J.J.duistermaat, V.W.guillemin - The Spectrum of Positive Elliptic Operators and Periodic...

Inventiones math. 29, 3 9 -7 9 (1975) �9 by Springer-Verlag 1975

The Spectrum of Positive Elliptic Operators and Periodic Bicharacteristics

J.J. Duistermaat (Utrecht) and V.W. Guillemin (Cambridge, Mass.)

Introduction

Let X be a compact boundaryless C '~ manifold and let P be a positive elliptic self-adjoint pseudodifferential operator of order m > 0 on X. For technical reasons we will assume that P operates on half-densities rather than functions. (We will denote the half-density bundle over X by g2�89 We will also assume that P is a classical pseudodifferential operator in the sense that on every coordinate patch its total symbol ae(X, ~) admits an asymptotic expansion

3) ~ pr._j(X, j=O

with Pm_j(X, ~) homogeneous of degree m - j . We recall that the principal symbol p of P is equal to Pm on local coordinates, and the subprincipal symbol is equal

1 t~Zp to P,,-1 2i ~ axa c?{ s

Let 21, 2 2 . . . . be the eigenvalues of P. It was remarked by Chazarain in [6]

and by ourselves in [11] that the sum ~ e - i ~ ' is well-defined as a generalized function of t and that if T is in its singular support then the Hamiltonian vector field

(~q ~ ~q (') q =~/p

Hq= (~J 63xJ (~xj g~S '

has a periodic integral curve of period T. The purpose of this article is to analyze the nature of the singularities at these T. The analysis of H6rmander [16] of the "big" singularity at T= 0 leads to an asymptotic expansion of the form

~ p(l~--pS)"~(2n)-" E Ckg "-l-k, ~s=*~ (0.1)

as p - ~ + oo, for an appropriate class of Schwartz functions p. The Ck'S are the integrals over the cosphere bundle of polynomial expressions in the symbol of P and its derivatives, and are independent of p. (See Proposition 2.1 and (2.16).) In Section 2 we show how they are related to the residues at the poles of the zeta function of P and to the coefficients occuring in the asymptotic expansion of the trace of the heat kernel at t =0. From this we obtain rather easily results of Seeley [22] on the zeta function and Minakshisundaram-Pleijel [18] on the trace of the heat kernel (just for scalar operators, however). We note in passing that the asymptotic expansion of the trace of the heat kernel involves logarithmic terms unless P is a differential operator. The existence of these terms seems to have

40 J.J. Duistermaat and V. W. Guillemin

been neglected in the literature. Section 2 concludes with a priori estimates for the spectrum which follow from (0.1) and which are used in Section 3.

There we study operators P for which all Hq solution curves are periodic with the same period. Specifically we show that if the Hq flow is periodic with

period T there exists a constant fi such that most of the spectrum of ~/P is con-

centrated near the lattice points 2~ k + fi, k = 1, 2, We show that conversely r ....

if this "clustering" occurs then the Hq flow is periodic. In fact we show that if a few pathological examples are excluded then for non-periodic Hq flOW the spectrum is rather equally distributed. Also the spectral estimate of H6rmander [16] can be

n--1

slightly improved in this case, and an error term of order 0(2 " ) replaced by an n - - 1

error term of order o(2 " ). See Theorem 3.5. In Section 4 we begin our analysis of the singularities of ~ e i"/~.~, at periods

T+ 0. Our main result, Theorem 4.5, is that whenever the map exp THq: T* X \ 0 T*X'-,O has a clean fixed point set (in the sense of Bott), then an asymptotic expansion of the form (0.1) is valid in a neighborhood of T; moreover, the leading term in this asymptotic expansion can be computed from such data as the length of the period and the eigenvalues of the Poincar6 map. Chazarain obtains results similar to ours in [6] but without the explicit formula for the leading term. As a corollary of Theorem 4.5 we obtain the following residue formula in case all the periodic Hq solution curves of period Tare isolated and non-degenerate:

limr(t T) E e-imv~'= - To'; - Z - ~ - i~' II - P~I - ~ (0.2)

the sum taken over all integral curves ~ of period T. Here Toy is the smallest positive period of y, 6~ is a Maslov factor (explained in Section 6) and P~ the Poincar6 map around ,/.

For the proof of Theorem 4.5 we need some results concerning composition of Fourier integral operators under "clean intersection" assumptions, generalizing results of H6rmander [17, Chapter 4]. They are discussed in Section 5 and proved in Section 7. Similar results have been announced by Weinstein at the Conference on Fourier Integral Operators in Nice, May 1974.

If all the periodic Hq solution curves are isolated and non-degenerate and only one such curve, 7, or tWO such curves, 7 and - 7 , occur for each period 1, then from (0.2) one can determine ] I - Pyk] for all k. (Just apply (0.2) to the k-fold iterate of ~/.) It turns out that these data almost suffice to determine P7 itself. In fact it determines all the eigenvalues of P~ of modulus 4= 1 and, up to multiplication by roots of unity, all the eigenvalues of modulus 1. This result is due to Harold Stark; and he has generously allowed us to publish it here in an appendix.

Many of the results of this paper extend to operators operating on vector bundles providing the eigenvalues of the symbol are of constant multiplicity. We hope to discuss these results in a future article. We will content ourselves here with mentioning a typical result concerning the Laplace operator on k-forms. For this

This is the generic case if P is a differential operator; for pseudo-differential operators generically only one periodic solution curve occurs for each period.

The Spectrum of Positive Elliptic Operators and Periodic Bicharacteristics 41

opera tor the residue formula (0.2) is still valid except that the residue associated with ~ is

T0, i ~',' ] I -P~[ -+ trace H~,

2g

H~.: A k ~ A k being the h o l o n o m y along ?,.

In conclusion we would like to thank Iz Singer and Michael Atiyah for helping us to clarify the relations among heat equation, wave equation and zeta function asymptotics; and we would like to thank Harold Stark for proving for us the result described above concerning the Poincar6 map. Our main inspiration for writing this paper was the beautiful article of H6rmander [16] on the spectral function of an elliptic operator. We would also like to thank Alan Weinstein for helpful conversations concerning the material in Section 3. Formal resemblances with the methods used by Colin de Verdi&e [7] and Cotsaftis [8] were an incentive to the computation of some of the coefficients in the asymptotic expansions in Theorem 4.5.

1. The Operator e-itpl/m and Its Trace

Let Q = p1/m be the m-th roo t of the unique self-adjoint extension of P in L2(X, Q~), given by the spectral theorem. Accord ing to Seeley [22], Q is a pseudo- differential opera tor of order 1 with principal symbol

q(x, ~)=p(x, ~)1/,,, (1.1)

so again elliptic (and positive). Note that q(x, ~)= 1 if and only if p(x, ~)= 1 and 1

Ho = ~ - Hp on S* X = {(x, ~) e T* X \ 0; p(x, ~) = 1 }. It follows that for P = - A + c

on a Riemannian manifold X, Hq coincides on S* X with the geodesic spray, so the projections in X of the Hq solution curves are just the geodesics with unit velocity.

As an opera tor acting on ~-densities, P has a subprincipal symbol which is an invariantly defined homogeneous C + function of degree m - 1 on T * X \ O given by

sub P=p,,_I - ( 2 i ) - I ~ ~2p,,/?~x~ (l.2) j=l

on local coordinates (Duistermaat and H 6 r m a n d e r [9, Section 5.2]). We have

sub (P~) = Op ~- 1. sub P, (1.3)

in part icular sub Q = 0 if and only if sub P = 0. For 0 equal to a positive integer (1.3) follows from

sub (A o B)= ( sub A). b+a. (sub B)+(2i) -~ {a, b}, (1.4)

here a, resp. b are the principal symbols of A, resp. B. (1.4) is an immediate consequence of the formula

i aA(X, q) . as(Y, ~) ,,= x ..(x, :I+ ~ i+ 2

for the total symbol of the p roduc t of pseudo-differential operators on local coordinates. (1.3) for all positive integers ~9 implies (1.3) for all rat ional 0 and then for all complex ~9 by analytic continuat ion.


Using the formula

92 k

for the total symbol of the adjoint (on local coordinates) it follows that sub A is real if A=A*. Using the proof of [9, Proposition 5.2.l] one obtains for self- adjoint A:

(A(e '~ w), e '~ w)=z ~. f a(x, dq}(x)), lw(x)l 2 (1.5)

+z ~-~ ~(subA)(x, dq}(x)).lw(x)12+O(~ ~ 2) a s ~ - , o o ,

Here ~, resp. a are the order, resp. principal symbol of A. q) is a real-valued C ~ function on X, d~o(x)#O for xesupp w. Note that tw(x)l 2 is a density of order 1 in X if weC~(X, f2~), so can automatically be integrated over X. Formula (1.5) gives an alternative characterization of the subprincipal symbol of a self-adjoint pseudo-differential operator, from which it is obvious that it is real. Note also that sub A = 0 if A is a real self-adjoint differential operator of even order.

The operator

�9 U ( t ) = e - " Q (1.6)

is unitary in L2(X, f2~), and using that it commutes with Q * it is also bounded as an operator in H{,)(X, f2~), uniformly in t, for all selR (H~Srmander [16, Section 3]). Using repeated partial integrations with respect to t it follows that

p~--~ S e-"~ (1.7) - c o

is a continuous linear mapping: 5e(IR)~ Cco(X x X, f2~). In particular

Trace U: p ~--~Trace S e-it(2 p(t) dt (1.8) - c o

is a tempered distribution on Ill. Here the trace of an operator with smooth kernel K is defined by

Trace K = ~ K(x, x) (1.9) x E X

see Atiyah and Bott [3, Section 7] for a justification of this terminology.

The positive elliptic operator Q on the compact manifold X has a discrete spectrum/~j = 2}/" --, on as j--~ on and corresponding orthonormal eigenfunctions efiC~(X, f2~). (Use that Q-~ is a pseudo-differential operator of order - 1 , so compact as an operator in L2(X, ~21) and Q - # j is hypo-elliptic.) The distribution kernel of U(t) can be written as

U ( t , x , y) =

and therefore

• e-i,,j. Q(x). ej(y) (1.10) j = l

Trace U(t)= S ~ e-ituJlej(x)l 2= ~ e-i'"r (1.11) x e X j = l j = l '


is equal to the Fourier transform & of the "spectral distribution"

oo

a(/~)= ~ 6(Ft-#j). (1.12) j - I

Note that lej(x)[ 2 is a density of order 1 in X and therefore can be integrated over X in an invariant way. Because Fourier transformation is an isomorphism: 5P'(IR)--~ .9~'(lR) the identification of Trace U with b is justified, proving at the same time that ~r is a tempered distribution, that is

4~ {j; ~tj</~} < C[Ftl u for all p > 0 , {1.13)

for some constants C, N. Better estimates will follow from the more detailed description of U given below.

As a distribution in (IR x X) x X, U can alternatively be characterized as the kernel of the operator assigning to UoS C~(X, f2~) the solution us C~(lR x X, s'2~) of the following Cauchy problem:

i - l (?ufl?t +Qu=O, u(O,x)=Uo(X). (1.14)

Theorem.l . l . U is a Fourier integral operator ~?f class 1-4(IR x X, X; C) defined by the canonical relation~

C = {((L ~), (x, O, (Y, q)); (x, ~), (y, ~)s 7"* X "-0, (t, ~)s T* IR \ 0 , ~ + q(x, ~)=0, (x, ~)= q~'(y, r/)}. (1.15)

Here ~' denotes the H q .[low in T* X \ O.

Proof Using [9, Section 5.3] one can find UoSI-�88 x X, X; C) such that (i-10/(?t+Q)oUo has a smooth kernel and Uo(0)=identity in C~ Q~). Imi- tating Duhamel's principle this operator can be used to solve the inhomogeneous

problem i -10u/Ot+Ou=.[; u(x, 0 )=0 (1.16)

for small It[ , the solution u s C ~ ( I R x X , ~ ) depending continuously on .re C~(IR x X, ~21). It follows that by adding an operator with smooth kernel, U o can be changed to an operator U such that (i -~ c)/Ot+Q)o U = 0 for small ]t} and still U(0)=identity in C~(IR x X; s The local existence theorem for (1.16) also leads to a local uniqueness theorem for (t.14) using a standard duality argument (note that Q is self-adjoint here). In view of U(t a + t2)= U(q)o U(t2) and using the local uniqueness the local solutions automatically piece together to a global solution U which differs from Uo by an operator with a smooth kernel and therefore is of class I-~(lR x X, X; C).

For more details, see for instance [10, Chapter 5.1]. A technical complication here is that [] = i -~ ~?/?t+Q is not a pseudo-differential operator in IR x X because is not pseudo-local there, as observed in Nirenberg [19, Section 9]. How- ever its wave front relation WF'(U]) outside the diagonal in T * ( I R x X ) \ 0 x T* (IR x X ) \ 0 (where [] is pseudo-differential) only contains points of the form (((t, x), (~, 0)), ((t, y), (~, 0))). In taking the product with U o these points do not contribute to WF'([] o Uo) (H6rmander [17, Theorem 2.5.15]) so in the construction of U o (and U) we may treat Q as if it were a pseudo-differential operator in IR x X.


Theorem 1.1 implies that U(t): C~ Y2~)~C~(X, 0~) is a Fourier integral operator of order 0 defined by the canonical transformation 4~' (for each teN). This characterization sounds more natural, but the formulation of Theorem 1.1 is slightly more precise. More explicitly, Theorem 1.1 means that U(t, x, y) is a locally finite sum of integrals (converging in distribution sense) of the form

I (t, x, y) = (2~)-" ~ e i~''x" ~''~) a(t, x, y, ~) d~,. (1.17)

Here q~ is a non-degenerate phase function as in H6rmander [17, Section 1.2], ~eIR"'-. {0} and

d~ ~o(t, x, y, 0 = 0 ~ ((t, d,~), (x, dx~0), (y, - d, ,~))e C. (1.1 S)

The amplitude a is a symbol of order 0, having an asymptotic expansion of the form

a ~ ~ a_j, (1.19) j=O

with a ~ homogeneous in ~ of degree - j . That we never need more than n frequency variables ~=(~ . . . . . ~,) follows from the fact that the tangent space of C never intersects the tangent space of the fiber in T*(IR x X x X) more than n-dimensional. (See H6rmander [17, Theorem 3.1.4 and the integration away of frequency variables in Section 3.2].)

From the construction of U it follows that the principal symbol of U is nowhere zero, implying that WF'(U)= C and consequently sing supp (U)= {(t, x, y)e IR x X x X; (x, d)= q~t(y, r/) for some ~e(TxX)* \ {0}, r/e(T~,X)* "-. {0}}. Noting that ~b' commutes with multiplication in the fibers with positive scalars because q(x, ~) is homogeneous of degree 1 in ~, this description of sing supp U does not change if we restrict (x, ~), (y, q) to S*X. So for Q = ( - A + c) ~ on a Riemannian manifold the singular support of U is equal to the set of (+t , x,y)eIR x X x X such that x and y can be joined by a geodesic of length t.

Using our more detailed knowledge of U we can also give another inter- pretation of the distribution 3-=Trace U. If A denotes the diagonal map (t, x)~--,(t, x, x): I R x X - * I R x X x X , then the pull-back A* of functions on IR x X x X to functions on ]R x X is a Fourier integral operator (of order �88 defined by the canonical relation

Wr'(A*) = {((U, 0, (x, ~ +,)) , ((t, ~), (x, ~), (~, ,1)))}. (1.20)

Because r + 0 when ((t,v), (x,~), (x,~l))eWF(U) we can apply H6rmander [17, Theorem 2.5.11'] to find that A* U is a well-defined distribution in IR x X and

WF(A*(U))c {((t, r), (x, ~ -q)); ~ +q(x, ~)=0, (x, ~)= q~'(x, t /)}. (1.21)

Let next/7 denote the projection (t, x) w-, t:lR x X ~ IR. Then integration over x is equal to the pushforward/7.= transposed operator of H*, so it is a Fourier integral operator (of order ~ -~n ) defined by the canonical relation

WF' ( /7.)= {((t, r), ((t, ~), (x, 0)))}. (1.22)

Applying the same theorem it follows that 8-=Trace U = Ft.(A*(U))is a well- defined distribution in IR and

WF(3.)c {(t, z); ~ <O and (x, ~)=q)'(x, ~) for some (x, ~)}. (1.23)


In particular this proves

Corollary 1.2. 3- is C ~ in the complement of the set of periods of periodic H q solution curves in T * X \ O . (For Q = ( - A +c) -~ on a Riemannian manifold this means that the singular support of 8 is contained in the set of periods of periodic geodesics.)

Corollary 1.2 has been proved in the same way by Chazarain [6]. A similar argument occurs in the extension of the trace to operators of the form f * P with f a differentiable mapping transversal to the identity and P a pseudo-differentiai operator (acting on vector bundles) in Atiyah and Bott [3]. It might also be noted that up to this point the calculus of Fourier integral operators is not really needed because we only used that WF'(U)= C and this follows already from the theorem on propagation of singularities in terms of wave front sets of H6rmander [26]. If P is a real-analytic operator on a real-analytic manifold X then the same results for U and & hold with "singular support" replaced by "'analytic singular support". This follows from the theorem on propagation of singularities of Sato [20], Sato, Kawai and Kashiwara [21, Section 2.1]. (A proof of this theorem not using the sheafrg has been given by Andersson [I]).

A more explicit proof of Corollary 1.2 can be given using the local representa- tions (1.17) of U. It follows that for/3eC[(IR) the integral

(2tO-' ~ e i"' fi(t) a(t) dt = ~ p(ix-pj) (1.24) j=1

is a finite sum of integrals of the form

(2~) -1 -"ISIe'" ' e '~" . . . . . r f,(t) a(t, x, x, 0 d~ dx at (1.25)

= (2r t)- ' -" Ix" 555 ei"('+ oct . . . . . r ~(t) a(t, x, x, p~)d~ dx dt.

Applying the principle of stationary phase it follows that this is rapidly decreasing as ix - , co unless

dcqo(t, x, x, ~)=0,

dx ~o(t, x, x, O + d~.qo(t, x, x, 0 = 0 , (1.26)

1 +d, fp(t, x, x, ~)=0

for some tesupp/3, x e X , ( e l R " \ {0}. In view of (1.18), (1.15) this just means that p(x, ~) = 1, (x, ~) = 4~'(x, ~) if we write { = dxrp(t, x, x, O.

A priori different Hq solution curves with the same period T might lead to a cancellation of the singularity of 8 at T. Only a more detailed examination of the asymptotic expansion of (1.24) as ix * oo can show, using additional geometric assumptions on the Hq flow if necessary, that such cancellation actually does not take place. See Sections 2, 4 below.

No statement like Corollary 1.2 holds if we replace the first order operator (2 by a higher order operator P. For instance the function

./'(t) = ~, e - i , . ~ nffZ

can be regarded as the trace of e -ne for a second order operator P on the circle. On the other hand f is an automorphic function (cf. Serre [23, Chapter 7, w 1, 2


and 6]). The orbit of the origin under the fractional linear transformations

a z + b (a ~)~ SL(2, 7t) Z ~--~ c z -t-~d- '

is dense on the real axis, and this shows that the singular support of f is equal to the whole real axis. This example was brought to our attention by Professor Atiyah.

2. The S ingular i ty of U (t, x, x) at t = 0

Because every Hq solution curve is periodic with period 0, a big singularity of U(t,x, x) can be expected at t=0 . We start with a Fourier analysis of this singularity. Note that the canonical density in T * X divided by the projection: T * X - ~ X defines a density in the fiber (TxX)*, the value of which is a density in T~X. Dividing again by dq leads to a T~X-density-valued density on S*X = {~e(TxX)*; q(x, 3)= 1}.

Proposi t ion 2.1. There exists a sequence col, o)2 . . . . of real valued smooth densities in X such that for every peS,~(lR) with supp fi contained in a sufficiently small neighborhood of O and ~ = 1 in another neighborhood of O:

•p(# - ~j) l ejl 2 ~ (2~z)-" ~ cokft"-k-I (2.1) j k - O

for #--~ oo and rapidly decreasing as p ~ - ~ , asymptotics in the C~176 •l)- topology. Here

coo(X) = vol (S* X), co,(x) = 0, (2.2) and

col (x) = ( 1 - n) - ~ subQ. s*x

Moreover, !f P = Q" is a d!fferential operator, then

co,~l,,(x)=0 for l = 1 , 2 . . . . and COk(X)=0 f o rkodd . (2.4)

Proof For small It[, U(t, x, y) can locally be represented by an integral of the form (1.17) with the phase function

~o,(t, x, y, U)= O( x, Y, U)- t . q(y, U) (2.5)

introduced by H6rmander [-16, Section 3]. Here 0(x, y, y/) is, as a function of the first variable, the local solution near x = y of the Cauchy problem

q(x, d~b(x, y, ~1))= q(Y, ~I),

~,(x, y, , ) = 0 when ( x - y , r / ) = 0 , (2.6)

and finally

dxO(x, y,~)=~ 1 for x=y .

Substituting r /=p-co .~, co>0, q(y, ~)= 1, and treating the integration over t, co with the method of stationary phase,

(2n) -~ ~ e jut f~(t) U(t, y, y) d t=(2n) -" iz "-1 ~ b(y, ~/,/z) d~/, s*x


with

b 0 ~ o 1 p-m( 1 ) [~)(t)a(t,y,y,#oofl ) n-1 b(y,~l, ~ i- /72 , .co ]t=o ~ t O eJ o,=1

as/~ ~ , uniformly in (y, Fi)~S*X. [-[ere the integral over the unbounded t, co- domain is split into one over a compact neighborhood of t = 0, ~o = l and a domain without stationary points for the phase function, using a partition of unity. The latter integral is shown to be rapidly decreasing using repeated partial integrations as in H6rmander [17, Section 1.2]. This proves (2.1) with

k ook(y)= 5 a_k(O, y, y, f l)dfl+ ~ ( n - k + r - 1 ) ... ( n - k )

S~c X r= l (2.7)

~x 1 a , "s 7~.,(i -1 c?/ct) a k+,(O, y, y,~l)d ~.

If/z is replaced by -/~ then the phase function has no stationary points for co>O and we obtain rapid decrease in (2.1) as # - - + - m, reflecting the positivity of the spectrum.

In order to compute the ok, observe first that we need

(2 7t) "5 e ie'~,~''"~ a(O, x, y, rl) do = g(o, x, y) = 6(x - y). (2.8)

From (2.6) it follows that tp(x, y,t/) is a non-degenerate phase function locally defining the same Lagrange manifold as ( x - y , q), and d 2 O = d 2 ( ( x - y , r / ) )=0 at x = y . So according to H6rmander [17, Theorem3.1.6] there exists locally a homogeneous change of coordinates q = ~/(x, y, ~) such that r/0,, y, ~)={ and

~k(x, y, r/(x, y, ~))= (x - y , r (2.9)

Substituting this in the left hand side of(2.8) we see that U(0)= I if we choose

a(0, x, y, r/(x, y, ~)) = 1/Idet d~ q(x, y, ~)1. (2.10)

In particular %(0, y, y, r/)= 1 and a f t 0 , x, y, r/)=0 for all j > 0 , implying (2.2). Applying i -~ O/Ot+Q to (1.17) under the integral sign amounts to replacing

a by an amplitude a' such that

, 1 c~a a (t, x, y, t l ) ~ T ~ (t, x, y, q ) - q ( y , *1)" a(t, x, y, tl)

for ]tll ~ oo, here we have written

0(x + 2, y, r/) = t/,(x, y, r/) + (2, X(2, x, y, tl) ). (2.11)

The equation

a'_ ~(t, x, y, q)=0 (2.12)

is a first order linear partial differential equation for a ~ involving only the a_ k for k<j . With the initial condition (2.10) these successive "transport equations"

48 J.J. Duislermaat and V. W. Guillemin

for the a_j have a locally unique solution, leading to a local description of the solution U of (i- 1 c~/?~t + Q) U = O, U(O) = I, modulo integral operators with smooth kernels. Differentiating (2.12) sufficiently many times with respect to t at t=0 , x=y , and using (2.6), (2.9), (2.10), (2.11), the ~Ok(y ) in (2.7) can in principle be computed explicitly, although the computations become rapidly complicated for increasing k. For j = 0 this procedure leads to i -1 (?ao/Ot(O, y, y, r/)= - ( s u b Q) (y, t/), proving (2.3).

A funct ionfon S*X will be called even iff(x, - 3 )=f (x , 4), and odd iff(x, - 4)= - f ( x , 3). If P = Qm is a differential operator then m is even, and the hypothesis that the homogeneous part of degree j in aQ(x, 4) is an even, resp. odd function on S*X ifj is odd resp. even leads to the correct conclusion that the homogeneous part of degree k in ere(x, 3) is even, resp. odd if k is even, resp. odd. Because ere(x, 4) is determined locally in S*X by erp(X, 4), the hypothesis gives the correct solution for erQ(x, 4) at the antipodal points. By induction with respect to r it follows from (2.10), (2.12) that (O/Ot)raj(O,y, y, ~1) is an even, resp. odd function on S*X if r - j is even, resp. odd, so (2.7) implies that COg=0 ifk is odd.

Finally the assumption that P is a differential operator has the effect that applying it to (1.17) amounts to changing a into an amplitude a" satisfying again that a" ~(0, x, y, t /)=0 for j > 0 . As in the proof of m , = 0 it follows that the coefficient of/2-1 in the asymptotic expansion of (2n)- ~ ~ e i"' f3(t) PU(t, y, y) dt for /2-* oo vanishes. On the other hand P o U ( t ) = Q " e ite=(ic?/Ot)me gte which on the inverse Fourier transform side amounts to multiplication by/2". So the coefficient of/2-1-,n in (2.1) is equal to zero, that is co,+m=0. Replacing P by U in this argument proves that co,+~,, =0 for all positive integers I.

For P = Q", 2j-/2j- ", the (-function

Ze(s) = ~ 2} -s" lejl 2, (2.13) J

which is the restriction of the distribution kernel of P ~ to the diagonal, is a holomorphic function of s for Re s sufficiently large, with values in the space of smooth densities on X. (The distribution kernel of P-S is continuous for Re s sufficiently

d large and dx~ P- ~(x, x) is equal to the restriction to the diagonal of the distribu-

tion kernel of (8/Ox~)P-~-P-~(~?/~?x~), which is of the same order - R e s, so there is no loss of differentiability.)

Corollary 2.2. Z e has a meromorphic extension to the complex plane, having only simple poles at s = ( n - k ) / m with residues equal to (2n)-"Cok/m, k=0, 1, 2 . . . . . Moreover Z e is of at most polynomial growth on each half space Re s > c excluding neighborhoods of the poles.

Proof Let 0<e</21=f i r s t eigenvalue of Q, choose xeC~176 Z(/2)=0 for /2 < e, Z(/2) = 1 for/2 >/21. Writing Zs(/2) = Z(/2)/2- s, 2;(/2) = ~ 6(/2-/2j) [e~l 2, we have

J

zds) = <2;, z~> = <2, L>-

For t 4 = 0 repeated partial integrations give

,~,(t) = (2~) -~ j" e'" ' Z(/2)/2-~ d/2 = (2~) - I (i/t)k ~ e'" ' (~/c1/2) k [Z(/2)/2- s] d/2,


showing that

]t g 2s(t)l <= C(1 + is[) k. c-~Res +k- l l / (Re s 4- k - 1)

for Re s > - k + 1. Because this holds for all k and similar estimates hold for the derivatives of)~s(t) with respect to t it follows that (1 - 3 ) 2s is an entire functionofs with values in 09~(lR) having at most polynomial growth in half spaces Re s>c . Here p is as in Proposit ion 2.1. As a consequence

<Z, Z,> - <P *z, Zs> = <(1 - ~) 2, L>

is also entire in s, and of polynomial growth in half spaces Re s > c. I f f( t t) = O(/t" i k) f o r / ~ o o then ; f(/t)Z(/z)/~ Sd/z is holomorphic and bounded

in half spaces Re s > c with c > n - k . On the other hand ~ # , - 1 - k Z(/~)/~-~ d/ l=

0 ( s ) / ( s - ( n - k)), where O(s)= ~/~"- k-, dz d ~ (~) dlt is entire in s, and bounded in

half spaces Re s > c, because d U d p has compact support. Moreover

dz O(n-k) =j ~-(l,)all,= 1.

So (2.l) implies that ( p , X, Z~) can be extended to a meromorphic function of s having only simple poles at s = n - k with residue equal to (2 rt)--" O)k, and bounded in half spaces Re s > c. This proves the statements for Z e, for Zp they follow from Zp(s) = Za(ms) .

Note that Z e has no pole at s =0, and not either at s = - 1, - 2 . . . . if P is a differential operator�9 This proves part of the results of Seeley [22], who treats general elliptic operators on vector bundles and also gives a formula for Zp(O ). The latter has been used by Atiyah, Bott and Patodi [4] for an alternative proof of the Atiyah-Singer index theorem. However they work rather with the restriction

Oe(z) = ~ e - ~ lejl } (2.14) J

of the distribution kernel of the "heat opera tor" e - :P to the diagonal. This is a smooth density on X depending holomorphically on z for Re z>0 . For Op(z) we obtain:

Corollary 2.2'. There exist smooth densities v~, 1=0, 1, 2 . . . . on X such that

Y, �9 ( D k . 2 ( k - n ) / m m ( 2 ~ ) " O P ( Z ) ' ~ k , E l ~ \ m ]

Z~N (2.15)

+ L (-1)t+t L J - l o g z + �9 z t f o r z',~O. t = 1 l! " ( O n + I r a " ~ / = 0 VI

n ~ l m ~ Z

a)

4 lnventlones math.,VoL 29

P r o o f O p and Z e are related to each other by the inverse Mellin transform

1 z - s ze ( s ) r(s)ds, O e ( z ) - 2zci Re =c

50 J . J . Du i s t e rma a t and V. W. Gui l l emin

which holds for c sufficiently large. This follows from summing the formula

1 e - z ~ - ~ z - ' 2 - S F ( s ) d s

2gi Re ~=c b)

_ _ 1 ~ e_i~iog~zX)(z]t ) CF(c+i~) d~. 2re _~

In turn b) follows from the Fourier inversion formula if we view rx)

c) F ( c + i ~ ) = I t c+i~ e - ' d t = i s e i~ ' e ' ' e'dt 0 oe

as the Fourier transform at - ~ of the function t~-, e ~t-~'.

Shifting the path of integration in c) to the line Imr = c one obtains the estimate

d) i F ( c + i ~ ) l < e - ~ e ~ t - , ,, . . . . dt

for c > 0, t~l < rt/2. In combination with the well-known equation

1 e) F(s)= s(s+ 1) . . . ( s+k- 1) F ( s + k )

it follows that F(s) is exponentially decreasing in all vertical strips, excluding neighborhoods of the poles. Combining this with the growth properties of Ze(s ) it follows that Ze(s) . F(s) is exponentially decreasing in vertical strips, excluding neighborhoods of the poles.

This allows to shift the path of integration in a) by letting c',~ - oo. Obviously the integral in a) is O(z -r for z%0 so in the asymptotic expansion of Oe(z ) for z'-~0 modulo terms O(z") for all N we only pick up the contributions from the poles which are encountered in shifting the path of integration. This leads to (2.15), the logarithmic terms occurring whenever poles of Ze(s ) and F(s) coincide to produce a double pole. The v t are bijectively related to the values of Z e at - l (after taking off the pole of Z e if necessary), apparently no further information about these quantities can be obtained from the asymptotic expansion (2.1).

(2.15) generalizes the well-known expansion of Minakshisundaram-Pleijel [18] for the Laplace operator. For m = 1, (2.t5) can also be proved by viewing Ze-~"J' [ej[ 2 as the boundary value (in distribution sense) for z---, it of the analytic function OQ(z) in the half plane Re z > 0. The main ingredient in the proof is then the theorem that if the boundary value v (in distribution sense) of an analytic function u in a half plane is smooth in a neighborhood of the boundary point z o, then u is smooth up to the boundary in a neighborhood of z 0.

If P is a differential operator then no logarithmic terms occur in the asymptotic expansion (2.15), corresponding to the fact that F(s )Ze(s ) only has simple poles. However, for general pseudo-differential operators P poles of Zp(s) can occur at any point (n - k)/m, k "i: n. Iiadeed, replacing Q by Q + c, c > 0 amounts to replacing

b y / ~ - c in the asymptotic expansion (2.1), so for a suitable choice of c we have ~ok(x)4=0 for any k + n for the new operator if we had c%+i(x)4=0 for Q. Now according to (2.7) co,+ t (x) is an integral over S~ X of i Oa_ JOt(O, x, x, ~), which can be made arbitrarily large by a sufficiently large choice of Q_,(x , ~), keeping the homogeneous terms of degree > - n in the symbol of Q fixed.

The S p e c t r u m of Pos i t ive Ell ipt ic O p e r a t o r s a n d Pe r iod ic B icha rac te r i s t i c s 51

Integrating (2.1) over X leads to

E P (P- /~ )~(2~) - " L CkP"- ' -k J k=O

and rapidly decreasing as p -~ - oo.

Here

so in particular

for p - - ,~ , (2.16)

ok= ~(nk, (2.17) X

Co = VOl (S* X), q = ( 1 - n ) ~ subQ. (2.18) S * X

In view of Corollary 1.2, the asymptotic expansion (2.16) does not change if we replace p by any peSe(IR) such that t5 has compact support containing no period T :t=0 of a periodic/-/q solution curve, and all derivatives of 1 -/3 at t =0 vanish.

Integrating (2.13), resp. (2.15) over X leads to the analogue of Corollary 2.2, resp. 2.2' for the ~-function (e(s)= ~ j 2 f s, resp. the "heat function" O v(z ) = ~ e-4,=.

We conclude this Section by some basic estimates for the eigenvalues of Q which follow from a consideration of only the top order terms in (2.16).

Lemma 2.3. For any K >0 we have

# { j ; I p j - p t < K } = O ( l ~'-1) as l ~ O v . (2.19)

Proof Choose p~,Cf(IR) such that /5 has compact support, ~/5(t)dt+O. Re- placing p by p.t5 one obtains that p>0 , p(0)>0 and still /5 having compact support. Replacing /5 by tF--~)(t/6) amounts to replacing p by pF--~b.p(p3). Taking 3 small one gets supp/5 in any prescribed neighborhood of 0 and p >0 on [ - K, K]. An examination of the first part of the proof of Proposition 2.1 shows that the condition that ~ = 1 on a neighborhood of 0 is not necessary in order to obtain an asymptotic expansion of the for (2.16), possibly with other constants c k. Now (2.19) follows from

{J; IP j - P[ < K}- min {p(p); p e [ - K, K]} < ~ p ( I t - p ) = O(p"-~). J

Lemma 2.4. For every p s SP (IR ) and every e, > 0 there exists a number K such that

P(P- P) <=~ " lJ "- ~, (2.20) {J; [,% - u [ > Kt

Z I i p(~-~)d~ <~,v "-~, (2.21) {J;t~ s > v + 1(} l - o~

and

p(p - / J ) < ~ v"- t (2.22) {J;uj < v - K}

Jor all p, v >= 1.

Proof For every N there is a C u such that IP(P)I < CN(1 + I#l) -u for all/J. On the other hand (2.19) implies the existence of a constant C such that

#{j; lp~- ,u[<=�89189 "-a for all U. 4*

52 J . J . D u i s t e r m a a t a n d V. W. G u i l l e m i n

Combining these two estimates one obtains for # > 0, N > n:

I , u j - ~ t l > - K k = 0 K + k + l > l • j UI>_K+k

<=2C CN ~ (p+l)"-~ l -u / = K + I

<2 C CNj=~o.= p,-1 - j Kj-N+1/(N _ j_ 1).

So (2.20) follows by taking K sufficiently large. (2.21), resp. (2.22) follow by writing

v v-/~j

p(lt-#j)dp= ~ p(p)d#, - - a o - ,~. ,

resp. "X3

~p(p-l~j) d # = ~ p(#)d# v v - / ~ j

and using the rapid decrease of ~ ~--,~_~ ~ p(ff) d#, resp. a ~--, S~ ~' p(,u) dp as a--, - ~-.), resp. ~ -~ Go.

Corollary 2.5.

~{j ; #j<v}=(21t)-".vol(B*X).v"+O(v "-1) as v - ~ . (2.23)

Proof Integrating (2.16) from - o o to v one obtains

~ p(#-#j)d#=(2rc)-" "~lCkV"-k/(n--k)+O(1 ) (2.24) -oo j = l k=O

for v - , Go. On the other hand

--co j = l {j;t4j>v+K} - ~

{ J ; I v - - #.11 ~ K } - ~ { j ; . j < v - K } -

{j;#j<v-K} v

Now ; p (# -/~j) d# = 1

- o o

and the other terms are all O(v ~-1) as v ~ oo in view of (2.21), (2.19) and (2.22), so

~{j;#~<v-K}=(2M-"cov"/n+O(v ~-~) for v---,oo.

Replacing v by v + K in this formula gives (2.23), using that

1

vol {(x, ~ )e r* X; q(x, ~)< 1} = vol(S* X) ~o~ ~-1 do=co/n. 0

The Spec t rum of Posit ive Elliptic Opera to rs and Periodic Bicharacter is t ics 53

Remarks. Applying the same argument to (2.1) leads to

~. [ej(x)[ 2 =(2 r0-"-vol(B* X). Yn~-o(yn-1), Y - - ~ ~ (2.25) I~j~ V

uniformly in x, here B* X = { ~ ( T ~ X ) * ; q(x,~)__<l}. This is the main result of H6rmander [16]. Of course (2.23) follows from (2.25) by integration over X.

We also remark that (2.16), (2.20) easily imply that there exist K > 0 , 6>0, v such that

4~{j; ]p~- /~ l<K}>6.~ t "-~ for all p > v . (2.26)

3. Periodic H~ Flow

Theorem 3.1. Suppose that all Hq-solution curves are periodic" with period T >0 and that the average of sub Q over these curves is equal to a constant 7. Write

Vk=~-(2n k+�88 k = l , 2 , .... (3.1)

here ~ is the index of the O/3t + Hq solution curves of period T given by the Arnol'd cohomology class in Hi(C, 7l) (see Arnold [2], H6rmander [17, (3.2.15)]), (~ is the projection int~o T*((IR rood T) x X • X) of the canonical relation C defined in (1.15).

Then there exist positive constants K, K', f such that

# {J; IPj- P[ <= K, [pj - Vk{ < B k- ~ for some keN} (3.2)

>=(1-K'/B2) �9 #{ j ; Ipj-lz] =< K} for all B>0 , p>-ft.

Moreover, if sup Q=O then 3:=0 and the number Bk -~ in the left hand side of (3.2) can be replaced by Bk -1.

Proof cbr+T=cI9 ' for all t ~ T implies that U(t+ T, x, y) is defined by the same canonical relation C as U(t, x, y). The principal symbol u of U satisfies the transport

equation i- ~ Lf~e/0 , + n,) U + (sub Q). u = 0 (3.3)

([9, Theorem 5.3.1]), and it follows that the principal symbol of U(t+ T, x, y) is equal to i - ' e - ~ ' T u . Indeed, noting that ? T is the integral of subQ over the (~/(3t+Hq solution curve, t running from 0 to T, we see, in the notation of [17, Section 3.2], that

Sj : i*~,3-, +. . + a , , o e - i ) , T SO ,

whereas on the other hand

~---~ a0 ,1 -[- " '" "qt- r 1,j = - - ( a j , j _ 1 -b "'" -~- O'1, 0)

if the ~0 k, k = 0 . . . . . j are phase functions defining (~' on patches along a ~/Ot + Hq solution curve of period T. Writing �89 n a + V T= fl T it follows that

U(t, x, y ) - e ipr U(t + T, x, y)

is a Fourier integral operator defined by C of one lower order - � 8 8 so the integral

(2 ~) ' ~ e iut ~ (t) [6 ( t ) - e il3 T ~ (t + T)] dt (3.4)


can be written in the form (1.25) with the amplitude b now of order - 1, and we see that (3.4) is of order O(/tn-2). Choosing p >0 and taking real parts in (3.4) gives

Y~(1-cos((~-~)r)) . ,( l ,- ,~)__<A~ "-2, ~>__1. (3.5) J

for some constant A. Here it is used that t~--~8(t+T) is equal to the Fourier transform of

J

Now define for each B>0 , k~]N'

N(k, B)= ~ {j; Bk -~ < T[/~j- Vk[ < rt }. (3.6)

Using that l - c o s ( 2 u k + x ) > � 8 9 2 for Ixl<rt and substituting ~ = v k in (3.5) one obtains the estimate

n(k, B). �89 B 2 k - ' . rain {p(/z); I/~l-<_ re~T} __< A' k"- 2 (3.7)

for all k~N and some A'. Taking p such that p ( / 0 > 0 for I/~[<u/T(see the proof of Lemma 2.3) this leads to N(k, B)< A" B-2 kn-I for all B>0 , k~IN and still another constant A". Combining this with (2.26) the estimate (3.2) readily follows.

If sub Q = 0 then it follows from the self-adjointness of Q that the homogeneous part Qo(x, ~), resp. Q_i(x, ~) of order O, resp. - 1 of the total symbol of Q on local coordinates is purely imaginary, resp. real. Representing U(t, x, y) locally

by integrals (1.17) with ~ t ( t , x, y, ~)+q(x, d x q)(t, x, y, ~)) being identically zero

(instead of only on dr x,y, ~)=0), as can always be done, one obtains for (i- 1 ~/cgt + Q)o U a similar local representation with a replaced by some amplitude 5, again of order 0. It turns out that 50 = i-~ L a o, with L a real first order linear partial differential operator. Because we can choose ao(0, x, y, ~)= 1 as we have seen in the proof of Proposition 2.1 it follows that locally solving ~o=0 leads to an a o which on the k-th patch along the ~/?~t+Hq solution curve is equal to i~.~-,+...+ . . . . times a real function. With this choice of the a o, and using that iQo, Q-1 are real, one obtains 5_1= i - 1 L a l+ . f with i -( . . . . . . +'"+ . . . . ~..f real- valued. Because we have seen in the proof of Proposition 2.1 that a ~(0, x, y, () can be chosen purely imaginary, solving f i ~ =0 locally leads to an a ~ such that on the k-th patch i -~*~ . . . . + + .... i a is purely imaginary. It follows that in the --1 representation of (3.4) we may assume that b ~ is purely imaginary. However, taking real parts in (3.4) then shows that in (3.5) we may replace kt "-2 by /~n-3 and the argument goes through with k -~ replaced by k -~ in the definition of N(k,B). This proves the last statement in the theorem. (Of course the proof would have been much cleaner if we would have had an invariant analogue of [9, Theorem 5.3.1] for a subprincipal symbol of the product.)

The estimate (3.2) means that (by choosing B sufficiently large) an arbitrary large fraction of the eigenvalues in long and far away intervals lies in the intervals around the v k with decreasing radius B. k-~(B �9 k -1 if sub Q=0). There is also a converse to Theorem 3.1.


Theorem 3.2. Let X be connected and n > 2. Suppose that there is a constant K such that/or each e>0 there is a number fi such that

{,J; [P~- ~ t < K, ]/*j- Vk[ < e .]'or ,some k e N } (3.8)

>=(1 - ~). # {j; f/Aj-- ~l ~ K} .for all # ~ .

Here vk=2~k/T+ fl, keN, fl a fixed real number. Then all H q solution curves are periodic with period T, the O-T-average of sub Q over these curves is equal to a constant y, fl=�89 7zct/T+y modulo 2~/T, and (3.8) can be replaced by the stronger estimates of Theorem 3.1.

Proof (3.8) implies that

E(1--ei(P-u~)T) .p( f l - -pj)=O(l f l ~-1 ) for /~--. oo. (3.9) J

For the proof, split (3.9) as a sum over the j such that ] p - p i l < K , I~-vkl>__~ for all ke/N, a sum over the j such that I~j-vkl <~ for some keN, and the rest. Each of the summands can be estimated from above by e' . /P-1 for arbitrary e '>0, the last one by applying (2.20) and choosing K sufficiently large, the second one by observing that 1 - e x p i ( f l - ~ ) T becomes small and using that ~ j p(p-/z~)= O(/p-1), and finally the first one by combing (3.8) with (2.19), taking e sufficiently small and p>~ , /~ sufficiently large.

(3.9) can also be written as

(2 7~) -1 ~ e iu' ~( t )[&(t)-e i~r b(t + T)] d t=o( l l n- l ) (3.10)

for p--~ ~ . /3 (t) e i~ T U(t + T, x, y) can be written as a finite sum of integrals of the form (t. 17) with phase function (p = (p~ and amplitude a = a" in the r-th summand. Using only the top order term in (2.16) it follows as in the beginning of the proof of Proposition 2.1 that

E ~ eiUt~(~'~ q(x,o=l (3.1t)

=vol(S*X)+o(1) for p ~ m .

Here V(x, ~) is equal to the local solution t of the equation (p~(t, x, x, ~)=0 and a~ is the zero order part of a ~.

Using the principle of stationary phase the only contributions to (3.11) which are not o(1) f o r / t ~ m come from the points (x, () where all derivatives of t ~ of order > 1 vanish. Because the estimate (3.11) holds for suppt3 contained in an arbitrarily small neighborhood of 0 it follows that

~ do(O, x, x, ( )dx d ( = vol(S* X), (3.t2) r O r

here f2" is the set of (x, ~)eS* X such that t" vanishes of infinite order at (x, (). From (1.18), (1.15) it follows that q~T has a contact of infinite order with the identity at (x, d x (p~ (0, x, x, ~)) such that (x, ~)e f2 ", and in view of the proof of Theorem 3.1.6 in 1-17] we see that ~0" can be chosen such that it has arbitrary order of contact with the phase function ~o in (2.5) at the points (O,x,x,~') such that (x,()eO ". In fact, the contributions to (3.11) which are not o(1) as #--, ~ can be represented


by only one integral with such a phase function (p~, for which we then have

a~ (0, x, x, ~) ,Ix d ( = vol (S* X). (3.13) 12"

Because with the phase function r of (2.5) the amplitude in the integral representation of U(0, x, x) has top order term equal to t, it follows in view of the transport Eq. (3.3) for the principal symbol of U that

aCo(O,x,x,~)=i-~e -i';Ix'-r for (x, ()~ (2", (3.14)

here 7(x, () . T is equal to the 0 - T-integral of sub Q over the Hq solution curve starting and ending at (x, ()et2 ~. It follows that (3.13) can only be possible if �89 rr i a+ y (x , ~)" T is equal to 0 modulo 2 7r for almost all (x, ~)ct2 ~, and moreover S*X',,~2 ~ has (2n-1)-dimensional measure equal to 0. Because t2 ~ is closed, t2~=S * X, that is ~ r = I . Because 7 is continuous and S* X is connected (if X is connected and n > 1) it follows that y is constant (note that ~ is locally constant by definition) and the theorem is proved.

If we make some additional assumptions on the Hq flow (excluding some rather pathological situations) the above results can be improved considerably. The idea is that if we want a sharp estimate for # {j; ]p j-/~[ < �89 ~} from ~j p ( I t - p j), then it is better to let p resemble the step function

, f l / e I Z~(P)=~0 elsewhere.f~ I~1<~, (3.15)

2, is a smooth function with )~ (0)= l, but it has no compact support, so we multiply it by fia: t~ -~( t .~), with p > 0 , / 3 ~ C~~ fi(0)= 1, in order to cut it off. On the inverse Fourier transform side this amounts to taking the convolution &,~-- X~*P~- Of course &,~e~9~(lR) and for 3 ~ 0 it converges to X~. In fact:

Lemma 3.3. For every N there is a constant M > 0 such that

- i 8 �9 ~ {J; I&--~l ~�89 (1 - (&;')N)

<~p (# ) = e,3 --~/ j

J

< g - l . ~,{j; Iflj-~Ll=<l(~.+g)} +(~/~;)N �9 ~"

(3.16)

Jbr all p, 6, e,e; such that p_>l_>e_>d>_M6>0.

Proofi Note that (v+ }~)/6

v~,Av) = ~ - ~ S p(~)d~. ( v - �89

For the first inequality use that &,a > 0 and that 1 - ~_ ~ p(#) dp is rapidly decreasing as v-~ oo. For the second inequality the summation is split up into one over the j such that [ /a j -p l< �89247 and the rest into sums over p - � 8 9 /l i < p - �89 + d) - k, resp. p + �89 + e') + k < pj < p + �89 + e') + (k + 1). The latter sums are estimated using (2.19) and the rapid decrease of ~-~ p(p)d/~ and S~ ~ p(/a)d/~ as v ~ oo, and then the result is summed over k = 0, 1, 2 . . . . .

The Spectrum of Positive Elliptic Lperators and Periodic Bicharacteristics 57

As a first application of the use of these cut-off functions we improve the estimates in Theorem 3.1. Note that we allowed there a big constant K, which makes it possible than in fact many of the intervals Jgj- Vkl < Bk ~ are quite empty. This would happen for instance if the Hq flow is periodic with period To>0 and T = I . T o for some integer I> 1. Then the majority of the eigenvalues will only cluster around the v k for k = l, 2 l, 31 . . . . and not around the other v k. Now let T be the minimal positive common period of all Hq solution curves. Then a solution curve is called subperiodic if it is periodic with a positive period which is not an integer multiple of T, it is then automatically periodic with a period T/l for some integer l>0.

Theorem 3.4. Assume that the Hq.[Iow is periodic with minimal positwe period T and that the union of the subperiodic orbits has measure zero in T* X \ O. Assume also that the average of sub Q over the O-T-solution curves oJ the Hq flow is equal to a constant 7. Then there exist K', k o such that

I t j ; ~ > ~ l p j - V k t > B k -~ < K ' B -2 '~-~ (3.17)

[or all B>0, k >=k o. On the other hand

lira ~{.J;ll~j--Vkt<Bk-~}/v2 I = T I(27r)I-"vol(S*X). (3.18) B , k ~ ~)

Here the v k are as in (3.1) and again we may replace B . k -~ by B . k -I in the left hand sides of(3.17) and (3.18)/fsub Q=0.

Proof The set of (x, ~ )eS*X such that 4~t(x, ~)=(x, () for some t which is not an integer multiple of T has measure 0 in S* X. Using computations as in the beginning of the proof of Proposition 2.1 it follows that the contribution to (1.24) coming from such periods of Hq solution curves is of order o(#"-~) as / t - , oo. So modulo terms of order o(#"-~) we get only contributions from neighborhoods of t = l . T, le7Z, and iterating (3.10) and using the estimate (2.16) for the contribution from the singularity at t = 0, it follows that

~p( l l - - l t j )=~e"U-~)~r .D(IT) . (2rO-" .Co. l~"- t+O(l~"-~) as / l ~ (3.19) j f e z

for every ~e C~(IR). (Note that the sum in the right hand side of(3.19) is finite.)

Substituting p = p~,~, :t = v k and using that

~(I T) ~ ( I T)=(~ T) -1. e T ~ ~t(1. e T) . D6(l " r, T) l e Z l e e

-~(e,T)- lS~l( t) .~6(t)dt+06(1) as e, "~ 0

and ~,(t)~(t)dt-,2~ as ~ 0 ,

it follows from Lemma 3.3 that for every ~ > 0 one has

2 ~ . . - t +o(v .k- l ) # {J; lug- vkl <_-~} = ~ - . (2~)- Co v~ (3.20)


for k ~ oo. Using (3.2) this implies (3.17) and using (3.17),(3.20) can be improved to (3.18).

Remark. Weinstein [24] obtained a very precise estimate about the resemblance between the spectra of operators P, P' such that the principal symbol of P' is given by p o C, C a homogeneous canonical transformation, p = principal symbol of P. For a Zoll surface this leads to the conclusion that all the eigenvalues of the Laplacian lie in the union of intervals of the form [ j z - j - B , j z - j + B ] , j E N , and the number of eigenvalues in such an interval is equal to j2 _ j for sufficiently large j. Comparing this with (3.17), (3.18) for Q = ( - A + c) ~ (sub Q = 0 in this case), we see that we have a similar estimate for the intervals in which the majority of the eigenvalues lie, but that Weinstein's results for both the number ofeigenvalues inside these intervals and those outside are much more precise. (On the other hand our estimates are valid for more general operators.)

Theorem 3.5. Suppose that the set of (x, ~ )~S*X such that ~t has a contact 0[" infinite order with the identity at (x, ~).['or some t +O, has zero measure in S*X. Then .[br every ~ > 0:

lira ~ {j; I/~j-/~[ <�89 = e. (2zt)-" vol(S* X). (3.2t) I t ~ c~C

Furthermore,

{j;/~j<v} =(27t) ". vol(B*X), v"

- (27 r ) - " . ( ~ s u b Q ) . v " - l + o ( v "-~) as v~oQ, (3.22) S*X

improving (2.23)just a tiny little bit.

Proo[2 The assumption implies that all contributions to (1.24) coming from t in a neighborhood of periods T + 0 are of order o(/L "-1) a s / ~ oo. Substituting p = p~,~, using (2.16) and (3.16) and letting (5 "-~ 0, (3.21) readily follows.

Secondly, if :~ ~ C~ ~ (IR), 0q~ supp ~, then

p(# - p~) d/~ = lim (21r) -~ ~ ~e '"t ~(t) 8(0 dt d~ ~ , j a ~ ~ a

= (27z)-I ~ e i ~t(i t)-I &(t) ~(t) dt

- lira (27z) -1-feint(it) -1 &(t)&(t)dt a ~ - - ~

=(2rc)-l~ei~t(it) l ~:(t)~(t)dt

=o(v "- I ) as v--,oe.

So replacing p by pa in (2.24) leads to

i' ~, po(/ l- /~) d/~ =(270-" c o . v"/n+(27r)-" c~ . v"- t / (n - 1) - o 9 j

+oo(v "-1) as v ~ .

Note that the assumption implies that n > 1.


and

Now, for any e > 0 we can obtain both

{J, l~-,,IZ <=,c} _i~ Pa(#-&)dla < # {j; fl~j-vl< K} ._~ Ipa(#)l d / t<c , v "-~

1# {J;l~i< v - K } - # {j; /~j<v}l= # {j; v--K </~j <v} <e. v "-1

by choosing K > 0 sufficiently small (independent of 6) and v sufficiently large, using (3.21).

With this choice of K, the quantity

can be estimated from above by a constant times

co ~-I y IP(U)IdF 'do)= W((V-bl~)"-(v+K)").lp(u)[dla, v + K - o : - �9

and therefore is smaller than e. v"-~ if 6 > 0 is sufficiently small and v sufficiently large. Similarly ~c

{J; U j < v - K }

if ~ > 0 is sufficiently small and v sufficiently large.

Combining these estimates gives

f #{j; /x~<v} <4~:. Vn--1 oe j

for 3 > 0 sufficiently small and v sufficiently large, completing the proof of (3.22).

(3.21), (3.22) express that for large v the eigenvalues of Q are quite evenly distributed on the real axis, in sharp contrast with the clustering described in Theorem 3.1, 3.4. If X is a real-analytic manifold and q is a real analytic function on T*X \ 0 , then the assumption in Theorem 3.5 holds as soon as there is no T#:0 such that all Hq solution curves are periodic with period T.

Question. What happens (in terms of estimates for finite/t j-intervals) if there is a T4=0 such that all Hq solution curves are periodic with period T, but the average of sub Q over these curves is not constant?

4. The Singularities of ~'(t) for t # 0

We recall that the singularities of 3" occur at the set of periods of the periodic Hq solution curves (Corollary 1.2). If the Hq solution curves of period T form a "nice" submanifold of S*X we can obtain more precise information on the singularity at T. We first need a definition (due to Bott [5]).

Definition 4.1. Let M be a manifold and let 45: M--~M be a diffeomorphism. A submanifold Z c M of fixed points of q~ is called clean if for each z ~Z the set of fixed points of dq~ z: T= M-~ T~ M equals the tangent space to Z at z.


We will show that if M and q~ are sympletic, Z possesses an intrinsic positive measure. This depends on the following three elementary lemmas.

Lemma 4.2. Let V be a symplectic vector space with two-lbrm Q and let P: V-* V be a symplectic linear mapping. Then k e r ( l - P ) and c o k e r ( I - P ) are canonically paired by (2.

Proof If v e k e r ( I - P ) then v=Pv , so v ~ k e r ( l - P - t ) . This implies f2(v, (I - P )w)=0 for all w, or that veim(I - P)• Hence ker(l - P) and coker(I - P) are canonically paired. Q.E.D.

Now consider the exact sequence

0-+ ker-~ V j - P > V ~ c o k e r - - , 0 . (4.1)

Letting I ]~ be the functor that assigns to each vector space Vits one-dimensional subspace of a-densities,

Ikerl+|174174 +~ 1. (4.2)

Since [VI+| and Icoker[+-~lkerl -+ by the lemma, we get l ke r ]~ l ; so we've proved:

Lemma 4.3. I f P is a symplectic mapping of V onto V then k e r ( I - P) possesses a canonical density.

From I - P we get a linear map

(I - P)+ : V/ker ~ V/ker, ker = ker(l - P). (4.3)

If(4.3) is onto, then V= k e r ( l - P)O I m ( I - P). (4.4)

Moreover, by Lemma 4.2, f2 restricted to both factors is non-degenerate. Substituting the left hand side of (4.4) for V in (4.2) and making obvious can- cellations we get

Lemma 4.4. U (4.3) is onto, the canonical density just defined on k e r ( l - P ) is equal to

Ide t ( I - P)+ [-+ t~2rl (4.5)

where r =�89 dim ker(I - P).

Let M be a symplectic manifold, 4~ a symplectic diffeomorphism and Z a clean fixed point set of 45. By Lemma 4.3 the tangent space to Z at each z e Z possesses an intrinsic positive density, or what is the same thing, Z possesses an intrinsic smooth positive measure, which we'll denote by d/t z. In particular let Z be a submanifold of S*X consisting of periodic Hq solution curves of period T. If Z is a clean fixed point set for 4~r: S*X---, S ' X , then Z ' = {(x, )+~), 2elR +, (x, d)~Z} is a clean fixed point set for 4~r: T * X \ 0 - - , T * X \ 0 ; so d/a z, is defined. Dividing by ]dq[ we get an intrinsic measure d/~ z on Z. We now state a result relating the integral of this measure over Z to the singular behavior of 6(t) at t = T.

Theorem 4.5. Assume that the set o[" periodic Hq solution curves o[ period T is a union of connected submaniJblds Z1, Z 2 . . . . . Zr in S ' X , each Zj being a clean fixed point set .['or ~r o[ dimension dj. Then there is an interval around T in which no


other periods occur, and on such an interval we have 8( t )= ~ = 1 f l j ( t - T) where

~o

flj(t) = ~ o~(s) e -i~' ds . - o o

with \(dj - 1 )/2

~ j ( , ; ) ~ [ ~ / ) / S i-~rJ ~O~j,kS-k as S---*OO. k=O

(4.6)

(4.7)

Here we interpret (1/i) (dJ-1)/2 = e -~iCaJ-1)/4 (f dj is even, and a~ is an integer on which we will comment Ji~rther at the end of Section 6. Finally

1 - - - - y e - i t " , (4.8) ~xj, o - 2 n ;' d'uz~

with 7(x, ~)= average o f sub Q over the periodic Hq solution curve passing through (x, ~ )~S*X . O f course o~j(s)~O as s --~ - oo.

If d j = 1 then the cleanness condition reduces to the condition that 1 -P j is invertible. Here P~ denotes the differential of 4 ,r at (x, ~)EZj reduced modulo the eigenvector Hq(x, ~). P~ will be called the ( l inear) Poincark map. By Lemma 4.4, dl~j = II - P~l- ~ Idt[~ and integrating over Zj gives

To e - i r . : , , l I _ P j ] % o = 2 ~

where T o denotes the primitive period of the Hq-solution curve having Zj as its orbit.

In particular, if Q = (A + c) ~, then, for generic Riemannian metrics, the Poincar6 maps of periodic Hq solution curves have no eigenvalues equal to 1 and for each period T=t=0 there is not more than one antipodal pair of orbits of periodic H 0 solution curves with that period. (See Klingenberg and Takens [25].) This situation has also been considered by Colin de Verdi6re [7]. The contributions from the

i 6 antipodal orbits are equal; and, recalling that sub Q = 0 and f~ e -is ' ds = - - + jo one gets from (4.6)-(4.8) the residue formula t 4n"

TO ~r lim(t - T) a ( t ) = - - - i- ~ f I - Pj1 - �89 . (4.9) t ~ T TC

Note that the primitive period, T o , is equal to the length of the geodesic covered by the periodic Hq solution curves with period equal to T.

5. Clean Intersection Theory

The proof of Theorem 4.5 will require some facts concerning composition of Fourier integrals under hypotheses less restrictive than those considered in H6rmander [17]. First, however, we'll need some elementary facts about symplectic vector spaces.

Let V and W be symplectic vector spaces, let F be a Lagrangian subspace of V x W, and let A be a Langrangian subspace of W. Let F o A be the set of vectors v~ V, such that there exists (v, w ) e F with w e A .


Lemma 5.1. F o A is a Lagrangian subspace of W.

Proo[~ Let p and 7z be the projections of F on V and Wrespectively. Consider the diagram

F , . . . . . . F

71: I

W, ; A

F being the fiber product: {(a, b)eF • A, re(a)= i(b)}. Associated with this diagram is an exact sequence

O - - * F - , F O A - ~ , W--~coker r - ~ 0 (5.1)

where z(a, b) = rt(a)- i(b). F o A can be defined as the image of the composite map

F - * F ~ E

Denoting this composite map by ~, we get an exact sequence

0 ~ k e r ~--~F ~ F o A--~0. (5.2)

We will now show that ker ~ and coker r are dually paired by the symplectic structure on W. Note first of all that ker ~ consists of all pairs (a, w) in the fiber product for which p(a)=0. We can write a e Vx Was a pair (v', w'). To say that (a, w) is in the fiber product says that w '=w, and to say p ( a ) = 0 says that v ' = 0 ; so ker ~ can be identified with the set of w e W such that

i) weA

and

ii) (0, w)eF.

Suppose now that u is in the image of r, i.e. u=w~ +=(v 2, wz) with (t12, Vc2)GF. Then fJw(W, w 0 = 0 by i) and f2w(W, We)=0 by ii) so f2w(W, u)=0. Since F and A are maximally isotropic, this argument works backward to show that (ker ~)s= (Im r) in W. It is easy now to see that the dimension of F o A is half the dimension of V using the exact sequences (5.1) and (5.2). It is clear that F o ,4 is isotropic; thus it is Lagrangian, proving Lemma 5.1.

Lemma 5.2. Let ~: F--~ FoA be the mapping defined by (5.2). 7hen there is a canonical mapping of half-densities

I A I ~ | IF o Al~| cq.

Proc?s From (5.1) we get an identification

IFl-~|174174174

and

From (5.2) we get

IF[~| W[~| --~ ~ IFI~|

IFb = IF Al~| ~1 ~. (5.3)(i) ,r


The symplectic structure on Wgives us a trivialization

IWl+~ 1 (5.3)(ii)

and finally the dual pairing of ker ~ and coker r via the symplectic structure of V gives us a mapping

Icoker r l - -~ ~ [ker ~l ~. (5.3)(iii)

Putting this all together we get the assertion of Lemma 5.2. Q.E.D.

Remark. Let F' be the set of vectors {(v, -w) , (v, w)eF}. This is a Lagrangian subspace of V x W' where W' = Was a vector space but has - f2 w as its symplectic two-form. The Lagrangian subspaces of Wand W' are the same; so Lemmas 5.1 and 5.2 are true for F' as well as F.

Given manifolds X, y Z and maps./: X - * Z and g: ~ - , Z , f a n d g are said to intersect cleanly (see [53), if the fiber product

F = {(x, y ) eX x Ef(x )=g(y)}

represented by the diagram

X , F y i

Z , g y

(5.4)

is a submanifold of X x Y, and in addition for each peF, p=(x, y),

T ~ X ~ - - - TvF

d f x I i

is a fiber product diagram, i.e. TpF is the fiber product of TxX and TyY (For example, i f / and g intersect transversally, they intersect cleanly.) We associate to the diagram (5.4) a non-negative integer, e, called its excess:

e = dim F + dim Z - (dim X + dim Y).

Note that e = 0 if and only if the clean diagram (5.4) is transversal.

Now let X and Y be compact manifolds and F and A closed homogeneous Lagrangian submanifolds of T*(X x }/)',0 and T* Y \ 0 respectively. Let F'= {(x, d, y, r/), (x, ~, y, - t/)eC} and let C' o A = {(x, ~), ~ (x, ~, y, ~l)eF', (y, ~/)eA }. Let p and ~ be the projections of F' on T* X and T* Y. Assume neither F' o A nor r~(F') contain zero vectors.

Lemma 5.3. f f the fiber product diagram

F'~- . . . . F

T * Y ~ - - - A


is clean, then F' o A is an (immersed) Lagrangian submaniJbld of T* X ",.0, and the composite map, ~,

F - - ~ F ' ~ F ' o A

is a fiber mapping with compact fiber.

Proo[i The first part of the Lemma is just a rephrasing of Lemma 5.1. The fibers of ~: F ~ F ' o A must be compact , for otherwise in view of the homogenei ty of F' and A, F ' o A would have to contain zero vectors. Q.E.D.

Given half-densities on F and A, then by L e m m a 5.2 we get an object on F which is a half-density in the horizontal direction times a density in the fiber direction. Integrat ing this over the fibers, we get a half-density on F 'o A which we will call the composite of the given densities on F and A. It is easy to check that if a and z are homogeneous densities, then so is their composite, a o ~, and

(dim Y - e) degree a o z = degree a + degree v - 2 ' (5.6)

e being the excess in the d iagram (5.5). (The excess comes in because of the pairing (5.3)(iii), which is homogeneous of degree e/2.)

We now state a "clean intersection" theorem on composi t ion of Fourier integrals. For basic facts concerning oscillatory integrals and the spaces, P(Y, A), see H 6 r m a n d e r [ 17].

Theorem 5.4. Let k be a generalized half-density on X x Y and K the operator associated with it. (K maps compactly supported half-densities on Y to generalized half-densities on X.) Let F and A be as above. I f k belongs to Im(X x Y, F); then K maps Is(Y, A) into

r e m + s + ~ - - -

I (X, F' o A) and a(K/~)= (27ri) 2 a(k) o a(/0

modulo Maslov factors.

We will briefly describe how to handle the Maslov factors. Let L r, L A and L r,oA be the Maslov bundles of F, A and F ' o A. Let L r, be the pull back of L r to F' via (x, 3, Y, t/) --, (x, 3, Y, - r/). Let i: F ~ F' • A be the inclusion map and ~: F ~ U o A defined above. We claim

i*(Lr' [] LA) -~ ~* Lr'oA. (5.7)

Grant ing (5.7) it is clear how to define o(k)o a(#) when a(k) and a(/z) are half- densities tensored with sections of the Maslov bundle. In fact the product of a(k) with a(/~) is now an object on F which is a density vertically and a half-density times a section of ~* Lr,o A horizontally. Integrat ing over the fiber we get an object on F ' o A which is a half-density tensored with a section of Lr, o A.

To prove (5.7) let ~o 1 = ~01(x, y, 3) be a phase function for F and ~1 = qJl (Y, q) a phase function for A. Then ~o 1 +~91 is a phase function for F ' oA (with ~o=(y, 4, t/) as phase variables). 2 If ~02 = q~2(x, y, 3') and ~2 = ~'2(Y, ~') are another pair of such

2 qj +~p can be degenerate; however it is a "clean" phase function in the sense of w 7. Such phase functions can be used to give a Qech-theoretic definition of the Maslov class, just as H6rmander does in w of [17].


phase functions, then on the overlaps of their domains of definition

+ ,/, V' " sgn(qh ~-lJ~,~-sgn(~P2 +r (5.8)

H t~ tt tt = sgn (~o 1)r162 - sgn (q~ 2)r r + sgn(~l) , , - sgn(~b 2),','-

(Proof (5.8) is obvious when ~o l, ~0 2 and ~b 1 , ~O 2 are equivalent. It is also true if we modify such equivalent pairs by adding to ~0 2 and ~b 2 quadrat ic forms in new phase variables. By H/Srmander [17] w this covers all possible cases.)(5.8) implies that the transit ion functions of Lr,o A are the products of the transit ion functions of L r and L A. (5.7) follows.

6. Proof of Theorem 4.5

This will be by applying Theorem 5.4 to the distr ibution U(t, x, y) of Theorem 1.1. For this we need to know the symbol of U. We know already from Theorem 1.1 that UEI- I ( IR • X x X; C'), C being the set (1.15). Its symbol is a section of the half-density bundle of C' times a section of the Maslov bundle of C'. To see what it is, first note that both these bundles have intrinsic trivializations. In fact, for the half density bundle, the project ion ~" C -~ (T* X \ 0) x IR, ~(t, z, x, ~, y, r/) = (t, x, 4) is a dif feomorphism; so

a = n*(]dt[~| [dx A d~l ~) (6.1)

is a nowhere vanishing half-density.

Consider next Lc,, L c, is a locally constant line bundle (i.e. its transit ion functions are constant); so we are interested in finding a trivialization given by a constant section. Such a section, if it exists, is unique up to scalar multiples. To see that such a section exists note that the subset t = 0 of C' is identical with N * A = { ( x , r 1 6 2 1 6 2 Moreover L c, restricted to N*A is its Maslov bundle. Since N*A is a normal bundle, LN, ~ possesses a canonical constant section, s, by H6rmande r [17] w Now extend s to a global section of L c, by requiring it to be constant along each bicharacterist ic:

(t, r, x, 4, Y, - q ) , (x, 4)= ~bt(y, r/) (6.2)

with - ~ < t < ~ . It is clear this can be done in one and just one way.

Now let a be the section of f2~| c obtained by tensoring (6.1) with the constant section of Lr just described.

L e m m a 6.1. I

at the point ((t, z), (x, ~), (y, - t/))e C'.

Proof. At t = 0 the left and right hand side are equal since both are equal to the symbol of the identity map. Moreove r a is invariant under the Hami l ton flow so the right hand side satisfies the t ranspor t equat ion for a(U) induced by the equat ion (i -1 ~?fi?t + Q). U = 0. (See [9], Theo rem 5.3.1.) This proves the equality for all t.

5 lnvent iones math . , Vol. 29


Let A: IR x X-- , IR x X x X be the diagonal map. Given a half density u on lR x X x X we can pull it back to the diagonal and multiply the two half density factors in X to get an object A*u which is a density in X times a half-density in IR (at each point of X x IR). This object can be integrated over X to get a half-density on IR which we will denote by n . A*u(n: IR x X -* IR being the projection map). Since n . A * is an operator from half densities on IR x X x X to half densities on IR its Schwartz kernel, k~.a., is a half density on lR x X x X x IR. In the following lemma, identify lR x X x X x IR with (IR x X) x (IR x X) via (t, x, y, s) --, (s, x, t, y).

Lemma 6.2. k~,d, = k~d, k~ being the Schwartz kernel of the identity map.

Proof Both k~,~, and k~a can be viewed as multi-linear functionals on

C~(IR, ~ ) x C~(X, ~2~) x C~(X, Q~) x C~(IR, ~ ) .

It is clear that they are identical with the multi-linear functional given by the pre-Hilbert structures on C~176 f2~) and Coo(N, (2~). Q.E.D.

Let F be the conormal bundle to the diagonal in IR x X x IR x X. We will show, as we did above for C, that both the half-density bundle and the Maslov bundle of F are trivial. To see that the first of these assertions is true, note that the projection map n: F ~ T * ( l R x X ) \ O , ~ ( t , z , x , ~ , x , - ~ , t , - r ) = ( t , r , x , ~ ) is bi- jective and set

a'=n* [dt /x dz /x dx /x d~[ ~ (6.3)

a nowhere vanishing half density on E

The Maslov bundle of F is trivial since F is a conormal bundle; however, it will be helpful to see explicitly why. Over any coordinate patch, F is describable by the phase function

= ( x - y). ~ + ( t - s)z. (6.4)

It is clear that any two such t/"s are equivalent; so the set of all such ~U's determines a unique non-zero constant section of L r.

Lemma 6.3. k~,~,~I~ x X x X x IR, F) and a(k~,~,)= a'.

Proof Both these statements are well-known for k~d. See, for example, H6r- mander [17], Section 4.

Going back to (1.10) of Section 1 we have

U(x, y, t) = ~ e ~"; ej (x) e i (y)

where ej~ C~(X, (2~) is the j-th eigenfunction and u~ the j-th eigenvalue of Q. Setting x = y and integrating over X we get

~ e-i",' ( ej, ej)L2= ~ e-i"J';

so this proves

Lemma 6.4. ~ e -i"~t = n , A * U .


We are now in position to apply the clean intersection theory of Section 5. To do so we need to know that the fiber product diagram

FP.~___ _ 1 ~

! l T*(lR x X x X)~- C

(6.5)

is clean. The map of F' into T*(IR x X x X) is an embedding, and the image consists of the points {(4 r, x, ~, x, - ~)}. To apply Theorem 5.4 we need to know this intersects C cleanly. By (1.15) the intersection is the set of all (T, x, ~) for which crpr(x, ~)= (x, ~), i.e. the fixed point set of q~r; and by definition, F' intersects C' cleanly if and only if this fixed point set is clean. Suppose the fixed point set is a d-dimensional submanifold of S*X. d is the excess in diagram (6.5); so combining Theorem 5.4, Lemma 6.3 and Lemma 6.1 we get

d 1 g , A * U e I 2 4 ( A t ) ,

where Ar = {(T, ~), zelR }.

d 1

By definition, I z 4(At) consists of scalar multiplies of the distribution

1 o0 d - 1

S 2 e is(~-rlds (6.6) 2g o

plus similar distributions of lower order. The coefficient of (6.6) is the symbol of re, A* U. By Theorem 5.4 this is equal to ~(k~,~,)o~(u). The half-density part of this is easily computed to be

d - 1

s 2 l / ~ e i , r d p z (6.7)

(just observe that (4.2) is a special case of Lemma 5.2 with F equal to the diagonal and A equal to graph P).

To compute the Maslov contribution to (6.9), represent U(t, x, y) locally by oscillatory integrals (1.17) with phase functions of the form

~o(t, x, y, q)= Z(t, x, r/) - (y, r/). (6.8)

This can be done by taking Z(to, x, r/) as the generating function for the canonical transformation cb t~ that is

{(( x ) (d~Z(to, X,~l))) ; x, t/~IR"}, (6.9) graph(q~,o)-1 = dx~(t 0, x, r/) ' r/

and then taking g for nearly t as the solution of

d, z(t, x, rl) + q(x, dxz(t, x, r/)) =0 . (6.10) 5*

68 J . J . D u i s t e r m a a t and V. W. G u i l l e m i n

If we do this near a period t o= T then the leading term in (1.25) is equal to a positive constant times eS~i/4a, where s is equal to the signature of the matrix

d~.,t,,,rp(t,x,x,q)=~dtd~z d2z d . d . z - I ] . (6.11)

\dtd,)~ d ~ d , x - I dZz /

One can choose the local coordinates such that at the periodic point one has O/Ox~ =Hq and t~/c?~l is tangent to the cone axis. Differentiating (6.10) and the equation q(x, d~z(t, x, r/))=q(d,z(t, x, r/), q) it follows that (6.11) has the matrix

00 on the space spanned by ~?/~?t, (?/~x 1, ~?/~?~, and that the vectors (?/Ox s, (?/O~j for j > 1 are orthogonal to this space with respect to (6.11). So s is equal to the signature of the matrix

[ d:x Q=\d~d~x-I d~z ] (6.12)

restricted to the space spanned by c')/~x s, 0/~ ~s with j > 1.

Next we have to investigate the Maslov factors picked up by the amplitude a. Let ~ s = X s - ( y , r/>, j = 1, ..., J be a sequence of phase functions of the form (6.8) related to a chain of neighborhoods in C' along the H~+q~ solution curve; denote the corresponding amplitudes by a s. Let t j, j = 1 . . . . . J - 1, be times at which the curve is in the intersection of the j-th and ( j+ 1)-th neighborhood. Using H6r- mander [17, p. 148] we have at time tj:

_ .~g~a~oj- ~g.d~o2 +~) aj (6.13) a s + 1 - - 1

modulo positive factors, so the amplitude picks up the factor i k with

J - 1

k= ~'5(sgn d.Z ~Ps - sgn d2 q~J+ 1) �9 (6.14) j = l

2 __ 2 Now d, r d, Z j, and according to (6.9)

{ dzZs ; 3r/6lR"} (V) \ 6r/fir/) = (dq~,j)_l

where V denotes the "vertical space" 3 ~ ; 6x = 0 , that is, V is the tangent space

to the fiber in T* X. Denote by H s the "horizontal space" 6 ~ ; f ~ = 0 for the

local coordinatization of X used in thej- th local representation of U(t, x, y). Then (d4VJ) - 1 (V) is the graph of the linear mapping d~ Zs: V--, Hi, and it follows that

d. ~ Zj = - Q(v, Hi; (d~'~)- i (V))


in the notation of [-12, Section 2]. Using [12, (2.10') and (2.13)] it follows that

�89 d 2 q~i- sgn d 2 ~pj+ 1)

= �89 {sgn Q(V, Hi; (d~")- I(V)) - sgn Q(V, H~+~ ; (dq~'0-1 (V))}

= (v), v; Hi, 1) = s(/4:, Hj+I ; (V), V).

This integer in turn is equal to the Maslov-Arnol'd index of the loop consisting of an arc yj of Lagrange spaces going from (dcbt,)-l(V) to V transversal to H i followed by an arc y~ of Lagrange spaces going from V to (dqJt~)-l(V) transversal t o Hj+ 1.

Denote by 6j the curve (dq~t) - I (V), t running from tj_ 1 to tj, 6t the same one for t running from 0 to t 1 and 6 s this curve for t running from t s_. to the period T. By construction fi~(t) is transversal to Hj for tel,t j_ 1, t~] and we can arrange that 61, 6j both are transversal to H I = H s. Write fi for the curve (dq~') -a (V), t running from 0 to Tand s for the curve 6j o Ys- 1 ~ Y J- 1 ~ 6j_ ~ . . . . . 3 2 o ~)1 o ~'1 o ~1"

~~ 6 j ( t ) = ( d ~t) - l ( v ) (6.15)

Because the loop yj, o 5jo ' /)_l is contained in the contractible set of Lagrange spaces transversal to H i its intersection number with any Lagrange space H vanishes. Because 6j, 7~_~, 71, fil are transversal to H 1 it follows that the intersection number [s: H1] of the curve e, with H 1 vanishes too. On the other hand

s- 1 H 1] + [5: H~ ] = k + [6: H 1 ]. We conclude that the symbol I-s: H1] =~, j =1 f'~"l o ~)j: of lr, A* U is equal to (6.?) multiplied by

n i

e 2 {[,6" H 1 ] - � 8 9 -~i(d-1)/4 (6.16)

if we write

a = [ 6 : H~] + ind Q - ( n - I)

=[,6: H t ] + �89 (rank Q - s g n Q ) - ( n - 1)

= [6" H1] - �89 sgn Q - � 8 9 nullity Q,

and note that nullity Q = d - 1.

This completes the proof of Theorem 4.5. The integer a appearing in the ex- ponent of i can be identified with the Morse index for periodic geodesics in the following case. Assume that d~q is positive definite on Ker dr at points (x, 3) on the orbit of the periodic Hq solution curves. This means that the sphere q = 1 in the fiber is strictly convex (as a boundary of the ball q < 1) at these points (x, 3).


Taking p=q"~ for some m> 1 it follows that dip is positive definite and _ t Hp is m

related via the Legendre transformation to the Euler-Lagrange flow for a variational problem with finite index (see [t2]).

1 Denoting the - - Hp flow by ~t, the matrix of (d~ t)- a on the space spanned by

1 Hp and the direction of the cone axis is of the form with c > 0. (Note m

that ~,' = q~t on S* X.) If ~7. is represented as in (6.9) with a generating function 2, then ind Q = ind Q + 1 if we write

( d2~ d.d~,2-I] onlR2. Q= \dxd,2- I d~ ~ ]

and T > 0. So ~ = [6: Hal + ind 0 - n and according to [12, Proposition 4.6 and formula (4.5)] this number is equal to the Morse index for the variational problem with periodic boundary conditions. This identification was suggested to us by the expansion of Colin de Verdi~re [7] who used the trace of the heat operator rather than the wave operator.

7. The Proof of Theorem 5.4

We begin by considering oscillatory integrals of the type considered by H/Sr- mander in [17], i.e. integrals of the form

p(x)=5 a(x, O) e ir o) dO (7.1)

where a(x, O) and ~0(x, 0) are smooth functions on X x R N with the properties

(1) ~o(x, 0) is homogeneous of degree one in 0 and d(p 4= 0. (2) a(x,O)=O near X x{0} and is homogeneous of degree d+(n-2N)/4

for 0 large.

Finally there is a third condition which H6rmander imposes on q):

(3) I f ~ f f - ( x , 0)=0 then at (x,O) the differentials d c~q~ , . . . , d a0N are

linearly independent. &o

Let C~o be the set of points where-~0-=0. Then condition (3) implies that

Cr is an n-dimensional submanifold ofX x R ~ and that the map

&0 C~,---, T ' X , (x, 0)--, ~ (7.2)

immerses C~ as a Lagrangian submanifold, A, of T*X. U(X, A) is defined to be the space of all distributions which have a local representation of the form (7.1). The symbol of (7.1) is defined as follows. Let d c be the density on C~o obtained by pulling back the delta function on IR N via X x IRN~ IR N, (x, 0)--*q~ 0. The half density part of the symbol is the image of

N

(2n)-" (2hi) - T a l /~c

with respect to (7.2), and the Maslov part is the section ofL A determined by (p.

Tbe Spectrum of Positive Elliptic Operators and Periodic Bicharacteristics 71

For our purposes we need to consider distributions of the type (7.1) satisfying 1) and (2), but with (3) replaced by a weaker "cleanness" condition, to wit:

(3)' C~, is a submanifold of X x R N and at each point of C o the tangent space

Suppose that the dimension of the space spanned by these differentials is N - e.

Lemma 7.1. The map Co-+ T * X defined by (7.2) has as its image an immersed Lagrangian ma~lifold A c T * X and the map Co-*A is a fiber mapping of fiber

dimension e. Moreover the distribution I~(X) defined by (7.1) is in I (X, At.

Proof To see that the image of Cois a Lagrangian manifold, we apply Lemma 5.3 of Section 5 with A replaced by A =graph do and F replaced by the normal bundle to the graph of ~: X x R N--, X, which we will denote by H. It is simple to check that the fiber product diagram (5.5) of Section 5 is clean if and only if (3)' holds. To prove the last assertion we write (7.1) in polar coordinate form. Set O=s~o with t o e s N-1. Then

I~(X)=~ sk a(x, eo)ei'~ ~') dcods where k = d - l +(n+ 2N)/4. (7.3)

We can assume that a(x, co) has its support in a coordinate patch (on Ss-1), hence that when the integrand in (7.3) is non-zero, (o is in a compact subset of RU 1. The "po la r" critical set C'~, in X x R u 1 is defined by the equations

q) = 0, = 0, = 0 . . . . , 0. (7.4) ?(O1 0(02 0coS- I

By (3)' the differentials of these functions are the defining equations for the t normal space at each point of C o. By a change of coordinates we can assume that

the first N - e of these differentials are linearly independent, and that

( ~o ~, d ( e~o ~,dcol ' .,dcoN 1, I=N- -e , d,a, d ~(o~ f .... \ aco,_l ! " -

are linearly independent. This implies that C'~ is locally defined by the first I equations of (7.4) and that C~, intersects the surface cot=const, cot+, =const, ..., coN_ 1 =const transversally. Let ~o' denote the first l - 1 coordinates and co" the remaining e coordinates. Then for m"= c, the function q~(x, m", c) is a non-degenerate phase function on X x R z-: (in HiSrmander's sense), and its critical set is just the intersection of C' o with co"= c. So in particular its associated Lagrangian manifold in T* X is A. Now write

kt(x) = (5 s k a(x, co', co") e ''p( . . . . " ~'") do)') dco". (7.5)

For fixed (o" the inner integral is in I (X, A); therefore, so is/~ itself. To compute the symbol of (7.5), identify the compact subset of IR t ~ where co'

is defined with a subset of S ~-1 by stereographic projection and set 0'~IR ~ equal - to so.)'. Let

a,~,,(x, O')= s~ a(x, (o', co") and

~o ,o,,(x, O')= s~o(x, oY, co").


Then the symbol of (7.5) is equal to

a(o,, do)" (7.6)

where ao,,, is the image of the half density

. (2~) (27ti) ~,,,, with respect to

(dq~o,,,):,: Co,,, ---, A.

This symbol can be described more intrinsically as follows. Let 3 be the graph of d~o in T * ( X x IRN)\0 and let H be the conormal bundle to the r~L~h of n: X • IRN~X in T * ( X x (X • IRN))\ 0. Let a be the half density a 1/dx dO pulled back to A. Identify H with the subset, {(x, {, 0, 0)}, of T * ( X x 1R N) and let fi= (d]SdxAd{) ]//dO. By assumption 3 and H intersect cleanly, so by Lemma 5.3 the composite half density, fl o ~, on A = H o 3 is well defined.

N - e

Lemma 7.2. (7.6) is equal to (27z) -N (2rci) 2 fi o ~.

Proof. When e = 0 this is just a paraphrase of the formula above. If %~,, is independent of ~o" the equivalence of the two terms reduces to the case e=0. The general case can be reduced to the case where ~0,o,, is independent of (o" by HOr- mander [17], Theorem 3.1.6. (An inspection of H6rmander's proof shows that it applies to families of phase functions.) Q.E.D.

We now proceed to the proof of Theorem 5.4. With the notation of Theorem 5.4, let/z e I~. Represent/~ and k by oscillatory integrals:

k(x, y)=~ a(x, y, O) e i~ r, o) dO,

p(y) = [. b(y, ~) e iq'(y ~-) d~.

Then K -/~ is represented by the oscillatory integral

K~z(x) =~ a(x, y, O) b(y, ~) e i{~ r, o)+ o(y. r dO d~ d y. (7.7)

We claim that this is in f " + k ~ z ( x , F'o A). The proof is practically identical with the proof of Theorem 4.2.2. in H6rmander [17], so we will just sketch its main outlines. The integral (7.7) is of course an oscillatory integral in the phase variables (~, 0, y) but neither its amplitude nor its phase function satisfy (1) and (2). In the first place the amplitude and phase function aren't homogeneous in ~o=(~, 0, y), for IcoJ large and in the second place they are not even homogeneous in (~, 0) alone for I~[+10] large (i.e. there are problems when 0 is close to zero or

is close to zero). The first difficulty can be handled by using polar coordinates like in the proof of Lemma 7.1. To handle the second difficulty note that the critical set of ~o + ~k with respect to co, is the fiber product of C o and Co:

C~, ~ C o +

/ i

l i T* Y ~ C o


This set can't contain (~, 0) with either ~ or 0 equal to zero since this would contra- dict the assumption that the projection of F' on T* Y and F' o A don't contain zero vectors. Therefore we can assume that there exists a constant K > 0 such that

K IOl < tel < ~ - tot

on a neighborhood of the critical set of ~0 + r Off this set, the integrand of (7.7) can be made as smooth as desired by integration by parts, and on this set ~o + 0 is a clean phase function in the sense of (3') as long as I~]+10l is large, so we can apply Lemma 7.1. (For further details we refer to H6rmander [17], Section 4.2.)

To compute the symbol of (7.7) we will need first to make some general ob- servations concerning composit ion of canonical relations. Let C 1 c: T*(X x Y ) \ 0 and C 2 ~ T * ( Y x Z ) \ O be Lagrangian manifolds. Assume they are properly situated with respect to each other in the sense of Theorem 4.22 of H/Srmander [17]. We will say they intersect cleanly if C' 1 x C 2 intersects cleanly the set of points {(x, ~, y, r/, y, q, z, 7)} in T * X x T* Y x T * Y x T*Z. This means that

A' , fiber product

+ , T * ( X x Y x Y x Z ) , C l x C 2

is a clean diagram in the sense of(5.5) where A is the normal bundle to the diagonal in T * ( ( X x Y x Y x Z ) x X x Z ) . By Lemma(5,3) C'~oC2=A'o (C lXC2) is a Lagrangian submanifold of T*(X x Z). It is just the set of all (x, ~ , z , - y ) such that ~ (y, ~/) with (x, ~, y, - ~/)e C t and (y, ~/, z, - 7)e C2. Given half densities a 1 and cr 2 on C 1 and C 2 then by Lemma 5.3 we get a half density on C1 o C 2 which we will call the composite half density and denote by rr I o a 2. We leave it as an exercise to show that this composit ion is associative in the following sense

Lemma7.3. Let Cl c T * ( X x Y)\O, C2 ~ T * ( Y x Z ) \ O and A c T * ( Z ) \ O. Assume C1, C 2 and A are all properly situated with respect to each other and all intersections are clean. Then

(C' loC'2)oA=C'lo(C'2oA) and (0"1 ~ 0"2) o 0" = 0"1 o (0"2 o 0")

,for half densities al, a 2 and a on C 1, C 2 and A.

Now let P be the graph ofd~0 in T*(X x Y• IR N) and A be the graph of d0 in T*(Y• IRM). Let H r be the normal bundle to the graph of n: X • Yx IR N ~ X x Y. (H r is in T*((X • Y• IR N) • X x Y)) and let Ha be the graph ofzt: Y• IR M -~ Y.. Let A, in T*((X • Y x Y) • X), be the normal bundle to the diagonal. Let oa be the half density on d obtained by identifying J with T*(X x Y). (Compare with Lemma 6.4.) Then F 'o A = A ' o ( F x A). Moreover if ~ and ~ are half densities on F and A,

Consider the half densities z(x, y, O ) ~ d O on/~ and b(y, ~) l /~yd~ on / [ . One way to get a half density on F' o A is to take the product half density on P x fi, apply H r x H A to it to get a half density on F x A then apply A to get a half density


on U o A. By Lemma 7.2 this is the half density associated with the integrand of (7.7). The other possibility is to apply H r to the half density on P (which by Lemma 7.2 is the symbol of k) apply H A to the half-density o n / [ (which is the symbol of/x) then apply z] to the product, which is the same as a(k) o a(/~). The assertion of Lemma (7.3) is that these two computations give the same answer, i.e. z[o (H r x HA) applied to the product symbol on F x A is the same as H r x H A

applied to this symbol followed by A-. One small adjustment has to be made in this argument. The computations of a(k) and a(/0 both involve generic phase functions but the computation of the symbol of (7.7) involves a clean phase function with excess e; so the factor of (2~i) occuring in a(K#) is not the same as the product of the factors occuring in a(k) and a(g) i.e. we get

e

a(Klt)=(2:zi ) z a(K) ~(t~).

Finally a word regarding the Maslov factors. The Maslov factor involved in (7.7) is sgn(cp + tp)~o, and the Maslov factors involved in a(k) and cr(/~) are sgn q)'0'0 and sgn tp~'~. Therefore the Maslov factor for a(Kt~ ) is the appropriate composite Maslov factor in view of (5.8).

Appendix

Suppose the periodic bicharacteristics are isolated, occur with distinct periods, and satisfy the genericity condition: I -P~ invertible, P~ being the Poincar6 map of the bicharacteristic 7. I lk7 is the k fold iterate of?; this means that I - P~ = I - pkT. is invertible for all k; so by (4.9) we can determine the numbers

Idet(I - PTk)[, k = 0, 1, 2 . . . . , (*)

for all k, from the spectrum of Q. To what extent do the numbers (*) determine the eigenvalues of PT? Being unable to answer this question satisfactorily ourselves we asked Harold Stark about it and obtained from him the following answer which he has kindly allowed us to reproduce in this appendix:

Theorem. Let P be a k x k symplectic matrix having no roots of unity .for eigenvalues. Then, knowing the numbers d e t ( I - pn) n = 1, 2, 3 . . . . . one can determine

a) the eigenvalues of P of absolute value #: 1

b) the eigenvalues of PN for some integer N.

In other words, one can determine the eigenvalues of P of absolute value = 1 up to multiplication by N-th roots of unity. Unfortunately, as will be seen below, there is no simple way of bounding N in b).

Corollary. One can determine a) and b).just from knowing Ide t ( l -P") l for n = 1 , 2 , 3 . . . . .

App,y thotheorom to (O O)

The Spec t rum of Posi t ive El l ipt ic Opera to r s and Per iodic Bicharacter is t ics 75

w I. The proof of part a) of the theorem is rather easy. Namely, form the "generating function":

f ( t ) = ~ P2~n tn, /~ ,=de t ( l -P" ) . n = 0 #n

If a 1 . . . . . a k are the eigenvalues of P, then

f ( t ) = 1-I (1 +aT) t" ~-.-. .=o i=I - = ~ t + - t - - a i t . . 1 - - a i a j t

re(co) (1.I)

- - ~ 1 --cot '

co being one of the numbers on the list: k

I-I@ e,i = 0, 1, (1.2) i ~ l

and re(co) the number of times it occurs as such a product. We already see from (1.1) that there can only be a finite number of eigenvalue combinations having the given/~,'s.

Let co o be a number on the list (1.2) of minimum absolute value (coo not necessarily unique). Suppose O,~o=l-I@, 61=0, 1. Consider the quotients co/co0. These are of the form

1-Ia? -6'= lq a 'I-I i=0,1. (1.3) 6, =0 6, =I

Let b i = a i if ~5i=0, and b i = a i I if 3i= 1. Then the numbers (1.3) are the set of all numbers

1-[ b7', e,=0, 1. (1.4)

We claim that knowing the products (1.4) one can determine the b~'s of absolute value + 1. In fact the products (1.4) are all of absolute value > 1, Suppose

t611 . . . . . ibsl=l and Ibs+i t , . . . , ]bk l> l .

Consider among the products (1.4) the ones of absolute value = t. These are the products

11Ib ~' e/=0, 1 (1.5) i=1

therefore we can extract these products from the list (1.4); and, in analogy with (1.1), determine the generating function

s

Y l-I(l +bT)t". i=1

We also know the generating function associated with the products (1,4) and comparing the coefficients of these two series, we get

k

2 I~ (l+bT)t" i = s + l


so, again in analogy with (1.1), we know all the products

J,

H b~, ~,=0,1. i - s + 1

Among these products the ones of minimum absolute value must be bi's. Since each bi is an a~ (P is a symplectic matrix so the inverse of an eigenvatue is an eigenvalue), we've located one eigenvalue, say a, of absolute value > 1. Divide the n-th coefficient of (1.1) by (1 +a")(1 + a - " ) and repeat the argument with the remaining a~'s. After a finite set of steps all the eigenvalues of absolute value 4= 1 can be determined.

w Given real numbers a and b let's say l a - b ] < e ( m o d 1) if there exists an integer n such that l a - b - n] < e. Recall Kronecker's theorem:

Theorem. Le t s 1 . . . . , s t be real numbers which are linearly independent over the

rationals. Then, given any set o f I real numbers Yl , Y2 . . . . . Yt and any e >0 there exis ts

an integer N such that IN si - yi[ < e (mod 1)for i = 1 . . . . . I.

(See Hardy-Wright, [15].)

Now suppose there exist two k x k matrices P and Q such that for all n

det(I - P") = det(I - Q"). (2.1)

Let aa . . . . , a k be the eigenvalues of P and b~ . . . . . b k the eigenvalues of Q. Assume as in w 1 that there are no roots of unity among the ai's and bs's. Assume also (an assumption justified by w 1) that the ai's and bi's are all of absolute value 1. Let b be a b i. Then by (2.1),

Hl(-l~a~) <2 k-~ for all n. (2.2)

Our main result will be:

Lemma. I f (2.2) holds then some a i is a power o f b t imes a root o f unity.

Before proving this lemma (the proof is a little complicated) let us explain why Kronecker's theorem is useful for analyzing estimates of the type (2.2). Suppose that not only is no a~ a power of b times a root of unity, but that the ai's and b together form a multiplicatively independent set, i.e. there are no non- trivial products of the form

tk b.,= 1. a 2.. ak

Then, by Kronecker's theorem, for every e > O, we can find an N such that

[a~+lL<~ and [ b n - l [ < ~ .

Comparing with 2.2 we get a contradiction

for e small; therefore the a~'s and b must be multiplicatively dependent.

T h e S p e c t r u m of Pos i t ive El l ipt ic O p e r a t o r s a n d Pe r iod ic B icharac te r i s t i c s 77

The proof of the lemma will be a refinement of this argument. Consider the numbers

1 t 1, 0 , - 2 n ] / ~ ] _ logan, 0 = 2 7 r ~ - 1 logb. (2.3)

The 0i's and 0 are, of course, only determined modulo 1. Assume that 1, 0, 0~, ..., 01 are linearly independent over the rationals, and the other 0~'s are linear combinations of them with rational coefficients. Then, for all i= 1, ..., k we can write

I .. oi= ri + Pi o+ q (2.4)

Multiplying (2.4) by n = Nq we get

l

nOi=--Pi NO+ 2 PijNOj (mod t). (2.5) j = l

Let # be a fixed integer (which in the argument below will be either 0 or 1). Choose real numbers x1,. . . , x l such that for each i= 1 . . . . . k for which the p~j's are not all zero

1

Zpijx~+ Pil~ ~0 mod 1. (2.6) ~=~ q

(Such a choice of x[s is clearly possible.) By Kronecker we can choose N = N(e) so that

NO --~- <g (mod 1 (2.7) q

and [NOj-xjl<e mod 1 j = l . . . . . l. (2.8)

We will now show that if the temma is not true (2.2) gets contradicted for this choice of N and n. First let # = 0 in (2.6) and (2.8). Then

n 0 - O (0 (mod 1) by (2.7) and

nOi -~ p o x i+O(e ) (mod I)

by (2.7) and (2.8). Suppose that for all i, some pij4=O. Then for all i, 1 - a ~ = 1 - e 2~r176 is bounded away from 0 by (2.6); however 1 - b " = 1 - e 2"v-~~ is of order O(e). This contradicts (2.2) for small e and therefore implies that for some i

q - - P, all pi~=0. Note that for such an i qO~=piO(modl) by (2.4) so a ~ - b , which is nearly the result we want. In fact the proof of our lemma would be finished if we could show q divides p~. To show this, let # = 1 in (2.7). Then for every i for which all p~j = O,

nOj=-~-+O(e) (mod 1) by (2.5), t /

and for the other 0i's !

nOi= Z PijXj + PiN +0(,~) i=1 q

mod 1.

78 J.J. Duistermaat and V. W. Guiltemin

W e st i l l h a v e

nO=NqO=-O(e) rood l

by (2.7). It is c lea r t h a t for s o m e i, for w h i c h all t he pu 's v a n i s h pjq m u s t be a n

i n t e g e r ; o t h e r w i s e as b e f o r e all t he t e r m s in t he n u m e r a t o r o f (2.2) will be b o u n d e d

a w a y f r o m 0 a n d t he d e n o m i n a t o r wilt be o f o r d e r ~. T h i s c o n c l u d e s t h e p r o o f

of t h e l e m m a .

T o c o n c l u d e t he p r o o f of t he t h e o r e m , go b a c k to (2.1) a n d c o n s i d e r t he e igen- v a l u e s o f p r a n d Qr i.e.

a~ . . . . , a [

a n d b r b r

l , " " 7 k "

For sufficiently large r every number in the second set, raised to some power, equals a number in the first set and visa versa. It is clear this can only happen if the two sets are identical.

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To appear in the Proc. of the C.N.R.S. Cotloque de Geom6trie Symplectique et Physique Math,5- matique. Aix-en- Provence, June 1974

25. Klingenberg, W., Takens, F.: Generic properties of geodesic flows. Math. Ann. 197, 323-334 (1972) 26. H6rmander, L.: Linear differential operators. Proc. Nice Congress, Vol. 1, pp. 121-133. Paris:

Gauthiers-Villars 1970

J. J. Duistermaat Mathematisch lnstituut Budapestlaan 6 De Uithof, Utrecht The Netherlands

V.W. Guiltemin Massachusetts I nstitute of Technology Mathematics Department Cambridge, Mass. 02139 USA

( Received October 20, 1974)

Note Added in Pro(f Colm de Verdi6re (S6minaire Goulaouic-Lions-Schwartz 1974-75) has shown that for P = ( - A +c) i the number rr m (6.16) can be identified with the Morse index for periodic geodesics by writing U(t, x, y) as an approximate Feynman integral.

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J.J.duistermaat, V.W.guillemin - The Spectrum of Positive Elliptic Operators and Periodic...

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