Discreteness of the spectrum of the Laplace-Beltrami operator

0.17 MB·English

by  M. Harmer

Discreteness of the spectrum of the
Laplace-Beltrami operator
Mark Harmer
Department of Mathematics
7
0 Australian National University
0
2 Australia

n email: [email protected]
a
J

0
2 Abstract

] We propose simple conditions equivalent to the discreteness of
A
the spectrum of the Laplace-Beltrami operator on a class of Rieman-
F
nian manifolds close to warped products. For this class of manifolds
.
h
we establish a relationship between discreteness of the spectrum and
t
a
stochastic incompleteness.
m
[

1 Introduction
1
v
4
6 We say that an operator has only discrete spectrum if the essential spectrum
5 of the operator is empty (i.e. the spectrum is discrete of finite multiplicity
1
0 with no accumulation points). Much work has been done on finding condi-
7 tions for the discreteness of the spectrum of the Laplace-Beltrami operator
0
on functions, see for instance [4, 1, 10, 3, 11, 12, 13, 14, 15] and references
/
h
therein. Some of the most general necessary and sufficient conditions were
t
a
found by Baider [1] who considers manifolds which are warped products with
m
the discreteness condition given as a condition on the spectrum of an oper-
:
v ator defined on one of the terms in the product. Kleine and Bru¨ning [10, 3]
i
X express the Laplacian as a Schr¨odinger operator with operator valued po-
r tential and derive general discreteness conditions for such operators. In [11]
a
multiply warped products are considered and various necessary and sufficient
conditions are derived in terms of the capacity. Very general conditions for
discreteness are given by [12, 13, 14, 15] in terms of capacity. Our condition
for discreteness is related to a condition on the capacity of subsets and may
be considered as a particular case of Maz’ja’s condition [15, 8].
For us an important condition for discreteness was given by Kac and Krein
[9] for the singular string. The main result in our paper may be considered
as a generalisation of the result of Kac and Krein to manifolds with ends
which are ‘close to’ warped products.
1
In this paper we prove a condition equivalent to discreteness of the spectrum
of the Laplace-Beltrami operator on a manifolds with ends having a certain
type of bound on the mean curvature. Analogous to the result of Kac and
Krein our condition splits into two cases depending on whether the end of
the manifold has finite or infinite volume. We use ideas which appear in
[9] and a paper by Muckenhoupt [16] on weighted Hardy inequalities. The
conditions equivalent to discreteness allow one to easily see the relationship
between volume growth down the ends of the manifold and discreteness.
We follow the main result with a discussion on the relationship between our
condition and the condition of Maz’ja based on capacity [15]. The paper
concludes by showing that, for the chosen class of manifolds, stochastic in-
completeness is sufficient for discreteness.
2 Preliminaries
Weconsideracomplete, noncompactn+1-dimensionalRiemannianmanifold
. We impose two hypotheses on .
M M
Hypothesis 2.1 There is an compact subset of such that con-
U M M\U
sists of a finite number of disjoint, noncompact ends . Each end is
i i
E E
diffeomorphic to R where is an n-dimensional compact manifold.
+ i i
× K RK
The diffeomorphism induces on the metric
+ i
×K
2 2 2
ds = dr +dθ (r) (1)
Ki
where dθ (r)2 is a smooth family of metrics on .
Ki Ki
Thedecompositionprinciple(seeProposition2.1of[4],Lemma2.3of[1]or[6]
for a precise statement) states that the essential spectrum of the Laplacian
on functions is invariant under compact perturbation. Consequently, the
discreteness of the Laplacian on is equivalent to the discreteness of the
M
Dirichlet Laplacianoneachoftheends. Forthisreasonwe dropthesubscript
i and consider questions of discreteness on a generic end in the sequel.
E
The form of volume on may be written
E
dµ = ω(r,θ)dθndr
while on the Laplace-Beltrami operator on functions has the form
E
1 ∂ ∂
∆ ω +∆ (r)
K
≡ −ω∂r ∂r
where ∆ (r) is the Laplacian on with respect to the metric dθ (r)2. We
K K
K
denote the norm in L ( ,dµ) by .
2
M k·k
The quantity
ω′
h(r,θ) = (r,θ)
ω
is the mean curvature of at the point (r,θ) [5]. In this paper ′ will denote
K
differentiation with respect to r. This leads us to our second hypothesis:
2
Hypothesis 2.2 The mean curvature h of satisfies
K
1
sup h hωdθn = sup h h¯ < c (2)
E (cid:12) − ω¯ ZK (cid:12) E −
(cid:12) (cid:12) (cid:12) (cid:12)
where ω¯(r) = ωdθn(cid:12)is the volume of(cid:12) at th(cid:12)e poin(cid:12)t r and h¯ is defined in
K (cid:12) (cid:12)K
the equality.
R
This assumption means that the end is very close to a warped product. This
allows us to effectively reduce the problem to a one-dimensional one: the
end is sufficiently well behaved that only the r dependence of the metric
E
is important for discreteness. The mean curvature is not an intrinsically
defined quantity so it is not clear whether hypothesis 2.2 is intrinsic or not.
We define the non compact subset , t 0, as the set diffeomorphic to
t
E ⊂ E ≥
(t, )
∞ ×K
underthediffeomorphismdefinedaboveandwedenotethenorminL ( ,dµ)
2 t
E
by .
t
Givken·kf C∞( ,R) we define (up to sign) the following average
0
∈ E
1
1 2
f¯(r) = f2ωdθn . (3)
ω¯(r)
K
(cid:18) Z (cid:19)
We say that such an averaged function is spherically symmetric.
Lemma 2.1 Given f C∞( ,R) on an end satisfying hypothesis 2.2 we
0 t
∈ E E
have
1 c2
¯′ 2 2 2
f f f . (4)
2 t − 4 k kt ≤ k∇ kt
Proof: We differentiate ((cid:13)3) (cid:13)
(cid:13) (cid:13)
1 2 1
f¯′ ff′ωdθn + f2ω′dθn h¯f¯2
≤ 2f¯ ω¯ ω¯ −
(cid:20)(cid:12) ZK (cid:12) (cid:12) ZK (cid:12)(cid:21)
(cid:12) (cid:12) (cid:12) (cid:12)1 (cid:12) 1 (cid:12)
(cid:12) (cid:12) 1 (cid:12)(cid:12)2 f2ωdθn(cid:12)(cid:12)2 (cid:12)(cid:12) (f′)2ωdθn 2 +su(cid:12)(cid:12)p h h¯ f¯2
≤ 2f¯"ω¯ K K K − #
(cid:20)Z (cid:21) (cid:20)Z (cid:21)
(cid:12) (cid:12)
1 (f′)2ωdθn 12 + cf¯. (cid:12) (cid:12)
≤ ω¯ 2
K
(cid:20) Z (cid:21)
Using (a+b)2 2a2 +2b2 we get
≤
1 f¯′ 2ω¯ c2f¯2ω¯ (f′)2ωdθn.
2 − 4 ≤
K
Z
(cid:0) (cid:1)
Then integrating over r and using
(f′)2ωdθndr (f′)2ωdθn + f 2ωdθn dr = f 2 dµ,
K
≤ |∇ | |∇ |
K K K E
Z Z Z (cid:20)Z Z (cid:21) Z
2
where is the gradient on , we get the result.
K
∇ K
3
For discreteness the stronger condition (2) is not necessary, all we need is
the condition (4) of the lemma, indeed we may choose this as our second
hypothesis. We refer to (4) as the ‘coerciveness’ condition (compare with
equation (1.13) of [3]). Ideally though we would replace hypothesis 2.2 with
an intrinsic condition.
We define the Dirichlet [18]
2
f
λ ( ) = inf k∇ kt : f,∆f L ( ) , f H1( )
0 Et ( f 2 ∈ 2 Et ∈ 0 Et )
k kt
and Neumann
2
f
µ ( ) = inf k∇ kt : f,∆f L ( ) , (∆f,u) = ( f, u) u H1( )
0 Et ( f 2 ∈ 2 Et ∇ ∇ ∀ ∈ Et )
k kt
Rayleigh quotients where H1( ) is the set of functions with u 2+ u 2 <
Et k kt k∇ kt
and H1( ) is the closure of C∞( ) in H1( ).
0 t 0 t t
∞ E E E
The following result ([1], theorem 2.2) is our main tool to prove discrete-
ness:
Theorem 2.1 The spectrum of the Laplace-Beltrami operator ∆ will be dis-
crete iff on each end the Dirichlet Rayleigh quotient satisfies
E
lim λ ( ) = + .
0 t
t→∞ E ∞
It is easy to see that the same result will not hold for the Neumann Rayleigh
quotient (consider a finite volume non compact end down which the Dirichlet
Rayleigh quotient is unbounded but where the Neumann Rayleight quotient
remains bounded because of the presence of the constant eigenfunction).
Nevertheless, in the infinite volume case we have:
Lemma 2.2 The spectrum of the Laplacian ∆ on a manifold satisfying
M
hypotheses 2.1 and 2.2 will be discrete iff on each finite volume end
lim λ ( ) = + (5)
0 t
t→∞ E ∞
and on each infinite volume end
lim µ ( ) = + . (6)
0 t
t→∞ E ∞
Proof: We need to establish that on an infinite volume end the unbound-
edness of the Neumann quotient is equivalent to the unboundedness of the
Dirichlet quotient. The Dirichlet quotient is bounded below by the Neumann
quotient so that unboundedness of the Neumann quotient implies unbound-
edness of the Dirichlet quotient.
Conversely, suppose that there exists a µ < such that for all t > 0,
∞
4
µ ( ) < µ. This means we can find a ψ satisfying the above conditions
0 t 1
E
such that
ψ 2
1 0
k∇ k < µ.
ψ 2
1 0
k k
Furthermore, since has infinite volume, we can choose ψ so that it has
t 1
E
compact support in r [0,t ) for some t . Repeating this at t = t we get a
1 1 1
∈
sequence ψ ,t such that ψ satisfies the above inequality and has support
i i i
{ }
in r [t ,t ). We average the ψ as in (3) to get a sequence of spherically
i−1 i i
∈ ¯
symmetrical functions ψ where, without loss of generality we may assume
i
that
¯ ¯′
ψ = 1, ψ = 0,
i r=ti−1 i r=ti−1
and we continue each ψ¯(cid:12)in r < t as(cid:12)ψ¯ = 1. In forming the average
i(cid:12) i−1 (cid:12) i
(defined up to signs) of a complex valued function, we take the average of
the real and imaginary parts separately.
¯
Clearly we can take a subsequence ψ ,t such that
{ kl kl}
¯ ¯
ψ > ψ .
kl+1 tkl kl tkl−1
(cid:13) (cid:13) (cid:13) (cid:13)
We drop the extra subscrip(cid:13)t and(cid:13)denot(cid:13)e thi(cid:13)s subsequence by the same nota-
tion ψ ,t so that
l l
{ } ¯ ¯
ψ > ψ .
l+1 tl l tl−1
Furthermore, it is clear from(cid:13)the d(cid:13)efinit(cid:13)ion(cid:13)and (4) that
(cid:13) (cid:13) (cid:13) (cid:13)
¯ 2
∇ψl tl−1 < 2µ+ c2 .
(cid:13) ψ¯ (cid:13)2 2
(cid:13) l (cid:13)tl−1
(cid:13) (cid:13)
We consider the sequence ϕ(cid:13)= ψ(cid:13)¯ ψ¯ defined on the whole end . This
l 2l 2l−1
− E
sequence has disjoint supports and
2 ψ¯ 2 + ψ¯ 2
k∇ϕlk0 = ∇ 2l t2l−1 ∇ 2l−1 t2l−2
2 2
ϕ (cid:13) (cid:13) ϕ(cid:13) (cid:13)
k lk0 (cid:13) (cid:13) k l(cid:13)k0 (cid:13)
¯ 2 ¯ 2
ψ ψ
∇ 2l t2l−1 + ∇ 2l−1 t2l−2 4µ+c2.
≤ (cid:13) ψ¯ (cid:13)2 (cid:13) ψ¯ (cid:13)2 ≤
(cid:13) 2l (cid:13)t2l−1 (cid:13) 2l−1 (cid:13)t2l−2
(cid:13) (cid:13) (cid:13) (cid:13)
In the first line we use the(cid:13)fact(cid:13)that ψ¯ (cid:13)and (cid:13)ψ¯ have disjoint supports
2l 2l−1
∇ ¯ ∇ ¯
while in the second line we use ϕ > ψ > ψ . Conse-
k lk0 2l t2l−1 2l−1 t2l−2
quently, according to [6], theorem 13 page 15, we have a point of essential
(cid:13) (cid:13) (cid:13) (cid:13)
spectrum. (cid:13) (cid:13) (cid:13) (cid:13) 2
We require one more result which will be used in the discussion of brownian
motion.
5
Lemma 2.3 Suppose that is a manifold satisfying hypothesis 2.1 with
M
only one end and such that the Laplacian has non empty essential spectrum.
In particular the metric down the end is of the form (1). Perturbing the
metric by an exponential factor to
ds2 = dr2 +e2cr/ndθ (r)2, (7)
K
where c > 0, the essential spectrum of the Laplacian will remain non empty.
Proof: Theassumptionofnonemptyessentialspectrumimplies, bytheorem
2.1, the existence of λ < and a family ϕ L ( ) such that
t 2 t
∞ ∈ E
2
ϕ
k∇ tkt < λ.
2
ϕ
k tkt
We define
ϕ = e−cr/2ϕ
t,c t
and denote the norm associated to the perturbed metric (7) by . Then
t,c
k·k
ϕ = ϕ and
t,c t,c t t
k k k k
ϕ 2 = ϕ′ 2ecrωdθndr + ϕ 2e−2cr/necrωdθndr
k∇c t,ckt,c t,c |∇K t,c|
ZEt ZEt
(cid:0) (cid:1) c 2
= ϕ′ ϕ ωdθndr + ϕ 2e−2cr/nωdθndr
t − 2 t |∇K t|
ZEt (cid:16) (cid:17) ZEt
c2
2 2
2 ϕ + ϕ
≤ k∇ tkt 2 k tkt
which implies that
2
k∇cϕt,ckt,c 2λ+ c2 .
ϕ 2 ≤ 2
k t,ckt,c
2
3 Main result
Theorem 3.1 The spectrum of the Laplacian on a manifold satisfying
M
hypotheses 2.1 and 2.2 is discrete iff for each end either
E
s ∞
−1
lim sup ω¯ dr ω¯dr = 0, (8)
t→∞ s>t Zt Zs
or
∞ s
−1
lim sup ω¯ dr ω¯dr = 0. (9)
t→∞ s>t Zs Zt
6
Proof: The proof is split into two cases: either ∞ω¯−1dr = and we
0 ∞
show that (8) is equivalent to the unboundedness of the Dirichlet Rayleigh
∞ R
quotient; or ω¯ dr = and we show that (9) is equivalent to the un-
0 ∞
boundedness of the Neumann Rayleight quotient. We refer to these as the
R
finite and infinite volume cases respectively.
1. Assuming ∞ω¯−1dr = we show that (8) is equivalent to (5).
0 ∞
We choose u C∞( ) assuming, without loss of generality, that u is
0 t
R ∈ E
real. Then
∞ ∞ ∞
2 2 ′
u dµ = u¯ ω¯dr = 2 ω¯dsu¯u¯ dr
ZEt Zt Zt Zr
s ∞ ∞ 1
−1 ′
= 2sup ω¯ dr ω¯dr u¯u¯ dr
rω¯−1ds
s>t (cid:18)Zt Zs (cid:19)Zt t
s ∞
2sup ω¯−1dr ω¯dr R
≤ ×
s>t (cid:18)Zt Zs (cid:19)
∞ 2 ν2(r,t) 21 ∞ ′ 2 21
u¯ dr (u¯) ω¯dr .
ω¯
(cid:20)Zt (cid:21) (cid:20)Zt (cid:21)
Here we have denoted
r −1
−1
ν(r,t) = ω¯ ds
(cid:18)Zt (cid:19)
ν2(r,t)
′
ν (r,t) =
⇒ − ω¯(r)
∞ ν2(s,t)
ν(r,t) = ds.
⇒ ω¯(s)
Zr
Using this in the first integral on the right hand side of our inequality
we have
∞ ν2(r,t) ∞
2 ′
u¯ dr = 2 u¯u¯ ν(r,t)dr
ω¯
Zt Zt
∞ 2 ν2(r,t) 21 ∞ ′ 2 12
2 u¯ dr (u¯) ω¯dr ,
≤ ω¯
(cid:20)Zt (cid:21) (cid:20)Zt (cid:21)
that is
∞ ν2(r,t) ∞
2 ′ 2
u¯ dr 4 (u¯) ω¯ dr.
ω¯ ≤
Zt Zt
Putting this back into the original inequality we have
s ∞ ∞
2 −1 ′ 2
u dµ 4sup ω¯ dr ω¯dr (u¯) ω¯ dr.
≤
ZEt s>t (cid:18)Zt Zs (cid:19)Zt
7
On the other hand, using lemma 2.1 we can write the Dirichlet integral
as
1 c2
′ 2 2 2
(u¯) ω¯dr u¯ ω¯dr u dµ.
2 − 4 ≤ |∇ |
Z Z Z
Putting these inequalities together we have
1 1 c2
2 2
u dµ u dµ
8sup sω¯−1dr ∞ω¯dr − 4 ≤ |∇ |
s>t t s !ZEt ZEt
(cid:0)R R (cid:1)
which proves that (8) is sufficient for discreteness.
For necessity we consider the family v(r;t,s,s ,s ), t < s < s < s , of
0 1 0 1
Lipschitz functions
Rtrω¯−1dr′ : t < r < s
Rtsω¯−1dr′
 1 : s < r < s0
v(r) =  2−(cid:18)Rtrω¯−R1ssd01rω′¯−−R1tsdωr¯′−1dr′(cid:19) : s0 < r < s1
0 : s < r
1



where, since ∞ω¯−1dr, we are able to choose s so that
0 1
R s1 s0
−1 ′ −1 ′
ω¯ dr = ω¯ dr .
Zs0 Zs
Putting this into the Rayleigh quotient we see that
v 2 sω¯−1dr′ s s0 −1
k∇ kt 1+ t ω¯−1dr′ ω¯ dr′ .
v 2 ≤ s0ω¯−1dr′
k kt (cid:18) Rs (cid:19)(cid:18)Zt Zs (cid:19)
Letting s gives tRhe result.
0
→ ∞
∞
2. Using the assumption of infinite volume ω¯dr = and the same
0 ∞
argument as above but with ω¯ ω¯−1 we see that for v C∞( )
0 t
↔ R ∈ E
∞ 2 −1 12 ∞ −1 s ∞ ′ 2 −1 21
v¯ ω¯ dr 2sup ω¯ dr ω¯dr (v¯) ω¯ dr .
≤
(cid:20)Zt (cid:21) s>t (cid:18)Zs Zt (cid:19)(cid:20)Zt (cid:21)
(10)
Now consider again a real function u C∞( );
0 t
∈ E
∞ ∞ r
2 2 ′
u dµ = u¯ ω¯ dr = u¯(s)ω¯(s)dsu¯(r)dr
ZEt Zt Zt Zt
∞ 1 r
= u¯ω¯ dsu¯′ω¯1/2dr
ω¯1/2
Zt Zt
1
∞ r 2 −1 2 ∞ ′ 2 21
u¯ω¯ds ω¯ dr (u¯) ω¯dr .
≤
"Zt (cid:18)Zt (cid:19) # (cid:20)Zt (cid:21)
8
r
Using (10) on the right hand side with v¯(r) = u¯ω¯ds we get the
t
inequality
R
∞ s ∞
2 −1 ′ 2
u dµ 4sup ω¯ dr ω¯dr (u¯) ω¯dr
≤
ZEt s>t (cid:18)Zs Zt (cid:19)Zt
analogous to the result in the finite volume case. Again using lemma
2.1 we get that (9) is sufficient for discreteness.
Fornecessityweconsiderthefamilyv(r;t,s,s ),t < s < s ,ofLipschitz
0 0
functions
1 : t < r < s
v(r) = 1 Rtrω¯−1dr′−Rtsω¯−1dr′ : s < r < s
 − Rss0ω¯−1dr′ 0
 (cid:16) 0 (cid:17) : s0 < r

in the Rayleigh quotient:
2
v 1
k∇ kt .
v 2 ≤ s0ω¯−1dr′ sω¯dr′
k kt s t
R R
Again letting s gives the result.
0
→ ∞
2
The proof of this theorem uses ideas from [2, 15], the proof of discreteness
for the singular string by Kac and Krein [9] and Muckenhoupt’s weighted
Hardy inequality [16]. The Kac and Krein result can be directly used to give
a condition for discreteness in the warped product case but, in the form it
is published, is not convenient to use in the more general context considered
here. Muckenhoupt gives a result very similar to this theorem (in somewhat
greater generality); however, he requires finiteness of the arguments of the
limits in (8, 9) which we are able to relax thereby simplifying the proof.
We note that the conditions (8,9) for discreteness have some resemblence to
the condition for discreteness given by Bru¨ning in the case of the warped
product (theorem 3.4 of [3]).
These conditions for discreteness can be further simplified.
Corollary 3.1 The spectrum of the Laplacian on a manifold satisfying
M
hypotheses 2.1 and 2.2 is discrete iff for each end either
E
s ∞
−1
lim ω¯ dr ω¯dr = 0, (11)
s→∞Z0 Zs
or
∞ s
−1
lim ω¯ dr ω¯dr = 0. (12)
s→∞Zs Z0
9
Proof: We consider the equivalence of (9) to (12), the other case follows
from a similar argument.
Assuming (9) we can, given ǫ > 0, find T such that for t T
≥
∞ s
−1
sup ω¯ dr ω¯dr < ǫ.
s>t Zs Zt
On the other hand (9) implies ∞ω¯−1dr < so that we can find S such
0 ∞
that for s S
≥ R∞ ǫ
−1
ω¯ dr <
M
Zs
T
where M = ω¯dr. Consequently, for s max(T,S) we have
0 ≥
∞ R s ∞ T s
−1 −1
ω¯ dr ω¯ dr = ω¯ dr ω¯dr+ ω¯dr < 2ǫ.
Zs Z0 Zs (cid:18)Z0 ZT (cid:19)
Now we suppose that (12) holds from which we have, for all s > 0,
∞ t
−1
lim ω¯ dr ω¯dr = 0
t→∞Zs+t Z0
following from the monotonicity of ∞ω¯−1dr. Consequently, for all s > 0,
t
∞ Rs+t ∞ t
−1 −1
0 = lim ω¯ dr ω¯dr ω¯ dr ω¯ dr
t→∞(cid:18)Zs+t Z0 −Zs+t Z0 (cid:19)
∞ s+t
−1
= lim ω¯ dr ω¯dr
t→∞Zs+t Zt
∞ s
−1
= lim ω¯ dr ω¯dr, s > t,
t→∞ ∀
Zs Zt
2
in particular (9) holds.
Ifweassumethatthemetricdowntheendsofthemanifoldissufficiently ‘well
behaved’ then it is possible to greatly simplify conditions (8,9) or (11,12).
Specifically, if we assume that the following limits exist (we will say that a
limit going to exists) then it is clear that (11,12) are equivalent to
±∞
∞
d
lim ln ω¯dr =
s→∞ ds −∞
Zs
or
∞
d
−1
lim ln ω¯ dr =
s→∞ ds −∞
Zs
respectively. This is a simple consequence of L’Hopital’s rule [17], in the case
of (12)
∞ s
−1
0 = lim ω¯ dr ω¯dr
s→∞Zs Z0
10