Essential Spectrum on Riemannian Manifolds

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the essential spectrum of −∆ on L2(µ), where µ is the Riemannian volume measure. .... where ρ is the distance function on Sd−1 from a fixed point ξ0, h ∈ C∞(R) with 0 ≤ h ..... 153–201. [20] Röckner, M. and Wang, F.Y., Supercontractivity and ...
Essential Spectrum on Riemannian Manifolds K. D. Elworthy and Feng-Yu Wang

Abstract. By using a measure transformation method, the essential spectrum of the Laplacian in a noncompact Riemannian manifold is studied. The principle result given includes the corresponding main theorems in both Kumura [16] and Donnelly [9]. The essential spectrum of the Laplacian on one-forms is also studied.

Key Words: Essential spectrum, Laplace operator, Riemannian manifold.

1. Introduction Let M be a complete connected Riemannian manifold of dimension d ≥ 2. Denote by ∆ the Laplace-Beltrami operator. The purpose of this note is to estimate σess (−∆), the essential spectrum of −∆ on L2 (µ), where µ is the Riemannian volume measure. Let k · kp denote the Lp (µ)-norm, p ≥ 1. Let us first recall some previous results in the literature. In 1981, Donnelly [8] proved σess (−∆) = [(d − 1)2 k 2 /4, ∞) provided M is a Hadamard manifold with sectional curvatures approaching −k 2 at infinity. In 1994, Li [18] proved σess (−∆) = [0, ∞) if M has nonnegative Ricci curvatures and possesses a pole, see also Escobar-Freire [14] for the case that M has nonnegative sectional curvatures. In 1997, Kumura presented the following result [16, Theorem 1.2]: let r be the distance function from a pole (more generally, a regular domain such that the outward-pointing normal exponential map induces a diffeomorphism), then σess (−∆) = [c2 /4, ∞) provided (1.1)

lim sup |∆r − c| = 0.

n→∞ r≥n

Kumura also proved that this result recovers all those mentioned above. Moreover, [9, Theorem 2.4] says that σess (−∆) = [0, ∞) provided there exists a C 2 exhaustion function γ satisfying ¤ R £ 1 (|∇γ|2 − 1)2 + (∆γ)2 vol(dx) = 0, b≤t t→∞ vol(γ ≤ t) (ii) |∇γ| + |∆γ| is bounded and there exist c > 1 and p ∈ M such that c−1 dist(p, x) ≤ γ(x) ≤ cdist(p, x) holds outside a compact set. (i) lim

Research supported in part by NNSFC(10121101, 10025105),TRAPOYT and the 973-Project. 1

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K. D. ELWORTHY AND FENG-YU WANG

As is pointed out in [7] these assumptions preclude exponential volume growth of the manifold. Essentially they therefore refer to a different class of manifolds from those covered by Kumura. We refer to [2, 3] for constructions of the function γ under certain curvature conditions. See [10] for further study concerning the absence of point spectrum and singular continuous spectrum. This paper is derived from the preprint [13] which was stimulated by Kumura’s work. Our main result, Theorem 1.1, improves and simultaneously generalizes the above results by Kumura[16] and Donnelly[9]. Some variations are given in Section 3 and we also briefly consider the one form case (see section 4). For a description of the essential spectrum of manifolds with ’multicylindrical ends’ obtained by very different techniques see [17]. See [10] for further study concerning the absence of point spectrum and singular continuous spectrum. Recently, the essential spectrum was also studied by using functional inequalities, see [22, 23] and references within. Let γ ∈ C(M ) be a function satisfying (1) γ is unbounded above and is C 2 -smooth in the domain {γ > R} for some R > 0. (2) µ({m0 < γ < n}) < ∞ for some m0 and any n > m0 , where µ is the Riemannian volume element. Theorem 1.1. For any t > 0 and c ∈ R, let Bt = {x ∈ M : γ(x) < t}, dµc = e−cγ dµ and Uc (s, t) = µc (Bt \ Bs ) for t ≥ s. If there exists c such that (1.2)

1 lim lim s→∞ t→∞ Uc (s, t)

Z

©

ª (∆γ − c)2 + (|∇γ|2 − 1)2 dµc = 0

Bt \Bs

and (1.3)

n o lim max Uc (m0 , t), Uc (t, ∞)−1 e−εt = 0

t→∞

for any ε > 0,

then σess (−∆) ⊃ [c2 /4, ∞). Especially, if (1.2) and (1.3) hold for c = 0, then σess (−∆) = [0, ∞). Moreover, if in addition (3) {γ ≤ n} is compact for any n > 0 and |∇γ| → 1 uniformly as γ → ∞. 2

Then σess (−∆+ c4 − 2c ∆¯ γ ) = [0, ∞) for any C 2 -smooth function γ¯ such that γ¯ = γ outside a compact set. To see that Theorem 1.1 recovers the above Kumura’s result, let γ := r be the distance function from a regular domain which is smooth when r > 0. Let ν be the Lebesgue measure induced by the outward-pointing normal exponential map, i.e. under the polar coordinates (r, ξ) ∈ [0, ∞) × Sd−1 one has dν = rd−1 drdξ. Let θ = dµ/dν be defined out of the domain, then (refer to formula (3) in [16])

ESSENTIAL SPECTRUM ON RIEMANNIAN MANIFOLDS 1

3

d−1 ∂ + log θ. r ∂r Thus, it is easy to see that (1.1) implies (1.2) and (1.3). Moreover, the decomposition principle due to Donnelly and Li [11] says that the essential spectrum is independent of the behaviour of the operator in any compact domain. Hence ∆r =

³ ³ c2 ´ c2 c ´ σess − ∆ + − ∆r = σess − ∆ − 4 2 4 provided (1.1) holds. Therefore, Theorem 1.1 recovers Kumura’s result stated above. Next, let c = 0. It is easy to see that (i) implies (1.2) and according to [9, Proposition 2.2], (i) and (ii) imply (1.3). Thus, by Theorem 1.1 one has σess (−∆) ⊃ [0, ∞) and hence σess (−∆) = [0, ∞) provided (i) and (ii) hold for some exhaustion function γ. Therefore, Theorem 1.1 recovers also [9, Theorem 2.4] (note that σess (−∆ + V ) = σess (−∆) for a function V going to zero uniformly as γ → ∞) . Indeed, it is easy to find an example such that Theorem 1.1 works but [9, Theorem 2.4] does not. Indeed, in the above example letting g(ξ, r) = r−2 for big r, we have ∆r → 0 as r → ∞ and hence Theorem 1.1 implies that σess (−∆) = [0, ∞). But according to [9, Proposition 2.2], there is no exhaustion function satisfying (i) and (ii) as the volume of the manifold is now finite. Similarly, it is not difficult to find in particular manifolds with θ := dµ depending dν only on r, such that (1.1) does not holds but Theorem 1.1 is valid. Here, we present the following example in which M is not rotationally symmetric. Example. Let o ∈ M be a pole and the metric is given as follows under the polar coordinates at o: ds2 = dr2 + g 2 (ξ, r)dξ 2 , £ ¤ g(ξ, r) = r exp (sin r)h((1 + r)ρ(ξ)) ,

r ≥ 0, ξ ∈ Sd−1 ,

where ρ is the distance function on Sd−1 from a fixed point ξ0 , h ∈ C ∞ (R) with 0 ≤ h ≤ 1, h|(−∞,1/2] ≡ 1, h[1,∞) ≡ 0. Then (1.1) does not hold but (1.2) and (1.3) hold with γ = r and c = 0. Proof. Obviously, we have n1 o ∂ log g = (d − 1) + h((r + 1)ρ) cos r + (sin r)h0 ((1 + r)ρ)ρ ∂r r which does not converges as r → ∞ by taking, for example, ξ = ξ0 . On the other hand, (1.3) holds since e−1 ν ≤ µ ≤ eν. It remains to prove (1.2). Obviously, there exists c0 > 0 such that d−1 + c0 1{(1+r)ρ(ξ) 0 such that ∆r = (d − 1)

volSd−1 (Ar ) =: volSd−1 ({ξ : ρ(ξ) ≤ (1 + r)−1 }) ≤ c1 (1 + r)−(d−1) for r > 0. Then, for any s > 0,

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K. D. ELWORTHY AND FENG-YU WANG

1 lim t→∞ µ(Bt ) − µ(Bs ) This implies (1.2).

Z

c(s) (∆r) dµ ≤ lim d t→0 t Bt \Bs

Z

t

2

s

rd−1 volSd−1 (Ar )dr = 0. ¤

Finally, to present the exact estimate of σess (−∆) for c 6= 0, we assume that Pt := et∆ is ultracontractive [7]: there exists T > 0 such that (1.4)

kPT k2→∞ =: C(T ) < ∞,

where k · k2→∞ is the operator norm from L2 (µ) to L∞ (µ). In particular, if either the Ricci curvature of M is bounded from below and the injectivity radius is positive (see [21], or [6, Corollary 2.4.3] and [19]), or the injectivity radius is infinite (see [1] or [5]), then there holds the Nash inequality µ(f 2 )1+2/d ≤ C(µ(|∇f |2 ) + µ(f 2 ))µ(|f |)4/d , f ∈ C0∞ (M ) for some constant C > 0, and hence (see e.g. [6, Corollary 2.4.7]) kPT k2→∞ ≤ C 0 (1 + T −d/4 ) for some C 0 > 0 and all T > 0. Thus, the above geometric conditions imply the ultracontractivity. See [6, 4, 22] for more descriptions of the ultracontractivity using functional inequalities. See [20] for explicit criteria for the ultracontractivity on Riemannian manifolds. Corollary 1.2. Suppose that (1.2) and (1.3) hold with γ satisfying (1) and (3), and (1.4) holds for some T > 0. If there exists f ∈ Lp (µ) for some p ∈ [1, ∞) such that 2 ∆γ − f → c uniformly as γ → ∞, then σess (−∆) = [ c4 , ∞). 2. Proofs Proof of Theorem 1.1. We prove the two assertions respectively. a) By (1.3), there exist sm ↑ ∞ and tn ↑ ∞ such that Uc (sm−1 , sm ) = 0, m→∞ Uc (sm , ∞)

(2.1)

lim

(2.2)

lim

n→∞

Uc (tn − 1, tn ) = 0. Uc (m0 , tn − 1)

Let λ ≥ c2 /4 be fixed. For any t > s + 1 > α := max{m + 1, 2 + R}, choose h ∈ C ∞ (R) 0 00 such that 0 ≤ h ≤ 1, h|[s,t−1] ≡ 1, h|(−∞,s−1] ≡ h|p [t,∞) ≡ 0 and kh k∞ , kh k∞ ≤ c0 for some constant c0 independent of s, t. Next, let λc = λ − c2 /4 and f1 (s) = e−cs/2 sin(λc s),

f2 (s) = e−cs/2 cos(λc s), s ≥ 0.

Finally, let ( gs,t =

R h(γ)f1 (γ), if Bt−1 \Bs sin2 (λc γ)dµc ≥ 21 Uc (s, t − 1), h(γ)f2 (γ), otherwise.

ESSENTIAL SPECTRUM ON RIEMANNIAN MANIFOLDS 2

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Then kgs,t k22 ≥ 12 Uc (s, t − 1). On the other hand, noting that fi00 + cfi0 + λfi = 0, i = 1, 2, we have © k(∆ + λ)gs,t k22 ≤ c(λ) Uc (s − 1, s) + Uc (t − 1, t) + k(∆γ − c)1Bt \Bs−1 k22 + k(|∇γ|2 − 1)1Bt \Bs−1 k22

ª

for some constant c(λ) > 0. Therefore, k(∆ + λ)gs,t k22 2c(λ)[Uc (s − 1, s) + Uc (t − 1, t)] ≤ kgs,t k22 Uc (s, t − 1) Z © ª 2c(λ) + (∆γ − c)2 + (|∇γ|2 − 1)2 dµc . Uc (s, t − 1)) Bt \Bs−1

(2.3)

By (2.1) and (2.2) we have Uc (sm − 1, tn ) Uc (sm − 1, sm ) = 1 + lim = 1, m→∞ Uc (sm , tn − 1) Uc (sm , ∞) Uc (sm − 1, sm ) + Uc (tn − 1, tn ) lim lim = 0. m→∞ n→∞ Uc (sm , tn − 1) lim lim

m→∞ n→∞

Therefore, by (2.3) we obtain k(∆ + λ)gs,t k22 k(∆ + λ)gsm ,tn k22 ≤ lim lim t>s+1>α m→∞ n→∞ kgs,t k22 kgsm ,tn k22 Z © ª 2c(λ) = lim lim (∆γ − c)2 + (|∇γ|2 − 1)2 dµc m→∞ n→∞ Uc (sm − 1, tn ) B \B tn sm −1 = 0. inf

(2.4)

2

This implies that λ ∈ σ(−∆) and hence σ(−∆) ⊃ [ c4 , ∞). b) Let L = ∆ − c∇¯ γ . Then L is symmetric with respective to µ ¯c := e−c¯γ µ and we shall 2 always consider it as an operator on the space L2 (¯ µc ). For λ ≥ c4 , let g¯s,t = gs,t ecγ/2 . We have, for big s, t > 0 so that γ¯ = γ on {γ ≥ s}, (L + λ − c2 /4)¯ gs,t = ecγ/2 {(∆ + λ)gs,t + c(∆γ − c)gs,t /2 + (|∇γ|2 − 1)c2 /4}. Then 2 c2 )¯ gs,t k2L2 (µc ) k(∆ + λ)gs,t + 2c (∆γ − c)gs,t + c4 (1 − |∇γ|2 )gs,t k22 4 = k¯ gs,t k2L2 (µc ) kgs,t k22 Z ª © 2 2k(∆ + λ)gs,t k22 c2 c (1 − |∇γ|2 )2 + (∆γ − c)2 dµc + 2 kgs,t k2 Uc (s, t − 1) Bt \Bs−1

k(L + λ −



2

since |¯ gs,t | ≤ 1Bt \Bs−1 . By a) and condition 3) we obtain λ − c4 ∈ σ(−L). Hence σ(−L) = [0, ∞) since −L ≥ 0. Finally, noting that

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K. D. ELWORTHY AND FENG-YU WANG

(∆ − c2 |∇¯ γ |2 /4 + c∆¯ γ /2)f e−c¯γ /2 = e−c¯γ /2 Lf,

(2.5)

f ∈ C 2 (M ),

2

2

we have σess (−∆+ c4 |∇¯ γ |2 − 2c ∆¯ γ ) = σess (−L) = [0, ∞). Therefore σess (−∆+ c4 − 2c ∆¯ γ) = [0, ∞) by condition (3) and the decomposition principle. ¤ Proof of Corollary 1.2. By Theorem 1.1, we need only prove that λ0 =: inf σess (−∆) ≥ c2 /4. For any t > s > R, let λs,t be the first Dirichlet eigenvalue of −∆ in Bs,t =: Bt \ Bs , with c the normalized eigenfunction φs,t ≥ 0. We extend φs,t ∈ C(M ) by taking φs,t |Bs,t ≡ 0. 2 c c 2 2 From (2.5) we know that −∆ + 4 |∇γ| − 2 ∆γ ≥ 0 in L (µ). Then Z

(2.6)

λs,t

c2 |c|ε(s) |c| = φs,t (−∆)φs,t dµ ≥ − − 4 2 2 Bs,t

Z Bs,t

|f |φ2s,t dµ,

where ε(s) = sup{|∆γ − f − c| + |1 − |∇γ|2 ||c|/2} → 0 γ≥s

as s → ∞ by assumption. Let (PTs,t )T ≥0 be the Dirichlet semigroup generated by ∆ on Bs,t . By (1.4) and noting that ∆φs,t = −λs,t φs,t on Bs,t , we obtain e−λs,t T φs,t = PTs,t φs,t ≤ PT φs,t ≤ C(T )kφs,t k2 = C(T ). Thus, kφs,t k∞ ≤ C(T )eλs,t T . Therefore, for any p ≥ 1, Z 2(p−1)/p

Bs,t

|f |φ2s,t dµ ≤ kf 1Bsc kp · kφs,t k22p/(p−1) ≤ kf 1Bsc kp · kφs,t k2

· kφs,t k2/p ∞

≤ C(T )2/p e2λs,t T /p kf 1Bsc kp . Hence, by (2.6) and the decomposition theorem we obtain λ0 ≥ lim λs,t t→∞

c2 |c|ε(s) |c| − − C(T )2/p e2λ0 T /p kf 1Bsc kp . ≥ 4 2 2

If f ∈ Lp (µ), then by letting s → ∞ we arrive at λ0 ≥

c2 . 4

¤

3. Extensions Recall that in Theorem 1.1, we consider the reference measures dµc = e−cγ dµ. In general, we may replace µc by any other measure with certain condition on the volume growth. This is the main idea of this section. For positive g ∈ C 2 (M ), let dµg = gdµ. Then we have the following result.

ESSENTIAL SPECTRUM ON RIEMANNIAN MANIFOLDS 3

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Theorem 3.1. Suppose that γ satisfies (1) and (2). Let Ug (s, t) = µg (Bt \ Bs ) for t ≥ s and Vg = g −1/2 ∆g 1/2 . If 1 lim lim s→∞ t→∞ Ug (s, t)

(3.1) and

½³

Z

h∇γ, ∇ log gi + ∆γ

´2

2

Bt \Bs

n o lim max Ug (m0 , t), Ug (t, ∞)−1 e−εt = 0

(3.2)

¾ + (|∇γ| − 1) dµg = 0 2

t→∞

for any ε > 0,

then σ(−∆ + Vg ) = [0, ∞). Proof. Let L = ∆ + ∇ log g. We have (−∆ + Vg )g 1/2 f = −g 1/2 Lf for f ∈ C 2 (M ) and using this conjugacy we can, and will, consider L as a self-adjoint operator on L2 (µg ). Then σ(−∆ + Vg ) = σ(−L). Hence, it suffices to prove σ(−L) ⊃ [0, ∞) since −L ≥ 0. Actually, for any t > s + 1 > α, let h be as in the proof of Theorem 1.1. For λ ≥ 0, set

kgs,t k2L2 (µg )

such that k(L + λ

2

)gs,t k2L2 (µg )



gs,t = 1 U (s, t 2 g

h(γ) cos(λγ) or h(γ) sin(λγ) 00 − 1). Noting that gs,t + λ2 gs,t = 0 on [s, t − 1], we have

½ ≤ c(λ) Ug (t − 1, t) + Ug (s − 1, s) ¾ Z h¡ ´2 i ¢2 ³ 2 + |∇γ| − 1 + ∆γ + h∇γ, ∇ log gi dµg Bt \Bs−1

for some c(λ) > 0. Then the remainder of the proof is similar to the proof of Theorem 1.1. ¤ Remark. Let us consider Rd with the Riemannian metric g(∂i , ∂j ) = gij , i, j = 1, · · · , d. Let g = (detgij )1/2 . Then the corresponding Riemannian volume measure is µ(dx) = g(x)dx and the associated Laplacian operator writes d X ¡ ij ¢ ∆g = ∂i g ∂j + (g ij ∂i log g)∂j , i,j=1 ij

where (g ) is the inverse matrix of (gij ). We have L :=

d X

∂i g ij ∂j = ∆g − ∇g log g,

i,j=1

where ∇g is the gradient operator induced by g. Then the proof of Theorem 3.1 shows that σ(−L) = [0, ∞) on L2 (dx) under the conditions of Theorem 3.1. From now on, we assume that r is the distance function from a regular domain such that the outward-pointing normal exponential map induces a diffeomorphism. Let ν be the Lebesgue measure induced by the exponential map and let θ = dµ/dν which is well

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K. D. ELWORTHY AND FENG-YU WANG

defined out of the domain. Then (3.1) and (3.2) hold with the choice g = θ−1 . Therefore, by Theorem 3.1 we always have σess (−∆ + θ1/2 ∆θ−1/2 ) = [0, ∞).

(3.3)

Then we can describe σess (−∆) by studying the potential Vθ =: θ1/2 ∆θ−1/2 . We emphasize this potential because it plays an important role in the heat kernel formula due to Elworthy and Truman, e.g. see [12]. Corollary 3.2. 1) We have σess (−∆) = [−c, ∞) if limn→∞ supr≥n |Vθ − c| = 0. 2) We have σess (−∆) ⊃ [−c, ∞) provided Z −d

(3.4)

(Vθ − c)2 dν = 0.

lim s

s→∞

Bs

Especially, σess (−∆) = [0, ∞) if (3.4) holds with c = 0. To prove this result, we need the following lemma. Lemma 3.3. For any λ ≥ 0 and n ≥ 1, there exists gn ∈ C0∞ (Bn ) which depends only on r, such that kgn kL2 (ν) = 1, kgn k∞ ≤ c(λ)n−d/2 for some c(λ) > 0 and (3.5)

lim k(Lr + λ)gn kL2 (ν) = 0,

n→∞

where Lr =

∂2 ∂r2

+

(d−1)∂ . r∂r

Proof. Choose h ∈ C ∞ (R) such that 0 ≤ h ≤ 1, h|(−∞,−1] = 1, h|[0,∞) = 0. Take √ gn = c−1 n h(r − n) cos( λr), where √ cn = kh(r − n) cos( λr)kL2 (ν) ≥ c(λ)−1 nd/2 for some c(λ) > 0. Therefore, there exists a constant C > 0 such that d−1 k(Lr + λ)gn k2L2 (ν) ≤ c−2 n Cn

which goes to zero as n → ∞.

¤

Proof of Corollary 3.2. We need only to prove 2). Let H = −∆+Vθ . For any λ ≥ 0, let {gn }n≥2 be chosen as in Lemma 3.3. Noting that H g¯n = θ−1/2 (−Lr )gn for g¯n = θ−1/2 gn , we have k(−∆ + c)¯ gn − λ¯ gn k2 ≤ k(Vθ − c)¯ gn k2 + k(H − λ)¯ g n k2 = k(Vθ − c)gn kL2 (ν) + k(Lr + λ)gn kL2 (ν) ≤ c(λ)n−d/2 k(Vθ − c)1Bn kL2 (ν) + k(Lr + λ)gn k2 which goes to zero as n → ∞ by (1.1) and (3.5). Hence, σ(−∆ + c) ⊃ [0, ∞).

¤

ESSENTIAL SPECTRUM ON RIEMANNIAN MANIFOLDS 4

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Corollary 3.4. We have σess (−∆) = [−c, ∞) if there exists f such that Vt heta−f → c uniformly as r → ∞ and one of the following holds: 1) (1.4) and (3.4) hold and f ∈ Lp (µ) for some p ∈ [1, ∞). 2) f ∈ Lp (µ) for some p > 2 and the following Sobolev inequality holds: kgk22p/(p−2) ≤ C(µ(|∇g|2 ) + µ(g 2 )),

(3.6)

g ∈ C0∞ (M ).

Noting that the sufficiency of condition 1) in Corollary 3.4 follows immediately from Corollary 3.2 and the proof of Corollary 1.2, we need only to prove the sufficiency of condition 2). We remark that (3.6) holds for p ∈ (d, ∞) ∩ (2, ∞) provided either the injectivity radius is infinite or it is positive but the Ricci curvature is bounded below, see references mentioned after (1.4). Proof of Corollary 3.4. It suffices to prove the result using condition 2). Suppose that f ∈ Lp (µ) for some p > 2. Let H = −∆ + f + c, then σess (H) = σess (−∆ + Vθ ) = [0, ∞) since Vθ − f − c → 0 uniformly as r → ∞. So, for any λ ≥ 0, we may choose {gn }n≥1 such that kgn k2 = 1, gn ∈ C0∞ (Bnc ) and k(H − λ)gn k2 ≤ n−1 .

(3.7) Then

Z

Z 2

Z

gn Hgn dµ − f gn2 dµ − c Z Z ≤ −c + λ + gn (H − λ)gn dµ − f gn2 dµ

|∇gn | dµ = (3.8)

≤ −c + λ + n−1 + 1 + kf gn k22 since |f | ≤ 1 + f 2 . Moreover, (3.9)

k(−∆ + c)gn − λgn k2 ≤ k(H − λ)gn k2 + kf gn k2 ≤ n−1 + kf gn k2 ,

On the other hand, it follows from the H¨older inequality, (3.6) and (3.8) that kf gn k2 ≤ k1Bnc f kp · kgn k2p/(p−2) ≤ c3 k1Bnc f kp (1 + kf gn k2 ) for some c3 > 0. This implies c3 k1Bnc f kp =0 n→∞ 1 − c3 k1B c f kp n

lim kf gn k2 ≤ lim

n→∞

since f ∈ Lp (µ). By combining this with (3.9), we obtain λ ∈ σ(−∆+c). Hence σess (−∆+ c) ⊃ [0, ∞) = σ(H). Similarly, we can prove the inverse inclusion. ¤ Remark. The proof of Corollary 3.4 shows that the second part in this corollary is also true with Vθ replaced by Vg provided (3.1) and (3.2) hold.

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K. D. ELWORTHY AND FENG-YU WANG

4. The essential spectrum on one-forms Let ∆p = −(dδ + δd) be the Laplacian on Λp (M ) for 1 ≤ p ≤ d. Escobar and Freire [15] proved that σess (−∆p ) = [0, ∞) if the radial sectional curvature, the curvature tensor and the first derivative of the curvature tensor decay to zero at certain speeds. See also [17] for results concerning manifolds with ‘multicylindrical ends’. Here, we try to extend Theorem 1.1 to the one-form case. Theorem 4.1. Suppose that γ satisfies (1) and (3) and let Uc be as in Theorem 1.1. If there exists c such that (1.3) holds and 1 lim lim s→∞ t→∞ Uc (s, t)

(4.1)

Z

© ª (∆γ − c)2 + |∇∆γ|2 + |∇|∇γ||2 dµc = 0, Bt \Bs

c2

then σess (−∆1 ) ⊃ [ 4 , ∞) and σess (−∆1 + γ¯ = γ outside a compact set. Proof. For λ >

c2 , 4

c2 4

− 2c ∆¯ γ ) = [0, ∞) for C 2 -smooth γ¯ such that

let f1 and f2 be defined as in section 2. We have f10 (s)2 + f20 (s)2 = λe−cs ,

(4.2)

s ≥ 0.

Next, let h be chosen as in section 2 for t > s + 1 > α with h000 also uniformly bounded. By (4.2) and condition (3), we may take either gs,t = h(γ)f1 (γ) or gs,t = h(γ)f2 (γ) such that λ Uc (s, t − 1), s >> 1. 4 Now, take ωs,t = dgs,t , we have in Bt−1 \ Bs , ½ ¾ ∂ 2 gs,t ∂gs,t 2 (∆1 + λ)ωs,t = d(∆gs,t + λgs,t ) = d (|∇γ| − 1) + (∆γ − c) . ∂γ 2 ∂γ By (1.3), (4.1) and the proof of Theorem 1.1 a), we prove the first assertion. As for the second assertion, let L = ∆1 − c∇∇¯γ , we have k∇gs,t k22 ≥

³

´ c2 c |∇¯ γ |2 − ∆¯ γ e−c¯γ /2 ω = −e−c¯γ /2 Lω, ω ∈ Λ1 (M ). 4 2 Hence, we need only to show that σess (−L) ⊃ [0, ∞) since −L ≥ 0. Let ω ¯ s,t = ec¯γ /2 ωs,t , we have − ∆1 +

³

½ ¾ c2 ´ c c2 c¯ γ /2 2 L+λ− ω ¯ s,t = e (∆1 + λ)ωs,t + (∆¯ γ − c)ωs,t + (|∇¯ γ | − 1) . 4 2 4

Then the remainder of the proof is similar to the proof of Theorem 1.1 b).

¤

Corollary 4.2. Suppose that r is the distance function from a regular domain which 2 is smooth in {r > 0}. We have σess (−∆1 ) = [ c4 , ∞) provided |∆r − c| + |∇∆r| → 0

uniformly as r → ∞.

Acknowledgment. The authors would like to thank the referee for his corrections.

ESSENTIAL SPECTRUM ON RIEMANNIAN MANIFOLDS 5

11

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