jj

x

0 ?(

t

0) jj 2

for a strictly positive



.

Let's see now the remaining terms. We know that jj

p

jj

C

jj

x

0

?

jj from (14) and from the Hamilton equations we havejj

x

_ 00

jjjj

d

H

~jj ;

we thus conclude that

jj

@

x 00

R

1

:

x

_ 00

:p

0 ? jj



8jj

x

0 ? jj jj

@

x00

R

2

:

x

_ 00

:

p

~ jj



8jj

x

0 ? jj

:

Moreover, because

R

1(0) =

Id

and jj

p

_ 0 ? jj



1 jj

x

0 ?

jjfor a strictly positive



1 from (16), we have jj(

R

1 ?

Id

)

:

p

_ 0 ? jj



8jj

x

0 ? jj

:

It remains to bound the term

R

2

:

p

_~. But

_

p

0 += ?

@

x 0 +

H

~ =?

V

3

:x

0 ? ?

V

4

:x

0 + ?

@

x 0 +

L

and jj

@

x 0 +

L

jj=O(jj

x

0 jj 2) ; thus jj

p

_ 0 + jj



jj

x

0 ? jj for a constant

 >

0. Similarly, _

p

00= ?

@

x00

H

~ = 12

< @

x00

R:p;p >

+12

< @

x00

d

2 x0

V

~(

o;x

00)

:x

0

;x

0

>

+

@

x00

L

and jj

@

x 00

L

jj=O(jj

x

0 jj) ; thus jj

p

_ 00 jj



1 jj

p

jj 2+



2 jj

x

0 jj 2+



3 jj

x

0 jj



4 jj

x

0 ? jj

for some strictly positive constants



1,



2,



3 and



4.

Using the fact that

R

2(0) = 0 we have: jj

R

2

:

p

_~ jj



8jj

x

0 ? jj

;

which nally gives us, by gathering all the inequalities:

x

0 ?(

t

0)

:

x

 0 ?(

t

0) 



2jj

x

0 ?(

t

0) jj 2

>

0

:

|

End of the proof of the proposition : We can assume that the of the preceding lemma is equal to the ball

B

(0

;

) (with

 >

0) and that for all

z

= (

x;p

) in

B

(0

;

) we have:

jj

z

_(

t

)jj=jj

dH

(

z

)jj

c

jj

z

jj (17)

for some

c >

0.

Let's take

U

T =

B

(0

;e

?cT



), and assume that there exists

= (

z

(

t

))

t2[0;T ], a periodic orbit of E c

with

T

2]0

;T

] and such that there exists a time

t

1 for which jj

z

(

t

1) jj

< e

?cT



.

>From the preceding lemma, there exist

t

0 such that jj

z

(

t

0)

jj=



. One can moreover choose

t

0 so

that 0

t

1

?

t

0



T

and such that for all

t

2[

t

0

;t

1], we have

z

(

t

) 2

B

(0

;

). Let's write

f

(

t

) =

e

2ct jj

z

(

t

)jj 2

:

Then, for all

t

with

z

(

t

)2

B

(0

;

), we have :

f

0 (

t

) = 2

e

2ct (

c

jj

z

(

t

)jj 2 +

z

(

t

)

:

z

_(

t

))  2

e

2ct (jj

z

(

t

)jjjj

z

_(

t

)jj+

z

(

t

)

:

z

_(

t

)) from (17)  0

;

so

f

does not decrease in [

t

0

;t

1].

We thus have

e

2ct1 jj

z

(

t

1) jj 2 

e

2ct0 jj

z

(

t

0) jj 2

which implies that,

jj

z

(

t

1) jj

e

c (t0?t1) jj

z

(

t

0) jj

e

?cT



3 Microlocal decomposition and Proof of the main statements.

In this section, we will show that (

E

c

;



h

) is equal, modulo O(

h

1), to a nite sum of oscillatory

integral with large parameter 1

=



h

. Some of these integrals will correspond to the ones encountered in the proof of the regular trace formula (with non-degenerate critical points). The others will have degenerate phases and will be studied in the next section. In order to study (

E

c

;



h

), we will have

to use the theory of FIO (Fourier Integral Operators) depending on a large parameter. We won't recall here the results of this theory and we refer the reader to the following non exhaustive list of references : [9, 10, 17] for the standard theory and [18, 7, 8, 20] when there is a large parameter. Dene (

h

) := (

E

c

;



h

) = X Ek (h)2[Ec?;Ec+]

'



E

k(

h

)?

E

c 

h



;

where ^

'

2

C

1 0 ([

?

T;T

]) for some

T >

0 and

 >

0.

If not explicitly mentioned,

T

,

'

and



will remain xed until the end of this paper. Let



2

C

1

0 ([

E

c

?

;E

c+



]) such that



1 in a neighborhood of

E

c and 0



1.

Then, with some standard semi-classical arguments (cf [8, 20]), we have (

h

) = X k0

'



E

k(

h

)?

E

c 

h





(

E

k(

h

)) +O(

h

1 ) = 12



Z T ?T ^

'

(?

t

)

e

i  htE cTr(

e

? i  ht ^ H



_{( ^}

_H

₎₎

_dt

₊O(

h

1)

Following the construction of [8], one can obtain ~

U

(

t

), a FIO which approximates the \localized" propagator :

U

(

t

) :=

e

?

i



ht ^

H



_{( ^}

_H

_{) in the}

_L

2-norm of linear operator and modulo O(

h

1). It is now

easy to verify that (

h

) is equal, moduloO(

h

1), to the composition of two FIO.

(

h

) = ~

U



A

+O(

h

1)

 The rst one is ~

U

(

t

).

Its lagrangian manifold is:

U =f(

t;

?

H

(

y;

)

;

t(

y;

)

;

(

y;

))

= t

2

IR;

(

y;

)2

IR

2n

g

;

and its principal symbol dened on U is



U(



) = (2





h

)?n=2

e

i  h R t 0L( s (y;))ds

e

i  2( t )



(

H

(

y;

))



 8



2U (18) where

L

(

y;

) =



2

=

2

?

V

(

y

) is the Lagrangian,



(

t) is the Maslov index of the path : f(

;

?

H

(

y;

)

;

(

y;

)

;y;

)g

2[0;t] and



 is the canonical density dened on U. We will

denote

K

U(

t;x;y

) for the local integral kernel of ~

U

(for more details, see [8]).

 The second one will be denoted by

A

.

Its lagrangian manifold is :

A=f(

t;

?

E

c

;x;;x;

)

=

(

t;x;

)2

IR

1+2n

g

:

and its kernel is

K

A(

t;x;y;



h

) = 12

_

(2





h

)?n Z

e

i  h ((x?y):+tEc)

'

^( ?

t

)

d;

Indeed, ~

U



A

= 12



Z

K

U(

t;x;y

)(2





h

)?n Z

e

i  h ((x?y):+tEc)

'

^( ?

t

)

ddtdxdy

= 12



Z

e

i  htE c

'

^( ?

t

) Z

K

U(

t;x;x

)

dxdt

= 12



Z

e

i  htE c

'

^( ?

t

)Tr(~

U

(

t

))

dt

= (

h

) +O(

h

1)

:

If (H1) and (H2) were assumed for

E

c, this would imply that the manifolds U A and
= f(

t;;x;;y;;t;;x;;y;

)

=

(

t;;x;;y;

) 2

IR

2+4n

g intersect cleanly (in the sense of Bott)

which is equivalent to saying that the composition of ~

U

and

A

is clean. As a result, we would obtain an asymptotic expansion of (

h

) in powers of 

h

.

In our case,

E

cis a critical value of

H

and we have to study more carefully the points where

this intersection is not clean.

By denition, UA and
intersect cleanly if and only if  (UA)\
is a submanifold of

IR

4+8n and  8



2(U A)\

;

T

((UA)\
) =

T

(UA)\

T



:

But (UA)\
= f(

t;

?

E

c

;x;;x;;t;

?

E

c

;x;;x;

)

=

H

(

x;

) =

E

c

;

t(

x;

) = (

x;

) and (

t;x;

)2

IR

1+2n g

;

and

T

(UA) = f(

t;

?

dH

(

x;

)

:

(

x;

)

;

(

J

r

H

(

x;

))

t

+

d

t(

x;

)

:

(

x;

)

;x;;

t;

0

;x;;x;

)

=

(

t;x;

) 2

T

t;x;

IR

1+2n g

:

Thus, if we denote

P

Ec = f(

t;x;

)2

IR

E c

= 

t(

x;

) = (

x;

) g

we get that U A and
intersect cleanly if and only if

C1

P

Ec is a submanifold of

IR

1+2n and

C2

T

(t;x;)

P

E c = f(

t;x;

)2

T

(t;x;)

IR

1+2n

= dH

(

x;

)

:

(

x;

) = 0 and (

J

r

H

(

x;

))

t

+

d

t(

x;

)

:

(

x;

) = (

x;

)g

:

Let's write (UA)\
=
1 [
2 [
3 [
4 where
1 = f(

t;

?

E

c

;x;

0

;x;

0

;t;

?

E

c

;x;

0

;x;

0)

=

(

t;x;

0)2([?

T;T

]nf0g)g

;

2 = f(

t;

?

E

c

;x;;x;;t;

?

E

c

;x;;x;

)

=

(

t;x;

)2

P

E c \([?

T;T

]nf0g)(E c n)g

;

3 = f(0

;

?

E

c

;x;;x;;

0

;

?

E

c

;x;;x;

)

=

(

x;

)2E c g

;

4 = f(

t;

?

E

c

;x;;x;;t;

?

E

c

;x;;x;

)

=

(

t;x;

)2

P

E c and

t =

2[?

T;T

]g

:

a) We don't have to study the points of
4 because for such points,

t =

2 supp^

'

and thus, they

will give no contribution to ~

U



A

.

b) The contribution of the points of
3 are calculated in [3] (cf Appendix B).

c) If we assume (H1bis), we easily get that on every points of
2, the intersection of U A

and
is clean and this gives us a contribution coming from the periodic orbits of Ec n in

the same way as in the regular trace formula.

d) We only need now to study the intersection of UAand
at a point

(

t;

?

E

c

;x;

0

;x;

0

;t;

?

E

c

;x;

0

;x;

0)2
1.

We rst use the result of the proposition 2.1 to say that there exist , a neighborhood in

IR

1+2n

of (

t;x;

0) such that

\

P

E c =

\f(

t;x;

0)2

IR

g

:

This implies that the condition (C1) is fullled locally at (

t;x;

0). Moreover, we have

T

(t;x;0)

P

E c =

IR



T

(x;0)

:

Since

dH

(

x;

0) = 0, we get that (C2) is fullled if and only if

8(

x;

) 2

T

(x;0)

IR

2n

;

(

d

t(

x;

0)?

Id

)

:

(

x;

) = 0)(

x;

)2

T

(x;0)

:

This is exactly the condition (5.7) of [14] where this case is studied.

Let's study more precisely the linearised ow

d

t(

x;

0) at a point (

x;

0) 2 to detect when

the condition (C2) is not fullled. We have

d

t(

x;

0) =

e

tJd2H(x;0) where

d

2

H

(

x;

0) =

Id

0 0

d

2

V

(

x

) ! and

J

= 0

Id

?

Id

0 !

:

d

2

V

(

x

) is a symmetrical matrix whose eigenvalues have multiplicities which does not depend on

(

x;

0) 2 (from (H5)). We can thus nd an orthogonal matrix

R

(

x

) which is

C

1 in

x

and such that _t

R

(

x

)

d

2

V

(

x

)

R

(

x

) = diag[



2 1(

x

)

;

;

2 r(

x

)

;

?



2 r+1(

x

)

;

;

?



2 s(

x

)

;

0

;

;

0]

:

with all the



i(

x

) strictly positive. We observe that with (H4), we necessarily have

n

?

s

= dim.

If we dene

P

(

x

) =

R

(₀

x

) 0

_R

₍

_x

₎

!

;

it is then easy to verify that

P

(

x

) is a symplectic and orthogonal matrix. We thus have

J

t

_P

₌

_P

?1

J:

Then

P

?1 (

x

)

d

t(

x;

0)

P

(

x

) =

e

tP?1 (x)Jd2H(x;0)P(x) =

e

tJtP (x)d2H(x;0)P(x) =

A

_C

(₍

x;t

_x;t

)₎

B

_D

(₍

_x;t

x;t

)₎ !

;

where

A

(

x;t

) =

D

(

x;t

) = diag[cos(



1(

x

)

t

)

;

;

cos(



r(

x

)

t

)

;

cosh(



r +1(

x

)

t

)

;

;

cosh(



s(

x

)

t

)

;

1

;

;

1]

B

(

x;t

) = diag[ 1



1(

x

) sin(



1(

x

)

t

)

;

;

1



r(

x

) sin(



r(

x

)

t

)

;

1



r+1(

x

) sinh(



r+1(

x

)

t

)

;

;

1



s(

x

) sinh(



s(

x

)

t

)

;t;

;t

]

;

(19) and

C

(

x;t

) = diag[?



1(

x

)sin(



1(

x

)

t

)

;

;

?



r(

x

)sin(



r(

x

)

t

)

;



r+1(

x

)sinh(



r+1(

x

)

t

)

;

;

s(

x

)sinh(



s(

x

)

t

)

;

0

;

;

0]

:

We nally see that

(

Ker

(

d

t(

x;

0)?

Id

)6=

T

(x;0))

,(9

i;

1

i



r;

9

k

2

ZZ = t

= 2

k



i(

x

))

:

These

t

will be called the periods of the linearised ow at (

x;

0) and we will write

f

t

j(

x

)g

?m(x)jm(x)= f

2

k



i(

x

)

= k

2

ZZ;

1

i



r

g\[?

T;T

]

;

for the nite set of periods of the linearised ow at (

x;

0) 2 which belong to [?

T;T

].

Of course,

t

0(

x

) = 0 is always a period of the linearised ow and this singularity is exactly

the one studied in [3]. We thus have to nd an asymptotic expansion for ~

U



A

in a microlocal

neighborhood of the points (

t

j(

x

)

;

?

E

c

;x;

0

;x;

0

;t

j(

x

)

;

?

E

c

;x;

0

;x;

0) with (

x;

0)2 and

j

6= 0.

Let's write precisely this microlocal decomposition.

Let (

t;E;x;;y;

) = (

t;x;y

) denote the projection on

IR

1+2n.

Since ~

U

is a FIO associated to the lagrangian manifold U, there exists

 f

U

g

2I a covering of open sets of U, 

 :

IR

1+2n+n 

?!

IR

(

t;x;y;

)7?! (

t;x;y;

)

a family of non degenerate phase functions associated to the

U

 and parametrizing U and



a

 :

IR

1+2n+n  +1 ?!

IR

(

t;x;y;



;



h

)7?!

a

(

t;x;y;



;



h

)

such that for all (





;



h

)2

IR

n

+1, we have

supp

a

(

:;



;



h

)(

U

)

and such that

K

U(

t;x;y;



h

) = X 2I Z IRn

e

i  h  (t;x;y;  )

a

(

t;x;y;



;



h

)

d

 = X 2I

K

(

t;x;y

)

:

We thus can write

(

h

) = X

2I Z

IR1+2n

K

(

t;x;y;



h

)

K

A(

t;x;y;



h

)

dtdxdy

+O(

h

1) = X 2I 1 2



Z IR1+n

e

i  htE c

'

^( ?

t

)

K

(

t;x;x;



h

)

dtdx

+O(

h

1) = X 2I

I

(

h

) +O(

h

1)

;

Since [?

T;T

]E

c is compact, we can assume that

I

is a nite set. Moreover, for the



such

that

U

 does not contain any point of the form (

t

j(

x

)

;

?

E

c

;x;

0

;x;

0), with (

x;

0) 2  and

j

2

[?

m

(

x

)

;m

(

x

)]nf0g, the asymptotic expansion of

I

(

h

) corresponds to the cases already studied :

cf for example [8, 3] and part 5 of [14].

We need now to calculate the asymptotic expansion of

I

0(

h

) for a



0 such that there exists a

(

x

0

;

0) 2 and a

j

0 2[?

m

(

x

0)

;m

(

x

0)] nf0g for which (

t

j0(

x

0)

;

?

E

c

;x

0

;

0

;x

0

;

0) belongs to

U

0.

Remark 3.1

Since the eigenvalues of

d

2

V

(

x

) do not cross when (

x;

0) stays on , we can assume,

if the covering f

U

g

2I is well chosen, that there exists a unique

j

=

j

0 such that there exists a

(

x;

0)2 with (

t

j(

x

)

;

?

E

c

;x;

0

;x;

0)2

U

0.

We thus have to nd an asymptotic expansion of

I

0(

h

) = 12

_

Z IR1+n+n 

e

i  h ( 0 (t;x;x;)+tEc)

a

0(

t;x;x;;



h

)^

'

(?

t

)

ddxdt

(20)

where the support of

a

0(

:;;



h

) is contained in a small neighborhood of (

t

j0(

x

0)

;x

0

;x

0), and 0

satises :

f(

t;@

t (

t;x;y;

)

;x;@

x (

:

)

;y;

?

@

y (

:

))

= @

 (

:

) = 0

;

(

t;x;y;

)2

IR

n 

g= U\ ~

with a neighborhood of (

t

j0(

x

0)

;x

0

;x

0) and ~ a neighborhood of (

t

j0(

x

0)

;

?

E

c

;x

0

;

0

;x

0

;

0).

Let's dene

N

:= 2

n

?dim =

n

+

s

the codimension of  in

IR

2n and

f

(

t;x

) =

_t

_n 1 ?N=2 Q ri=1sin( i (x) 2

t

) QN ?n i=r+1sinh( i (x) 2

t

)

:

for all (

t;x;

0)2

IR

 suciently close to (

t

j0(

x

0)

;x

0

;

0) and such that

t

6

=

t

j0(

x

).

We notice that

f

(

:;x

) is meromorphe with a nite pole at

t

j0(

x

) so that the distribution

f

(

t

?

i

0

;x

) is well dened by Theorem 3.1.11 of [16].

Theorem 3.2

As 

h

goes to 0,

I

0(

h

)

h

N=2?n 1 X j=0

d

j

h

j (21) with

d

0 = (2



) N=2?n2n?N

e

i=4 Z 

< f

(

t

?

i

0

;x

)

;

'

(

t

)



0(

t;x

)

> dx:

(22)

where



is an integer and



0 comes from the decomposition of unity associated to the open covering

f

U

g which means that



0 is a smooth function equal to 1 in a neighborhood of (

t

j0(

x

0)

;x

0).

The proof of this theorem, which is the central theorem of this paper, will be done in the next section.

End of the proof of theorem 1.3 : We can now end the proof of Theorem 1.3 by adding all the contribution coming from the microlocal decomposition. We use Theorem B.1 for the contribution coming from the singularity at

t

= 0, Theorem 5.6 of [14] for the contribution coming from  and far from the period of the linearised ow, Theorem 2.1 of [4] for the periodic orbits of Ec

n and nally

Theorem 3.2 for the contribution coming from the periods of the linearised ow. Since we assumed that the periodic orbits of Ec

n were of zero Liouville measure, their contribution does not appear

at the rst order of the dierent asymptotic expansions. Moreover, these asymptotic expansions must not depend on the microlocal decomposition, so that the addition of all the contributions must be \smooth" which implies that, in the case



= 0, we necessarily have from theorem B.1 :



=?

N

. |

Proof of theorem 1.5 : we recall that

d

= 2

n

?

N

.

The case

d

= 0 is just a corollary of the following lemma

Lemma 3.3

8

'

2S(

IR

) and 8



i

>

0 (1

i



n

), we have : X k1;


;k n 0

'

(



1(

k

1+ 12)+


+



n(

k

n+ 12)) = 1 (2

i

)n

<

Q 1 ni=1sin( i 2 (

t

?

i

0))

;

'

(

t

)

> :

Proof: Let's dene the tempered distribution

u

= X k1;


;k n 0



₁(k1+12)+


+ n (k n +12)

where



0 is the Dirac distribution supported inf



=



0

g.

We then have to show that

^

u

= 1₍₂

_i

_)nQ 1 ni=1sin( i 2 (

t

?

i

0))

:

Since

u

= lim_ !0+

e

?

u

= lim_ !0+ X k1;


;k n 0

e

?



1(k1+12)+


+n(kn+12)

;

we have ^

u

= lim_ !0+ X k1;


;kn0 Z IR

e

?it?



1(k1+12)+


+ n (k n +12)

d

= lim_ !0+ X k1;


;kn0

e

?i(1k1+


+ nkn )(t?i)?i(  12+


+ n 2 )(t?i) = lim_ !0+ n Y j=1 (X k0

e

?i  j 2 (t?i)

e

?i jk (t?i) ) = lim_ !0+ n Y j=1 (

e

?i  j 2 (t?i) 1?

e

?i j (t?i)) = lim_ !0+ n Y j=1 (₂

_i

_sin(j1 2 (

t

?

i

)))

:

|

For the case

d >

0, we use the fact that

F ?1  (

t

?

i

0) ?(n?N=2)  (



) =

e

i(n?N=2)=2 ?(

n

?

N=

2)



n?N=2?1 + (see [16]) and that F ?1 1 QN ?n i=1 sin( i (x) 2 (

t

?

i

0)) ! = (2

i

)N?n X k2IN N?n



k

(x) (see the previous lemma)

whereF

?1 denotes the inverse Fourier transform and the



k(

x

) are dened in theorem 1.5.

Since these two distributions have support contained in

IR

+, we can calculate the convolution which

gives F ?1 1 (

t

?

i

0)n ?N=2 QN ?n i=1 sin( i (x) 2 (

t

?

i

0)) ! (



) = F ?1  (

t

?

i

0) ?(n?N=2) 

?

F ?1 1 QN ?n i=1 sin( i (x) 2 (

t

?

i

0)) ! (



) =

e

i(n?N=2)=2 ?(

n

?

N=

2)(2

i

)N?n X k2IN N?n Z (



?



0)n ?N=2?1 +



 k (x)(



0)

d

0 =

e

iN=4 ?(

n

?

N=

2)2 N?n X k2IN N?n (



?



k(

x

)) n?N=2?1 +

:

Using the expression of

c

0 in the 3) of theorem 1.3 gives the desired result.

4 Proof of Theorem 3.2

In order to simplify the notations of the previous section, we won't write anymore the index



0

and we will write

t

0,

t

0(

x

),

R

,

P

and



i instead of

t

j0(

x

0),

t

j0(

x

),

R

(

x

0),

P

(

x

0) and



i(

x

0). We

will also use the following notations :

For any set of indices

K

[1

;n

], any

n



n

-matrix

A

and any

n

-vector

x

,j

K

jwill be the cardinal

of

K

,

A

K will be the j

K

jj

K

j-matrix obtained from

A

by suppressing the lines and columns

whose indices are not in

K

and

x

K thej

K

j-vector obtained in the same way.

K

c will be equal to

[1

;n

]n

K

.

In this section, we would like to prove the theorem 3.2 by studying the asymptotic expansion of the oscillating integral dened by formula (20) and which is, with the new notations,

I

(

h

_{) = 12}

_

Z IR1+n+n 

e

i  h ( (t;x;x;)+tEc)

a

(

t;x;x;;



h

)^

'

( ?

t

)

ddxdt

(23)

where the support of

a

(

:;;



h

) is contained in a small neighborhood of (

t

0

;x

0

;x

0), and satises : f(

t;@

t (

t;x;y;

)

;x;@

x (

:

)

;y;

?

@

y (

:

))

= @

 (

:

) = 0

;

(

t;x;y;

)2

IR

n



g= U\ ~

with a neighborhood of (

t

0

;x

0

;x

0) and ~ a neighborhood of (

t

0

;

?

E

c

;x

0

;

0

;x

0

;

0).

The proof will be divided in three parts : the rst one will consist in nding a new phase function, \equivalent" to +

tE

c, for which we will be able to identify the critical points and to

express the hessian at these critical points. In the second part, we will use a parameters depending Morse Lemma to express the phase function as the sum of a normal form and a remainder. We will then obtain an asymptotic expansion for

I

(

h

) with methods similar to the ones developed in Section 4 of [3]. Finally, the last part of this section will be devoted to the identication of the leading term of this asymptotic expansion.

4.1 The phase function

In order to diagonalise

d

2

V

(

x

0), we make the following linear change of variables in (23):

q

=

R

?1

:x

>From jdet

R

j= 1, we get

I

(

h

_{) = 12}

_

Z IR1+n+n 

e

i  h ( ~ (t;q;q;)+tEc)~

a

(

t;q;q;;



h

)^

'

( ?

t

)

ddqdt

where ~₍

_t;q;q

0

;

) = (

t;R:q;R:q

0

;

) and ~

a

(

t;q;q

0

;;



h

) =

a

(

t;R:q;R:q

0

;;



h

)

:

~ _{is a non degenerate phase function whose lagrangian manifold is} ~ = f(

t;@

t~(

t;q;q

0

;

)

;q;@

q~(

:

)

;q

0

;

?

@

q 0~(

:

))

= @

~(

:

) = 0 g = f(

t;@

t (

t;R:q;R:q

0

;

)

;q;@

x (

:

)

:R;q

0

;

?

@

y (

:

)

:R

)

= @

 (

:

) = 0g = f(

t;@

t (

t;x;y;

)

;R

?1

:x;

t

R:@

x (

:

)

;R

?1

:y;

?t

R:@

y (

:

))

= @

 (

:

) = 0g = f(

t;

?

H

(

y;

)

;P

?1

:

t(

y;

)

;P

?1

:

(

y;

) g

in a neighborhood of (

t

0

;

?

E

c

;R

?1

:x

0

;

0

;R

?1

:x

0

;

0). 8 (

x;

0) 2 , we denote



(

t;x

) = (

t;

?

E

c

;R

?1

:x;

0

;R

?1

:x;

0) and since

dH

(

x;

0) = 0, we obtain

T

(t;x) ~ = f(

t;

0

;P

?1

:d

t(

x;

0)

:

(

y;

)

;P

?1

:

(

y;

)) g = f(

t;

0

;P

?1

:d

t(

x;

0)

:P:

(

y;

)

;

(

y;

))g (24) and thus,

T

(t;x0) ~ =

f(

t;

0

;A

(

t

)

:y

+

B

(

t

)

:;C

(

t

)

:y

+

D

(

t

)

:;y;

)g

where

A

(

t

) =

A

(

x

0

;t

) is dened by (19).

B

(

t

),

C

(

t

) et

D

(

t

) are dened in the same way.

Let's dene

K

=f

k

2[1

;r

]

=

cos(



k

t

0)

6

= 0g

:

We can observe that

A

Kc(

t

0) = 0 and

A

K(

t

) and

C

K

c(

t

) are inversible when

t

is suciently close

to

t

0.

Lemma 4.1

 :

T

(t0;x0) ~ ?!

IR

2n+1 (

t;

0

;x;;y;

) 7?!(

t;x

K

;

K c

;

) is an automorphism.

Proof : We just need to prove that  is surjective. Let (

;v;w

)2

IR

1+2n. Dene

t

=



,



=

w

, and

y

=

A

?1 K (

t

0)

:v

K ?

A

?1 K (

t

0)

:B

K(

t

0)

:w

K

C

?1 Kc(

t

0)

:v

K c ?

C

?1 Kc(

t

0)

:D

K c(

t

0)

:w

K c !

:

We thus have (

t

0 is not written)

(

A:y

+

B:;C:y

+

D:

)

= (

A

K

:y

K+

B

K

:w

K

;A

Kc

:y

Kc+

B

Kc

:w

Kc

;C

K

:y

K+

D

K

:w

K

;C

Kc

:y

Kc +

D

Kc

:w

Kc)

= (

v

K

;A

Kc

:y

Kc+

B

Kc

:w

Kc

;C

K

:y

K+

D

K

:w

K

;v

Kc)

;

which shows that (

t;

0

;A:y

+

B:;C:y

+

D:;y;

) = (

;v;w

). |

~is a lagrangian manifold. By using the preceding lemma and a classical result of symplectic

geometry (cf Theorem 5.2 of [13]), we can nd a neighborhood of



(

t

0

;x

0) in

IR

4n+2 and

S

2 C 1(

IR

1+2n;

IR

) such that ~ \ =f(

t;@

t

S

(

t;q

K

;

K c

;

)

;q

K

;@

 K c

S

(

:

)

;@

q K

S

(

:

)

;

?



K c

;@



S

(

:

)

;

g\

:

(25)

We can notice that

S

is dened with an arbitrary constant. It is then easy to verify that if we dene

(

t;q;q

0

;

Kc

;

) =

S

(

t;q

K

;

Kc

;

)?

q

Kc

:

Kc?

q

0

:;

is a non degenerate phase function which parametrize ~ in a neighborhood of



(

t

0

;x

0) (in this

case, the ber of is



= (



Kc

;

)).

Denote



0 a point such that

@

~(

t

0

;q

0

;q

0

;

0) = 0 and

(

t

0

;@

t~(

t

0

;q

0

;q

0

;

0)

;q

0

;@

q~(

:

)

;q

0

;

?

@

q0 ~(

:

)) =



(

t

0

;x

0)

:

Then, we dene the arbitrary constant of

S

in order to have (

t

0

;q

0

;q

0

;

0

;

0) = ~(

t

0

;q

0

;q

0

;

0)

Now, since and ~are both non degenerate phase functions whose associated lagrangian manifold are identical in a neighborhood of



(

t

0

;x

0), and because we manage to make them coincide at one

point, we can use the proposition 1.3.1 of [10] to say that there exists a classical symbol

b

of order :

ord

(

b

) =?(

n=

2 + (

n

+j

K

cj)

=

2) such that

I

(

h

) =Z IR1+2n+jK c j

e

i  h ( (t;q;q; K c;)+tE c )

b

(

t;q;q;

Kc

;;



h

)^

'

( ?

t

)

d

K c

ddqdt

+ O(

h

1) (26)

where the support of

b

(

:;

Kc

;;



h

) is contained in a small neighborhood of (

t

0

;R

?1

:x

0

;R

?1

:x

0).

Remark 4.2

SinceRt 0

L

(



s(

x;

0))

ds

=

?

tE

c for all (

x;

0) 2 and from (18), we have



U(



(

t

0

;x

0)) = (2





h

) ?n=2

e

? i  ht0E c j

dtdx

K

d

K c

d

j 1=2

:

But since the principal symbol of a FIO is nothing but the leading term of a stationary phase expansion, we necessarily have

e

?

i  ht0E c =

e

i  h ~ (t0;q0;q0;0). We nally obtain (

t

0

;q

0

;q

0

;

0

;

0) = ~(

t

0

;q

0

;q

0

;

0) = ?

t

0

E

c

:

(27)

Let's denote the phase function of (26) by

(

t;q;

Kc

;

) := (

t;q;q;

Kc

;

) +

tE

c

and dene the set of indices of the \degenerate" directions by

K

D :=f

k

2[1

;r

]

=

cos(



k

t

0) = 1

g (28)

Let us look for the critical points of the phase function  of

I

(

h

). For (

t;q;

) close to (

t

0

;R

?1

:x

0

;

0), we have (

t;q;

Kc

;

) is a critical point of  , 8 > > > > > < > > > > > :

@

t

S

(

t;q

K

;

Kc

;

) = ?

E

c

@

qK

S

(

t;q

K

;

K c

;

) =



K



Kc = ?



K c

@

K c

S

(

t;q

K

;

K c

;

) =

q

Kc

@



S

(

t;q

K

;

Kc

;

) =

q

, ( (

t;

?

E

c

;q;;q;

)2 ~



Kc = ?



K c , 8 > < > :

H

(

P:

(

q;

)) =

E

c

P

?1

:

t(

P:

(

q;

)) = (

q;

)



Kc= ?



K c

(from proposition (2.1)) , (

P:

(

q;

)2



Kc = ?



K c , 8 > < > : (

q;

0)2

P

?1()



= 0



Kc = 0

:

We go on by calculating the hessian of  at a critical point (

t;q;

0

;

0) where (

q;

0)2

P

?1(). We have

d

2 qK;K c; K;K c;q K c(

t;q;

0

;

0) = 0 B B B B B @

@

2 qK

S

@

2 qK;K c

S

(

@

2 qK;K

S

?

Id

)

@

2 qK;K c

S

0

@

2 qK;K c

S

@

2 K c

S

@

2 K c; K

S

@

2 K c; K c

S

?

Id

(

@

2 qK;K

S

?

Id

)

@

2 K c; K

S

@

2 K

S

@

2 K;K c

S

0

@

2 qK;K c

S

@

2 K c; K c

S

@

2 K;K c

S

@

2 K c

S

?

Id

0 ?

Id

0 ?

Id

0 1 C C C C C A

>From now on, and to simplify the notations, we may forget the dependence in the variables (

t

,

x

, ...) in the various matricial expressions as it is the case in the following notation :

P

?1

:d

t(

x;

0)

:P

0 B B B @

y

K

y

Kc



K



Kc 1 C C C A := 0 B B B @

A

K

A

1

B

K

B

1

A

2

A

K c

B

2

B

K c

C

K

C

1

D

K

D

1

C

2

C

K c

D

2

D

K c 1 C C C A 0 B B B @

y

K

y

Kc



K



Kc 1 C C C A

>From (24) and (25), we thus have (

x

=

R:q

):

T

(t;x) ~= f(

t;

0

;

0 B B B @

A

K

A

1

B

K

B

1

A

2

A

K c

B

2

B

K c

C

K

C

1

D

K

D

1

D

2

D

K c

D

2

D

K c 1 C C C A 0 B B B @

y

K

y

Kc



K



Kc 1 C C C A

;y;

)g and

T

(t;x) ~ = f(

t;d

(

@

t

S

)(

t;q

K

;

Kc

;

)

:

(

t;q

K

;

Kc

;

)

;q

K

;d

(

@

 K c

S

)

;d

(

@

q K

S

)

;

?



Kc

;d

(

@



S

)

;

)g

:

Now, if we identify these two equalities, we must have

0 B @

q

K



Kc



1 C A= 0 B B B @

A

K

A

1

B

K

B

1 ?

C

2 ?

C

K c ?

D

2 ?

D

K c 0 0

Id

0 0 0 0

Id

1 C C C A 0 B B B @

y

K

y

Kc



K



Kc 1 C C C A :=

L

(

t;q

)

:

0 B B B @

y

K

y

Kc



K



Kc 1 C C C A

;

and thus

d

2 qK;K c;

S

(

t;q

K

;

0

;

0)

:L

(

t;q

) = 0 B B B @

C

K

C

1

D

K

D

1

A

2

A

K c

B

2

B

K c

Id

0 0 0 0

Id

0 0 1 C C C A

:

Moreover, since

A

K

A

1 ?

C

2 ?

C

K c ! (

t

0

;x

0) =

A

K(

t

0) 0 0 ?

C

K c(

t

0) ! is inversible,

for

q

close to

q

0(=

R

?1

:x

0) and

t

close to

t

0, we get that

L

(

t;q

) is inversible and thus

d

2 qK; K c;

S

(

t;q

K

;

0

;

0) = 0 B B B @

C

K

C

1

D

K

D

1

A

2

A

K c

B

2

B

K c

Id

0 0 0 0

Id

0 0 1 C C C A

:L

?1(

t;q

)

:

Then we have

d

2(

t;q;

0

;

0) = 0 B B B @ 0 B @

d

2

S

(

t;q

K

;

0

;

0) 1 C A 0 0 0 1 C C C A ? 0 B B B B B @ 0 0

Id

0 0 0 0 0 0

Id

Id

0 0 0 0 0 0 0 0

Id

0

Id

0

Id

0 1 C C C C C A

With some algebraic calculations and from (19) we nally obtain

d

2 (

t;q;

0

;

0) =

P

1(

q

)

:

0 B @

Id

?

A

(

x;t

) ?

B

(

x;t

) 0 ?

C

(

x;t

)

Id

?

D

(

x;t

) 0 0 0

Id

1 C A

:P

2(

t;q

)

;

where

P

1(

q

) = 0 B B B B B @ 0 0 ?

Id

0 0 0 ?

Id

0 0 ?

Id

Id

0 0 0 0 0 0 0 0 ?

Id

0 0 0 ?

Id

0 1 C C C C C A

:

0 B B B @ 0 B @

P

?1

:P

(

x

) 1 C A 0 0

Id

1 C C C A and

P

2(

t;q

) = 0 B B B @ 0 B @

P

?1(

x

)

:P

1 C A 0 0

Id

1 C C C A

:

0 B B B B B @

Id

0 0 0 0 0

Id

0 0 0 0 0

Id

0 0 0 0 0

Id

0 0 ?

Id

0 0

Id

1 C C C C C A

:

0 B B B @ 0 B @

L

?1(

t;q

) 1 C A 0 0

Id

1 C C C A

:

We can easily see that

P

1(

q

) and

P

2(

t;q

) are inversible if

t

and

q

are suciently close to

t

0 and

q

0.

Looking at the expressions of the matrices

A

,

B

,

C

and

D

in formulas (19), it is clear that we can nd two inversible matrices

P

3 and

P

4 which do not depend on

t

and

q

such that

0 B @

Id

?

A

(

x;t

) ?

B

(

x;t

) 0 ?

C

(

x;t

)

Id

?

D

(

x;t

) 0 0 0

Id

1 C A =

P

3

:

0 B @ 0n?s 0 0 0

M

1(

q;t

) 0 0 0

M

2(

q;t

) 1 C A

:P

4 with

M

1(

q;t

) = 0 B B @

L

i1(

q;t

) 0 ... 0

L

ijK D j(

q;t

) 1 C C A

M

2(

q;t

) = 0 B B B B B B @

tId

n?s

L

j1(

q;t

) 0 ... 0

L

jjK I j(

q;t

)

Id

jK c j 1 C C C C C C A

where we dene

K

D = f

i

2[1

;r

]

=

cos(



i

t

0) = 1 g := f

i

1

;

;i

jK D j g

;

K

I := [1

;s

]n

K

D = f

j

1

;

;j

jK I j g

;

L

i(

q;t

) := 1?cos(



i(

x

)

t

) ? 1 i (x)sin(



i(

x

)

t

)



i(

x

)sin(



i(

x

)

t

) 1?cos(



i(

x

)

t

) ! for

i

2[1

;r

]

;

L

i(

q;t

) := 1?cosh(



i(

x

)

t

) ? 1 i (x)sinh(



i(

x

)

t

) ?



i(

x

)sinh(



i(

x

)

t

) 1?cosh(



i(

x

)

t

) ! for

i

2[

r

+ 1

;s

]

:

Remark 4.3

If we write

t

0(

q

) instead of

t

0(

R

?1

:q

), we easily see that

M

1(

t

0(

q

)

;q

) = 0, that

M

1(

t;q

) is inversible if

t

6

=

t

0(

q

) (with

t

suciently close to

t

0) and that

M

2(

t;q

) is inversible for

all

t

and

q

. j

K

Djrepresents the set of indices of the degenerated directions of the phase function.

4.2 The asymptotic expansion

This part is concerned with proving the following result on

I

(

h

).

Proposition 4.4

As 

h

goes to 0,

I

(

h

)

h

N= 2?n 1 X j=0

d

j

h

j with

d

0= Z 

<

(

t

?

t

0(

x

) ?

i

0) ?jK D j

;c

0(

t;x;

0)

'

(

t

)

> dx

where

c

0 2

C

1 0 (

IR

) and does not depend on

'

.

We rst look for a change of variables in (26) in order to make

P

?1() at in a neighborhood

of (

q

0

;

0). Since

d

2

V

(

q

0) is a diagonal matrix in the

q

-coordinates, there exists a neighborhood

U

of (

q

0

;

0) in

IR

2n and a function

f

2

C

1 such that

P

?1 ()\

U

= f(

q

N

;q

T

;

0)2

IR

s

IR

n ?s 

IR

n

= q

N =

f

(

q

T)g\

U

with (

q

0)N =

f

((

q

0)T) and

df

((

q

0)T) = 0. Then, we dene

P

5 :

IR

n?!

IR

n (

q

N

;q

T)7?!(

y

N

;y

T) = (

q

N ?

f

(

q

T)

;q

T)

:

It is easy to see that in a neighborhood of

q

0,

P

5 is a dieomorphism and if we proceed to this

change of variables in (26), we get

I

(

h

) Z IR1+2n+jK c j

e

i  h 1 (t;y; K c;)

a

1(

t;y;

K c

;;



h

)^

'

( ?

t

)

d

K c

ddydt

(29)

where the support of

a

1(

:;

K

c

;;



h

) is contained in a small neighborhood of (

t

0

;q

0) and is of the

same order as

b

. Moreover, 1 satises

d

1(

t;y;

K c

;

) = 0 , 8 > < > :

y

N = 0



Kc = 0



= 0