An application of scattering theory to the spectrum of the Laplace-Beltrami operator

13 Pages·0.15 MB·English

by  Francesca Antoci

AN APPLICATION OF SCATTERING THEORY
TO THE SPECTRUM
3 OF THE LAPLACE-BELTRAMI OPERATOR
0
0
2
FRANCESCA ANTOCI

n
a
J Abstract. Applying a theorem due to Belopol’ski and Birman, we
0 show that the Laplace-Beltrami operator on 1-forms on Rn endowed
1 withanasymptoticallyEuclideanmetrichasabsolutelycontinuousspec-
trum equalto [0,+∞).

]
P
S
.
h
t
a
m
1. Introduction
[
The relationships between the geometric properties of a complete non-
1
compact Riemannian manifold and the spectrum of the Laplace-Beltrami
v
9 operator have been intensively investigated by many authors.
9 Among them, H. Donnelly since the late seventies studied the spectra of
0
the Laplacian and of the Laplace-Beltrami operators on particular mani-
1
0 folds, such as the hyperbolic space ([1]), manifolds with negative sectional
3 curvature ([3]), asymptotically euclidean manifolds ([2]). However, to our
0
knowledge, the case of the Laplace-Beltrami operator acting on p-forms on
/
h asymptotically euclidean manifolds has been left aside up to now.
t
a The purpose of this paper is to contribute to the investigation of this
m case. We study the absolutely continuous spectrum of the Laplace-Beltrami
: operator on Rn endowed with an asymptotically euclidean metric, that is
v
i with a Riemannian metric satisfying conditions (2.5), (2.6) and (2.7). The
X
tool employed is classical scattering theory in the wave operators approach.
r
In this case, the problem is reduced to a problem of scattering for vector-
a
valued operators.
Aninherentrestrictionoftheproof,however, thatmakesitdifficultanex-
tension to more general cases, is the use of the Fourier transform, which has
a crucial role in our considerations, particularly in connection with Lemma
3.2. The lack of this tool in the more general case of manifolds with an
asymptotically controlled Riemannian metric is a major obstacle to the ex-
tension of the theorem.
1991 Mathematics Subject Classification. 58G25, 35P25.
Key words and phrases. Scattering theory;differential forms.
1
2 FRANCESCAANTOCI
2. Preliminaries
Let(Rn,e)betheeuclideann-dimensionalspace,and(Rn,g)bethesame
space, endowed with a complete Riemannian metric g.
We will denote by Λ1(Rn) the vector space of all smooth, compactly
c
supported 1-forms on Rn, and by L2(Rn,e) the completion of Λ1(Rn) with
1 c
respect to the norm
(2.1) ω 2 = < ω,ω > dx,
k kL21(Rn,e) ZRn e
where dx denotes the Euclidean volume element and
< ω(x),ω(x) > = ω2(x)
e i
Xi
is the fiber norm for 1-forms induced by the Euclidean metric.
L2(Rn,e) is the Hilbert space direct sum of n copies of L2(Rn). The
1
Laplace-Beltrami operator ∆ on 1-forms ω Λ1(Rn) acts componentwise
e ∈ c
as:
∂2ω
k
(∆ ω) = .
e k − ∂x2
Xj j
It is well-known that ∆ is essentially selfadjoint on Λ1(Rn), and its closure
e c
H has purely absolutely continuous spectrum equal to
0
σ(H )= [0,+ ),
0
∞
with constant multiplicity.
We will denote by h [ω] the quadratic form associated to ∆ on Λ1(Rn):
0 e c
n ∂ω 2
(2.2) h [ω] = < ∆ ω,ω > dx = i dx.
0 ZRn e e ZRniX,j=1(cid:18)∂xj(cid:19)
L2(Rn,g) will stand for the completion of Λ1(Rn) with respect to the
1 c
norm:
(2.3) kωk2L21(Rn,g) = ZRn <ω,ω >g √gdx,
where√gdx,asusual,denotesthevolumeelementinducedbytheRiemann-
ian metric g and
< ω(x),ω(x) > = gij(x)ω (x)ω (x)
g i j
is the fibernorm for 1-forms inducedby g. (Here, as everywhere throughout
the paper, the repeated indices convention is adopted.)
The action of the Laplace-Beltrami operator ∆ = dδ + δd on 1-forms
g
ω Λ1(Rn) is given in local coordinates by the Weitzenbo¨ck formula:
∈ c
(∆ ω) = (gij ω) +Riω ,
g k − ∇i∇j k k i
where is the covariant derivative with respect to the connection induced
i
∇
by the the metric g, and Ri is the Ricci tensor.
k
ON THE SPECTRUM OF THE LAPLACE-BELTRAMI OPERATOR 3
Since the Riemannian metric g is complete, ∆ is essentially selfadjoint
g
on Λ1(Rn). We will denote its closure by H .
c 1
Moreover, we will denote by h [ω] the quadratic form associated to ∆
1 g
on Λ1(Rn)
c
(2.4) h [ω] = < ∆ ω,ω > √gdx =
1 g g
ZRn
= ω 2√gdx+ < Rω,ω > √gdx,
ZRn|∇ |g ZRn g
where
ω 2 = gijgαβ ω ω
|∇ |g ∇i α∇j β
and
< Rω,ω > = gαβRiω ω .
g α i β
In thenextsection we willshow how it is possible, undersuitable hypoth-
esisontheasymptoticbehaviourofg,togetinformationaboutthespectrum
of H from the knowledge of the spectrum of H , proving the following
1 0
Theorem 2.1. Let Rn be endowed with a Riemannian metric g such that
∂gil(x) is bounded and, for x >>0, there exists C >0 such that
(cid:12)∂xj (cid:12) | |
(cid:12) (1) (cid:12)for every i,j,
(cid:12) (cid:12)
C
(2.5) gij(x) δij <
| − | xk
| |
for some k > n;
(2) for every i,j,k,l
∂g C
(2.6) il <
(cid:12)∂x (cid:12) xk
(cid:12) j(cid:12) | |
(cid:12) (cid:12)
(cid:12) (cid:12)
∂2g C
(2.7) il <
(cid:12)∂x ∂x (cid:12) xk
(cid:12) j k(cid:12) | |
(cid:12) (cid:12)
for some k > n. (cid:12) (cid:12)
Then the Laplace-Beltrami operator ∆ acting on 1-forms has absolutely
g
continuous spectrum equal to [0,+ ):
∞
[0,+ ) = σ (H )= σ(H ).
ac 1 1
∞
In particular, it has no discrete spectrum. (There might be singularly con-
tinuous spectrum or embedded eigenvalues.)
The main tool for the proof is Belopol’ski-Birman theorem (see [5], [6]),
which provides a sufficient condition so that two selfadjoint operators have
the same absolutely continuous spectrum.
We recall it briefly:
4 FRANCESCAANTOCI
Theorem 2.2. Let H , H be selfadjoint operators acting respectively on
0 1
Hilbert spaces , , and let E (H ), E (H ), for Ω R, be the associ-
0 1 Ω 0 Ω 1
H H ⊂
ated spectral measures.
If J ( , ) satisfies the conditions:
0 1
∈ L H H
(1) J has a bounded two-sided inverse;
(2) for every bounded interval I R,
⊂
(2.8) E (H )(H J JH )E (H ) ( , ),
I 1 1 0 I 0 1 0 1
− ∈ I H H
where ( , ) = A ( , ) (A∗A)1/2 ( ) and
1 0 1 0 1 1 0
I H H { ∈ L H H | ∈ I H }
( ) denotes, as usual, the set of trace-class operators on ;
1 0 0
(3) fIorHevery bounded interval I R, (J∗J I)E (H ) is compaHct;
I 0
⊆ −
(4) JQ(H ) = Q(H ), where Q(H ) is the form domain of the operator
0 1 i
H , for i= 0,1,
i
then the wave operators W±(H ,H ;J) exist, are complete, and are partial
1 0
isometries with initial space P (H ) and final space P (H ), where P (H )
ac 0 ac 1 ac i
denotes, as usual, the absolutely continuous space of H , for i= 0,1.
i
As a consequence, the absolutely continuous spectra of H and H do
0 1
coincide.
Remark 2.3. We recall (see [4]) that if H is a densely defined, essentially
selfadjoint, positive operator on a Hilbert space and h is the associated
H
quadratic form, the form domain Q(H) of the selfadjoint operator H is the
domain of the closure h˜ of the form h, that is to say: Q(H) is the set of
those u such that there exists a sequence u D(H) converging to u
n
∈ H { }⊂
in such that
H
h[u u ] 0
n m
− −→
as n,m + .
→ ∞
3. Proof of Theorem 2.1
We will prove that, for a suitable J : L2(Rn,e) L2(Rn,g), H = ∆ ,
1 → 1 1 g
H = ∆ and J satisfy the conditions of Theorem 2.2, for = L2(Rn,e)
0 e 0
and = L2(Rn,g). H
1
H
We begin with the following
Lemma 3.1. Let g be as in Theorem 2.1. Then there exist C,C > 0,
1
D,D > 0 such that
1
(1) for every x Rn
∈
(3.1) C g(x) C ;
1
≤ ≤
p
(2) for every x Rn, v in the cotangent space at x, T∗(Rn)
∈ x
(3.2) D v2 gij(x)v v D v2.
i ≤ i j ≤ 1 i
Xi Xi
ON THE SPECTRUM OF THE LAPLACE-BELTRAMI OPERATOR 5
Proof. (3.1) follows immediately observing that, for every x, g(x) is
strictly positive and g(x) 1 as x + . p
→ | | → ∞
As for (3.2), sincepthe matrix gij(x), which expresses the Riemannian
metric g in contravariant form, is a continuous function of x and is positive,
its eigenvalues λ (x),...,λ (x) depend continuously on x and are strictly
1 n
positive. Hence, the functions f and h defined by
f(x) := infλ (x)
i
i
and
h(x) := supλ (x),
i
i
are continuous and strictly positive. Moreover, since the metric g is asymp-
totically euclidean, f(x) 1andh(x) 1as x + . Asaconsequence,
there exist D,D > 0 suc→h that, for ev→ery x |R|n→, ∞
1
∈
D f(x) h(x) D ,
1
≤ ≤ ≤
which yields (3.2). (cid:3)
Lemma 3.1 implies that there is a natural identification between L2(Rn,g)
1
and L2(Rn,e), and, moreover, (2.1) and (2.3) are equivalent norms. As
1
a consequence, the identity map on Λ1(Rn) extends to a bounded linear
c
operator
J :L2(Rn,e) L2(Rn,g),
1 −→ 1
with boundedtwo-sided inverse, and condition 1 of Theorem 2.2 is satisfied.
In order to prove (2.8), we need two Lemmas:
Lemma 3.2. Let A:ξ A be a n n-matrix-valued function on Rn, and
ξ
7→ ×
let be the linear operator
A
: D( ) L2(Rn) ... L2(Rn) L2(Rn) ... L2(Rn)
A A ⊂ ⊕ ⊕ −→ ⊕ ⊕
of the form
f(x)A( i ),
x
− ∇
where f(x) is a function on Rn and A( i ) is the operator
x
− ∇
A( i ) = Aˆ −1,
x ξ
− ∇ F ◦ ◦F
being the Fourier transform and Aˆ the multiplication operator
ξ
F
v Aˆ v
ξ
7−→
(Aˆ v)(ξ) = A(ξ)v(ξ).
ξ
Let L2(Rn) be the space of functions h such that
δ
khk2δ = k(1+|x|2)δ2h(x)kL2 < ∞.
If, for some δ > n, f(x) L2(Rn) and, for every pair of indices (α,β),
2 ∈ δ
Aβ(ξ) L2(Rn), then is a trace-class operator.
α ∈ δ A
6 FRANCESCAANTOCI
Proof of Lemma 3.2. It suffices to show that, for every fixed (α,β), the
operator α
Aβ
α : D( α) L2(Rn) ... L2(Rn) L2(Rn) ... L2(Rn)
Aβ Aβ ⊂ ⊕ ⊕ −→ ⊕ ⊕
ω = (ω1,...,ωn) 7−→ 0,...,0,f(x)Aβα(−i∇x)ωβ,0,...0
 α 
is trace-class. But this latter op|erator coinc{idzes with the}composition
I f(x)Aβ ( i ) P ,
α◦ α − ∇x ◦ β
(cid:16) (cid:17)
where P is the projection
β
P : L2(Rn) ... L2(Rn) L2(Rn)
β
⊕ ⊕ −→
ω = (ω ,...,ω ,...,ω ) ω ,
1 β n β
7−→
I is the immersion
α
I : L2(Rn) L2(Rn) ... L2(Rn)
α
−→ ⊕ ⊕
ω (0,...,0,ω,...,0),
7−→
α
and Aα( i ) is the operator | {z }
β − ∇x
D(Aα( i )) L2(Rn) L2(Rn)
β − ∇x ⊂ −→
Aα( i ) = (Aˆ )α −1,
β − ∇x F ◦ ξ β ◦F
where is the Fourier transform and (Aˆ )α is the multiplication operator
F ξ β
associated to the scalar function (A )α.
ξ β
TheconclusionfollowsfromthefactthatP andI areboundedoperators
β α
and (see [5], Theorem XI.21) any operator
L2(Rn) L2(Rn)
−→
of the form f(x)h( i ) is trace-class if f(x) and h(ξ) belong to L2(Rn).
− ∇x δ (cid:3)
Lemma 3.3. If f : Rn R is continuous and such that for some k > n
−→
C
(3.3) f(x) <
| | xk
| |
when x >>0, then f L2(Rn) for some δ > n.
| | ∈ δ 2
Proof. Choosing ǫ > 0 such that k > n+ǫ, then
C
f(x) <
| | xk
| |
ON THE SPECTRUM OF THE LAPLACE-BELTRAMI OPERATOR 7
for x >>0; as a consequence, a straighforward computation in polar coor-
| |
dinates shows that, for δ = n +ǫ,
2
f(x)2(1+ x2)δdx < + .
ZRn| | | | ∞
(cid:3)
Now, to prove (2.8), it suffices to see that for every bounded interval
I R,
⊂
(H H )E (H ) (L2(Rn) ... L2(Rn)).
1 0 I 0 1
− ∈ I ⊕ ⊕
Let Γα betheChristoffel symbols of the Riemannian connection inducedby
ik
g; then the difference H H is given by
1 0
−
δ2ω δω δω
(3.4) ((H H )ω) = ( gij +δij) k +gijΓα α +gijΓα k
1− 0 k − δxiδxj jk δxi ijδxα
δω δΓα
+gijΓα α +gij jkω gijΓαΓβ ω gijΓαΓβ ω +Riω .
ikδxj δxi α− ij αk β − ik jα β k i
A direct computation shows that conditions (2.5), (2.6), (2.7), and hy-
∂Γα
pothesis 3 in Theorem 2.1 imply that gijΓα , gijΓα , gijΓα , gij jk ,
| jk| | ij| | ik| (cid:12) ∂xi (cid:12)
gijΓαΓβ , gijΓαΓβ , Ri are all bounded from above by C (cid:12)for som(cid:12)e
| ij αk| | ik αj| | k| |x|k (cid:12) (cid:12)
constant C > 0 and some k > n.
(H H )(E (H )) is a sum of operators of type f(x)A( i ), with
1 0 I 0 x
f(x) −L2(Rn) in view of Lemma 3.3, and A(ξ) smooth an−d c∇ompactly
∈ δ
supported.
Thus, thanks to Lemma 3.2, (H H )(E (H )) is trace-class and condi-
1 0 I 0
−
tion 2 is fulfilled.
As for condition 3, first of all we observe that the adjoint of J
J∗ :L2(Rn,g) L2(Rn,e)
1 −→ 1
satisfies the equation
gijω φ √gdx = δijω (J∗φ) dx,
i j i j
ZRn ZRn
and therefore
(J∗φ) = δ gijφ √g.
k ik j
As a consequence, in local coordinates
((J∗J I)φ) = (√ggjk δjk)φ ;
k j
− −
now,
√ggjk δjk √g gjk δjk + (√g 1) δij.
(cid:12) − (cid:12)≤ | |(cid:12) − (cid:12) | − |
By (2.5), ther(cid:12)e exists C >(cid:12)0 such th(cid:12)at (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
C
gjk δjk
(cid:12) − (cid:12)≤ xk
(cid:12) (cid:12) | |
(cid:12) (cid:12)
8 FRANCESCAANTOCI
for some k > n for x >>0; moreover,
| |
1 1 K
(√g 1) = 1 g +o
| − | 2| − | (cid:18) xk(cid:19) ≤ xk
| | | |
for some K > 0 and some k > n as x + .
Thus,√ggjk−δjk belongstoL2δ(Rn)|. |H→ence∞(J∗J−I)EI(H0)isanoperator
of type f(x)A( i ), with f(x) in L2(R2) and A(ξ) smooth and compactly
− ∇x δ
supported; by Lemma 3.2, it is trace-class, and therefore it is compact.
As for condition 4, thanks to Remark 2.3, Q(H ) and Q(H ) can be charac-
0 1
terized as follows:
Lemma 3.4. Q(H ) is the set of those ω L2(Rn,e) for which there exists
0 ∈ 1
a sequence ω(n) Λ1(Rn) such that
{ } ⊂ c
(3.5) ω(n) ω in L2(Rn,e)
−→ 1
and
(3.6) h [ω(n) ω(m)] 0
0
− −→
as n,m + .
Analo→gousl∞y, Q(H ) is the set of those ω L2(Rn,g) such that there
1 ∈ 1
exists ψ(n) Λ1(Rn) for which
{ } ⊂ c
(3.7) ψ(n) ω inL2(Rn,g)
−→
and
(3.8) h [ψ(n) ψ(m)] 0
1
− −→
as n,m + .
→ ∞
We prove now that
(3.9) Q(H ) Q(H ).
0 1
⊆
For ω Q(H ), there exists a sequence ω(n) Λ1(Rn) satisfying (3.5)
∈ 0 { } ⊂ c
and (3.6). Due to the equivalence of the norms (2.1) and (2.3),
ω(n) ω in L2(Rn,g);
−→ 1
hence, in order to see that ω Q(H ) it suffices to prove that
1
∈
h [ω(n) ω(m)] 0
1
− −→
as m,n + .
→ ∞
To establish this fact, we consider first the curvature part of h [ω(n)
1
−
ω(m)],
(3.10) < R(ω(n) ω(m)),(ω(n) ω(m)) > √gdx.
g
ZRn − −
The following Lemma holds:
ON THE SPECTRUM OF THE LAPLACE-BELTRAMI OPERATOR 9
Lemma 3.5. There exists C > 0 such that
(3.11) |ZRn < Rω,ω >g √gdx|≤ Ckωk2L21(Rn,e)
for every ω L2(Rn,e).
∈ 1
Proof. Consider for every x Rn the quadratic form on T∗(Rn)
∈ x
ω gαβ(x)Ri(x)ω ω = Riβ(x)ω ω .
7−→ α i β i β
Sincethe matrix Riβ(x) dependscontinuously on x, its eigenvalues λ (x),...,
1
λ (x) are continuous functions of x. Hence the function
n
f(x):= supλ (x),
i
i
is continuous. Moreover, since the metric g is asymptotically euclidean,
f(x) 0 as x + . As a consequence, there exists C > 0 such that
f(x)→ C for|e|ve→ry x∞ Rn. This in turn implies
| |≤ ∈
Riβ(x)ω ω C ω 2
(cid:12) i β(cid:12)≤ k kTx∗(Rn)
(cid:12) (cid:12)
for every x Rn and fo(cid:12)r every ω (cid:12)T∗(Rn), which yields (3.11). (cid:3)
∈ ∈ x
Since ω(n) is a Cauchy sequence, the preceding Lemma implies that
{ }
(3.12) < R(ω(n) ω(m)),(ω(n) ω(m)) > √gdx 0
g
ZRn − − 7−→
as n,m + .
→ ∞
As for the gradient part of h [ω(n) ω(m)],
1
−
(3.13) (ω(n) ω(m))2√gdx,
ZRn|∇ − |g
we begin by proving
Lemma 3.6. There exist C,D > 0 such that
(3.14) C η 2dx η 2√gdx D η 2dx
ZRn| |e ≤ ZRn| |g ≤ ZRn| |e
for every smooth, compactly supported tensor η = η of rank 2, where
ij
n
η 2 = η2
| |e ij
iX,j=1
and
η 2 = gij(x)gkl(x)η η .
| |g ik jl
10 FRANCESCAANTOCI
Proof. Consider, for every x Rn, the quadratic form
∈
Rn2 Rn2 R
× −→
defined by
η = η a(x)[η] = gij(x)gkl(x)η η .
ij ik jl
7−→
Since this quadratic form is positive and depends continuously on x, its
eigenvalues λ (x), for k = 1,...,n2, are continuous, positive functions of x.
k
Moreover,
n
a(x)[η] η2
−→ ij
iX,j=1
as x + ,implyingthatλ (x) 1,foreveryk = 1,...,n2,as x + .
k
| |→ ∞ → | |→ ∞
As a consequence, there exist C,D > 0 such that
C η 2 a(x)[η] D η 2
| |e ≤ ≤ | |e
for every x Rn, η Rn2, which implies (3.14). (cid:3)
∈ ∈
Setting
(n) (m)
η = (ω ω ),
ik ∇i k − k
(3.14) yields
1
(3.15) (ω(n) ω(m))2dx (ω(n) ω(m))2√gdx
ZRn|∇ − |e ≤ C ZRn|∇ − |g
and
(3.16) (ω(n) ω(m))2√gdx D (ω(n) ω(m))2dx.
ZRn|∇ − |g ≤ ZRn|∇ − |e
Now,
∂ω
ω = k Γαω ,
∇i k ∂x − ik α
i
whence an easy computation shows that for every i,k = 1,...,n
(n) (m)
∂(ω ω )
ω(n) ω(m) k − k +
(cid:13)(cid:13)∇i(cid:16) k − k (cid:17)(cid:13)(cid:13)L2(Rn,e) ≤ (cid:13)(cid:13)(cid:13) ∂xi (cid:13)(cid:13)(cid:13)L2(Rn,e)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13)
+K ω(n) ω(m) .
(cid:13) − (cid:13)L21(Rn,e)
(cid:13) (cid:13)
As a consequence, (cid:13) (cid:13)
h [ω(n) ω(m)] 0
1
− −→
as n,m 0. Thus, Q(H ) Q(H ).
0 1
→ ⊆
We complete the proof of Theorem 2.1 showing that Q(H ) Q(H ).
1 0
⊆
For any ω Q(H ) there exists a sequence ψ(n) Λ1(Rn) such that (3.7)
∈ 1 { } ⊂ c
and (3.8) hold.