Annales de l’institut Fourier (AIF)

Vodev, Georgi

ANNALES DE L’INSTITUT FOURIER
GEORGI VODEV
On the distribution of scattering poles for
perturbations of the Laplacian
Annales de l’institut Fourier, tome 42, no 3 (1992), p. 625-635
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Ann. Inst. Fourier, Grenoble
42, 3 (1992), 625-635.
ON THE DISTRIBUTION OF SCATTERING POLES
FOR PERTURBATIONS OF THE LAPLACIAN
by Georgi VODEV(*)
1. Introduction.
In this note we study the distribution of the scattering poles
associated to second order differential operators of the form
G = c(x)-{- f; ^.(^(x)^.)+tfo,(x)^.4-a(x))
\ i,J=l J= l /
in R", n ^ 3, odd, where the coefficients are such that the following
conditions are fulfilled :
(i) The operator G admits a selfadjoint realization, which will be
again denoted by G, in the Hilbert space H = L2(Rn;c(x)dx)
with domain D(G};
(ii) There exists a constant po > 0 so that for any u e D(G) such
that u = 0 for \x\ < po we have ueH2^) and Gu = - AM,
A being the Laplacian in IR";
(iii) G is positively definite, i.e. (Gu,u)fj ^ 0, ^ueD(G).
In what follows || || will denote the norm in &(H,H), the space of
all linear bounded operators acting from H into H. It is easy to see
by (i) and (iii) that the resolvent R(z) = (G-z2)-1 e &(H,H) is well
defined and holomorphic in C+ = {z e C : Im z > 0}, and
(1.1) ||7?(z)|| < C(Imz)-2 for Imz > 0.
Choose a function ^eC^IR") such that / == 1 for \x\ ^ po + 1 and
set R^(z) = %R(z)^ for z e C+ . When
(iv) R^(zo) is a compact operator in &(H,H) for some Z o e C + ,
(*) Partially supported by Bulgarian Scientific Foundation under grant MM8/91.
Key words : Cutoff resolvent - Scattering poles.
A.M.S. classification : 35P25.
626 GEORGI VODEV
it is well known that the cutoff resolvent R^(z) admits a meromorphic
continuation from C+ to the entire complex plane C (see the analysis
in the next section). The poles of this continuation are known as
scattering poles or resonances and in our case they all are in C- ,
where C- = { z e C : Imz < 0}. Note that if (iv) holds for at least one
Zo, it holds for all Zo . Let {Kj} be the poles of ^(z), repeated according
to multiplicity, and set
N(r)= #{^. : |?i , |<r}.
When the operator G is elliptic, in [8] and [14] (see also [13]) it is
proved (without assuming (iii)) that
(1.2) N(r) ^ <y + C.
It also follows from the analysis in [8] and [14] that for hypoelliptic
operators, i.e. when we have the estimates
(1.3) 11/11^25^ C,(|[G/|L+||/V, V5^0,V/eZ)(G), G f e H 5 ,
where 0 < 8 < 1 and [| [ j , denotes the norm in the usual Sobolev space
H\ (again without assuming (iii)) the number of the poles satisfies the
bound
(1.4) N(r) < Cr^ + C.
Note that (1.3) implies (iv) at once. By (1.4) one actually concludes
that the less regular the operator G is, the worse bound for N(r) one
has. In this work we show that outside a conic neighbourhood of the
real axis the number of the scattering poles satisfies a much better
estimate than (1.4) no matter how regular the operator G is. It actually
has a bound of the type (1.2). To be more precise, given any
£, 0 < c « 1, set Ae == {z e C :8 ̂  arg z ^ Tc-8} and
N(s,r) = #{^.:|^. ^r .-^.eAJ.
Our main result is the following :
THEOREM 1. — Assume (i)-(iv) fulfilled. Then for any 8, 0 < 8 « 1,
there exists a constant Cg > 0 so that
(1.5) ^(8,r)^(y+ C,.
The estimate (1.5) shows that to study the counting function N(r)
modulo terms 0^) for positively definite selfadjoint hypoelliptic
operators it suffices to study the number of the scattering poles in a
conic s-neighbourhood of the real axis for any small 8 > 0.
ON THE DISTRIBUTION OF SCATTERING POLES 627
The idea for the proofs of polynomial bounds of the scattering
poles originates from Melrose [4] (see also [2], [5], [II], [12], [13], [14],
[17]). One first needs to find an entire family of compact operators,
K(z), so that (l-K(z))R^(z) is an entire operator-valued function and
1 - K(z) is invertible for at least one z e C. Thus one concludes that
the poles of ^(z), with multiplicity, are among the poles of (l-K(z))~1
and hence among the zeros of an entire function h(z) = det (l-K(zY),
where p ^ 1 is an integer taken so that K(zY is trace class. Thus the
problem is reduced to obtaining suitable estimates for \h(z)\.
To prove (1.5) we need to find a family K{z) as above so that
{\-K(z))~1 can be expressed in terms of R(z) for z e C + (see (2.5)),
and K(z) - K(-z) is trace class for any z e C. This enables us to
characterize the poles of R^(z) in C- , with multiplicity, as zeros of a
function h(z), defined and holomorphic in C- , such that for any y > 0
there exists a constant Cy > 0 so that
(1.6) \h(- z)| ^ C,exp(C,|z|") for I m z ^ y .
Then, we derive (1.5) from (1.6) and a classical result due to Carleman
(see Lemma 2).
Acknowledgments. The author would like to thank Vesselin Petkov
for his support and encouragement.
2. Representation of the cutoff resolvent.
Denote by Go the selfadjoint realization of — A in the Hilbert space
Ho = L2^) and let Ro(z) denote the outgoing resolvent of - A - z2 ,
z e C . Then Ro(z) = (Go-z2)-1 e Q(Ho,Ho) for z e C + and as is well-
known the kernel of Ro(z) is given in terms of HankeRs functions by
(2.1) Ro(z)(x,y) = W^zWx-y^-^H^^^x-y^.
It is easy to see that x^o(^)x e &(Ho,Ho) for all z e C and it forms an
entire family of compact pseudodifferential operators of order - 2.
Using this together with the assumption (iv) we shall build the
meromorphic continuation of the cutoff resolvent of G. Set Q = G - Go
and fix a Z o e C + . Clearly, for all z e C + we have
(2.2) R(z) = Ro(z) + R(z)QRo(z)
628 GEORGI VODEV
and
(2.3) R(z) = R(z,) + (z2-z^R(z)R{z,).
Combining (2.2) and (2.3) yields
R(z)(l-(z2-zl)QR,(z)R(z,)) = R(z,) + (z2-^)^)^)
for z e C + . Multiplying the both sides of this identity by 7, since
Q = X6» we get
(2.4) ^(z)(l - ̂ (z)) = ^,(zo) + K,(z) for z e C + ,
where
7C(z) == (z^z^e^zW^x
^(z)^2-^^)^)^
Moreover, since R(z) is well defined in C+ , it is easy to see by (2.4)
that 1 - K(z) is invertible in &(H,H) for all z e C + and
(2.5) (l-^(z))-1 = 1 + (z2-z§)e^o(z)(^o(Zo)+^o^o)e^(^)X
for z e C + . Now, since Ro(z) and 7?(z) are holomorphic in C+ with
values in &(H,H) and since QR^z) = QR^z^^-^-^-zDR^z)) for
z e C + , we deduce from (2.5) that (l—A^z))""1 is holomorphic in C+
with values in &(H,H). Moreover, by (1.1), which clearly holds with
R(z) replaced by Ro(z) as well, for any y > 0 there exists a constant
Cy > 0 so that
(2.6) ||(1 -^(z))-1!! ^ C,(l+|z|)4 for Imz ^ y.
Now let us see that the operator-valued functions K(z) and A^(z),
defined in C+ , extend analytically to the entire C with values in the
compact operators in &(H,H). We shall consider K(z) only, since K^{z)
is treated similarly. Using that R(zo) = Ro(zo) 4- Ro(zo)QR(zo) it is easy
to see that
(2.7) K{Z) = (z^z^e^z^o^xa+e^o))
for z e C + . Choose functions ^i, ^2^ C^R") such that Xi = 1 on
supp Q, X2 = 1 on supp 5Ci and ^ = 1 on supp ^2 • After a standard
computation (2.7) takes the form
(2.8) K(z) = (z2-zS)0^o(z)^o(zo)x^ + (z^z^o^Z^W^)
ON THE DISTRIBUTION OF SCATTERING POLES 629
for z e C + , where
^2 = 1 + [X2^o]^o(^)[Xl,^o]^o(^o)^^(zo).
K, = xi^o(^o)e + x2^o)Dci^o]^o)e.
Here [,] denotes the comutator. Clearly, we have A^, K^E&(H,H).
Further on, by a similar computation, for z e C + , one obtains
(2.9) (z^zDQR^R^z^
= (^+(^2-^2o)^5)x^)x - ̂ X^o)X
and
(2.10) (2^)X = (^4+(^-^5)X^)X + ^5,
where
K, = e^o)[Oo,Xi]^o)[^o,X2L
^5 = <2^o)Xi + e^o(^o)[^o,Xi]^o(^)X2.
Clearly, A:4, A^e 2(H,ff). Thus, by (2.8)-(2.10) we deduce
(2.11) 7C(z) = K,(z)Mz)^ + K,(z)Mz)^ + ^(z)
for z e C + , where
^)=^4+(^-^5,
^^^(Z^Z^^Z),
K,= K,R^z,),
W = - K^R^z^K, + (z^g^A^o).
Clearly, these four operators are analytic &{H,H)- valued functions. Now,
since ^Ro(z)^ forms an entire family of compact operators and by (iv)
so does A^(z), by (2.11) we can extend K(z) analytically to the entire
C. Then, since K(zo) = 0, by Fredholm theorem, (l-A:(z))~1 is a
meromorphic &(H,H)- valued function on C. Thus, by (2.4) we obtain
the desired meromorphic continuation of R^(z). Moreover, clearly the
poles of this continuation coincide, with multiplicity, with the poles of
(l—A'(z))~1 . Thus, since 1 — K(z) is invertible for z e C + , we have
that all the poles are in C-. Now, for z e C + , we have
(2.12) 1 - K{-z) = (l-^(z))(l-T(z)),
where
F(z) = (\-K(z)Y\K(-z)-K(z)).
630 GEORGIVODEV
By (2.11) we have
(2.13) T(z) = 7\(z)x^(zW + T,(zW(z)^K,
where
S(z) = ^o(-z) - ^o(z)
F^z)^!-^))-1^)
TO^l-^z))-1^).
By (2.6), for any y > 0, we get
(2.14) ||r,(z)|| ^ C,(l+|z|)8 for I m z ^ y , 7 = 1 , 2 .
On the other hand, by (2.1) and the well known properties of the
Hankel functions, we have the following formula for the kernel of
S(z):
(2.15) S(z)(x,y) = (i^zWx-y^-^J^^x-y})
--= O'/^Tir^z"-2 exp(fz<x-^,w»^w, x , ^e [R" ,
JS"-!
where S""1 denotes the unit sphere in IR". Denote by <S(z) the operator
with kernel given by the integral above. Now it is easy to see by (2.15)
that ^S(z)^ forms an entire family of trace class operators in 2(H,H).
Hence, by (2.13), T(z) is holomorphic in C+ with values in the trace
class operators in Q(H,H). Now, by (2.12) it is easy to see that
1 — T(z) is invertible in ^(H,H) for those z e C + for which so is
1 — K ( — z ) , and then we have
(2.16) (l-K(-z))-1 = (\-T(z))-\\-K(z}Y\
Since (\—K(z))~1 is holomorphic in C + , by (2.16) we conclude that
the poles of (l—K(—z))~1 lying in C + , with multiplicity, coincide with
the poles of ( l—T(z))~ 1 . Introduce the function
h(z)=det(l-r(z)) ,
which is well defined and holomorphic in C+ . Now, by the above
analysis we conclude that if ^, ^ e C _ , is a scattering pole, then — ^
is a zero of h(z) with the corresponding multiplicity. Thus we can
characterize the scattering poles as zeros of h(—z), z e C _ . Notice that
the fact that T(z) is trace class does not depend on whether (iv) is
ON THE DISTRIBUTION OF SCATTERING POLES 631
fulfilled or not. Hence the function h(z) is always defined, under the
conditions (i)-(iii), and holomorphic in C+ . Now we are going to study
the distribution of the zeros of h(z) without assuming (iv). Note that
in general the zeros of h(z) may accumulate at points on the real axis.
Let {Zj} c: C+ be the zeros of h(z), repeated according to multiplicity,
and given 0 < £, 6 « 1, r » 1, set
7V(£ ,8 , r )=^{z , :8^ |z , ^r,z,eA,}.
We have the following :
THEOREM 2. — Assume (i')-(nf) fulfilled. Then, for any £, 8, r as
above there exists a constant Cg § > 0, independent of r , so that
(2.17) ^(c,8,r) ^ C^ for r ^ l .
When (iv) is fulfilled the number of the scattering poles in
{ z e C : | z ^8} is finite for any 8 > 0, and hence (1.5) is obtained as
an immediate consequence of (2.17).
3. Proof of Theorem 2.
We start with the following :
LEMMA 1. — Under the assumptions (i)-(iii), for any y > 0 there
exists a constant Cy > 0 so that
(3.1) \h(z)\ ^ C,exp(C,[zr) for I m z ^ y .
Proof. — The estimate (3.1) is established in the same way as in
[13] (see also [17]). Here we shall sketch the proof. Given a compact
operator A, [ij(A) will denote the characteristic values of A, i.e. the
eigenvalues of (A* A)112, repeated according to multiplicity and ordered
to form a nonincreasing sequence. First, recall some well known
properties of [ij(A):
(3.2) H^4)^M||, V/-,
(3.3) {^(AB),^(BA)}^^(AW\, V/-,
(3.4) H/S^)^ Z^(^), V7,
\ i= i / 1=1
632 GEORGI VODEV
where j^ - [ j / k ] , [a] denotes the integer part of a. By (2.13)-(2.15) and
(3.2)-(3.4) it is easy to see that
(3.5) ^(T(z)) < C,(l+ IzD^^ec^z)^ for Im z ^ y .
On the other hand, clearly we have
(3.6) IIZ^)xll ^ Cexp(C|z|), V z e C .
Combining (3.5) and (3.6) yields
(3.7) H,( T(z)) ^ C, exp (C| z |), V/, for Im z ^ y.
Further on, we shall show that there exists a constant C > 0 so that
(3.8) ^i,(^(z)x)^ Ce-^j-^-^ if ; ̂  C|z|71-1, V z e C .
This is actually proved in [13], but for the sake of completeness we
shall repeat the key points. The key observation is the representation
(3.9) 5(z)=^(z)^(z),
where 5\(z) is the operator with kernel 5'i(z)(x,w) = exp O'z<x,w»,
5'2(z) is the operator with kernel S^z)(\v,x) = exp (-fz<x,w», x e HT,
weS""1 . Then, using (3.3) and (3.9) we have
(3.10) ^(x^OOx) ^ IIX^(^lll l l(l-A,)m^^)xll2^•((l-AJ-m), V/,
for any integer m > 1, where Ay, denotes the Laplace-Beltrami operator
on §"-1, || ||i and \\\\^ denote the norms in ^(L^S"-1),!/^")) and
^(L^IR^.L^S""1)), respectively. On the other hand, we have with a
constant C > 0,
(3.11) H/O-AJ-")^ CT2^
where I = dim S"~1 = n - 1, and
(3.12) I IX^ iOOII i < Cexp(C|z|),
(3.13) IKl-A.r^^XlIz < Csup ^(xXl-AJ^-^^)!
X,M;
< C^-^dz 2'"+(2m)2'")eclzl.
Thus, by (3.10)-(3.13),
(3.14) ^(ei2'^)^) < C2m+l(|^|2m+(2m)2'")eclzl7-2'"/',
ON THE DISTRIBUTION OF SCATTERING POLES 633
with a new constant C > 0 . Now, (3.8) is an easy consequence of
(3.14) (see [13], [17]).
Thus, by (3.5) and (3.8), we have
(3.15) H,(T(Z)) ^ O-^-^ if j ^ C l z j " - 1 , for I m z ^ y ,
with new constants Cy, 00. Now, it is a straightforward calculation
that (3.7) and (3.15) together with WeyFs convexity estimate imply (3.1)
(see [13], [17]. The proof of Lemma 1 is completed.
To derive (2.17) from (3.1), instead of Jensen's inequality, we shall
use the following classical result (see [9], Section 3, Carleman's theorem).
LEMMAI. - Givenr > r^ > 0,s^0 = {zeC-.r^ |z| ^ r . I m z ^ O } .
Let f(z) be a function holomorphic in 0 and let
y-i exp (Kpi), rz exp (Kpg), . . . , r^ exp (fq^) be the zeros off(z) in 0 repeated
according to multiplicity. Then,
k - - - r"Z (rj ' - r ^ r 2) sin (p, = (nr)~1 log \f(re^)\ sin (p Ap
^i Jo
+ (27r)-1 ( (^-r-^logl^O.A-OIA
Jro
- (Tcro)-1 logl/^o^lsinq)^.
Jo
Note that each term in the sum above is ^ 0. Fix e, §, 0 < 8,
8 « 1, and let r > 6. Let
Zi = FI exp O'cpi), Za = y-2 exp (((pa), . . . , z^ = r^ exp (up^)
be the zeros of h(z), repeated according to multiplicity, satisfying the
conditions : 3 ^ r,^ r / 2 ; e ^ % ^ TI - s. Clearly,
(3.16) N(^r/2) < k + 7V(8,8,3).
Set /(z) = A(z+ry) where y = sine. Clearly, /(z) is holomorphic in
C+ and by (3.1) we have
(3.17) |/(z)| ^ C,exp(CJz|71), V z e C ^ .
Moreover, z; = z, - 17, J = 1, . . . , fe , are zeros of /(z). Set r; = |z;|
and (p; = arg z;. It is easy to check that 2 ^ r; ^ 2r/3 and sin (p; ^
634 GEORGI VODEV
2~1 sin £, j= 1, . . . , fe . Hence
(3.18) (rf,-l-rf,r-2)sm^^(5y/12)r-\ 7 = l , . . . , fe .
Now, applying Lemma 2 to /(z) with y-o = 2 and using (3.17) and
(3.18), we get
k
(Sy/ll^^k^ Z^-1-^-2)^;.
j'-i
^ (w)"1 log \f(rei<y)\ sin (p d(p
Jo
+ (27T)-1 r^^-r-^loglAO^-OIA + C,
j2
^ y^+l) + C[ [ f^+l^ + Ce
j2
^ C^r-^^+l).
Hence
(3.19) ^(n^C^+l).
Now (2.17) follows from (3.16) and (3.19) at once.
BIBLIOGRAPHY
[1] C. BARDOS, J. GUILOT, J. RALSTON, La relation de Poisson pour 1'equation
des ondes dans un ouvert non borne. Application a la theorie de la
diffusion, Commun. Partial Differ. Equations, 7 (1982), 905-958.
[2] A. INTISSAR, A polynomial bound on the number of scattering poles for a
potential in even dimensional space IR", Commun. Partial Differ. Equations,
11 (1986), 367-396.
[3] P. D. LAX, R. S. PHILLIPS, Scattering Theory, Academic Press, 1967.
[4] R. B. MELROSE, Polynomial bounds on the number of scattering poles, J.
Func. Anal., 53 (1983), 287-303.
[5] R. B. MELROSE, Polynomial bounds on the distribution of poles in scattering
by an obstacle, Journees « Equations aux derivees partielles », Saint-Jean-
de-Monts (1984).
[6] R. B. MELROSE, Weyl asymptotics for the phase in obstacle scattering,
Commun. Partial Differ. Equations, 13 (1988), 1431-1439.
[7] J. SJOSTRAND, Geometric bounds on the number of resonances for semiclassical
problems. Duke Math. J., 60 (1990), 1-57.
ON THE DISTRIBUTION OF SCATTERING POLES 635
[8] J. SJOSTRAND, M. ZWORSKI, Complex scaling and the distribution of scattering
poles, J. Amer. Math. Soc., (1991), 729-769.
[9] E. C. TITCHMARSH, The Theory of Functions, Oxford University Press, 1968.
[10] B. VAINBERG, Asymptotic Methods in Equations of Mathematical Physics,
Gordon and Breach Sci. Publ., 1988.
[11] G. VODEV, Polynomial bounds on the number of scattering poles for
symmetric systems, Ann. Inst. Henri Poincare (Physique theorique), 54
(1991), 199-208.
[12] G. VODEV, Polynomial bounds on the number of scattering poles for metric
perturbations of the Laplacian in R", n ^ 3, odd, Osaka J. Math., 28
(1991), 441-449.
[13] G. VODEV, Sharp polynomial bounds on the number of scattering poles
for metric perturbation of the Laplacian in IR", Math. Ann., 291 (1991),
39-49.
[14] G. VODEV, Sharp bounds on the number of scattering poles for perturbations
of the Lapacian, Commun. Math. Phys., 145 (1992), to appear.
[15] M. ZWORSKI, Distribution of poles for scattering in the real line, J. Func.
Anal., 73 (1987), 227-296.
[16] M. ZWORSKI, Sharp polynomial bounds on the number of scattering poles
of radial potentials, J. Func. Anal., 82 (1989), 370-403.
[17] M. ZWORSKI, Sharp polynomial bounds on the number of scattering poles,
Duke Math. J., 59 (1989), 311-323.
Manuscrit recu Ie 24 juin 1991,
revise Ie 8 octobre 1991.
Georgi VODEV,
Institute of Mathematics
Bulgarian Academy of Sciences
Acad. G. Bonchev str. bl. 8
1113 Sofia (Bulgarie).


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