Math. Z. 242, 183–201 (2002) Digital Object Identifier (DOI) 10.1007/s002090100313
Lp -spectral properties of the Neumann Laplacian on horns, comets and stars Peer Christian Kunstmann Mathematisches Institut I, Universit¨at Karlsruhe, Englerstr. 2, D-76128 Karlsruhe, Germany (e-mail:
[email protected]) Received: 17 January 2000; in final form: 25 July 2000 / c Springer-Verlag 2001 Published online: 23 July 2001 –
Abstract. We study Lp -spectral properties of Neumann Laplacians on some planar domains and show by calculation that the essential spectrum of the Neumann Laplacian on certain horns depends on p. The proof uses ideas due to E.B. Davies and B. Simon for the reduction to one-dimensional operators and techniques involving Gaussian bounds. For domains looking like comets or stars, i.e. having countably many horn-shaped outlets, we prove a decoupling-reduction result. These results are used to construct planar domains for which the Neumann Laplacian has maximal Lp -spectrum in the class of generators of symmetric submarkovian semigroups. Mathematics Subject Classification (2000):35P05, 35J05, 47D07 1 Introduction Let Ω be an open subset of R2 . In L2 (Ω), the Neumann Laplacian ∆Ω N is defined via the form t ∇g ∇f d(x, y) n ¯ (f, g) := Ω
H 1 (Ω),
i.e. (−∆Ω ¯ (f, g). By Beurling-Deny with form domain N f, g) = n Ω ∆N generates a submarkovian semigroup. Hence there are consistent C0 semigroups of positive contractions (Tp (t)) in Lp (Ω), 1 ≤ p < ∞. We drop the subscript N and denote their respective generators by ∆Ω p , in particular Ω Ω we shall write ∆2 instaed of ∆N . In this paper we study spectral properties of ∆Ω p in Lp (Ω) for some planar domains Ω. We start with horns, i.e. regions Ω ⊂ R2 which are
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given by Ω := {(x, y) ∈ R2 : x > 0, ϕ1 (x) < y < ϕ2 (x)}
(1)
where ϕ1/2 : [0, ∞) → R are Lipschitz continuous functions satisfying ϕ1 (x) < ϕ2 (x) for all x ≥ 0. Further assumptions on ϕ1/2 will make Ω really look like a horn, see (2) below. Ω In many cases, σ(∆Ω p ) will coincide with σ(∆2 ). However, the results on p-independence of the spectrum using Gaussian bounds ([8], [1], [3], [9]) or weighted norm estimates ([13], [11]) cannot be applied here, since they would imply an embedding H 1 (Ω) → Lq (Ω) for some q > 2 which typically fails to hold for horn-shaped regions. On the other hand, the examples in [10] lead to the conjecture that the spectrum of ∆Ω p depends on p for ϕ1/2 (x) = ±e−x . We analyse in this paper the influence of the boundary curves ϕ1/2 on the spectrum of ∆Ω p in Lp (Ω), and show in particular that the above conjecture is true. The Neumann Laplacian in L2 of horn-shaped regions has been considered in [4] and we use the idea of reducing the problem to operators in Lp -spaces over (0, ∞). The new feature is that, although we might not have Gaussian bounds for the semigroup generated by ∆Ω 2 , we may use techniques involving (generalized) Gaussian estimates for the operators on (0, ∞). Since we are mainly interested in Lp for p = 2 we confine ourselves Ω to the essential spectrum rather than to treat σac (−∆Ω 2 ) and σsing (−∆2 ) in L2 which has been done in [4]. We exclude the case p = 1 which has its own difficulties (see Remark 19). We use the following definition of the essential spectrum (which is σe3 in [5]): λ ∈ σess (A) if and only if λ − A ∈ F where F is the set of all Fredholm operators, i.e. all operators having finite-dimensional kernel and (closed) finite-codimensional range. We use the following notation for measurable functions f : (0, ∞) → R: For 1 ≤ p, q ≤ ∞, we write f ∈ lq (Lp ) if f lq (Lp ) := (f 1[n,n+1] Lp )n∈N0 lq < ∞ and f ∈ c0 (Lp ) if limn→∞ f 1[n,n+1] Lp = 0. For 1 < p < ∞ we denote Qp := {ξ + iη : ξ ≤
1 1 η2 − }, − p2 p 4(p−1 − 2−1 )2
for p ∈ (1, ∞) \ {2} and Q2 := (−∞, −1/4], and for α ∈ R we denote Qp,0 := (−∞, 0] if α = 0 and Qp,α = α2 Qp if α = 0. Our basic assumptions for the boundary curves are that, with ϕ := ϕ2 − ϕ1 , lim ϕ(x) = 0 and ϕ1 , ϕ2 ∈ c0 (L∞ ). (2) x→∞
Then our first main result reads as follows.
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Theorem 1. Assume (1), (2), and ϕ /ϕ − α ∈ c0 (L2 ) Then, for 1 < p < ∞, Moreover
for some α ∈ R.
(3)
σess (∆Ω p ) = ∂Qp,α .
0 , λ ∈ Qp,α 1 , λ ∈ int (Qp,α ), 1 < p < 2 . ind(λ − ∆Ω p)= −1 , λ ∈ int (Qp,α ), p > 2
Remark 2. (i) The assumption ϕ /ϕ − α ∈ c0 (L2 ) means that ϕ (x)/ϕ(x) tends (in a weak sense) to a limit as x → ∞. The simplest case is of course ϕ(x) = eαx where α < 0. If α = −1 we get σess (∆Ω p ) = ∂Qp . (ii) It is clear that, in the situation of Theorem 1, σ(∆Ω p ) \ Qp,α is a discrete set of eigenvalues. A simple argument using spectral projections (see, e.g., the proof of [2, Cor. 1.6.2]) shows that this set does not depend on p ∈ (1, ∞) and is hence contained in (−α2 /4, 0]. For α = 0 this means that the spectrum σ(∆Ω p ) itself does not depend on p. For α < 0 we always Ω have 0 ∈ σ(∆p ) and 1Ω is an eigenfunction. It seems not to be clear if there may be other eigenvalues in (−α2 /4, 0). (iii) The behaviour of the Neumann Laplacian on horns resembles the behaviour of the Laplacian on certain Riemannian manifolds, in particular in hyperbolic space, see [2, Sect. 5.7] where the same parabola appears and the remark on [4, p. 107]. Note that, for φ1/2 (x) = ±eαx /2, α < 0, the volume growth m(r) in a uniform estimate |B(x, r)| ≤ m(r)|B(x, 1)| is sinh(|α|r)/ sinh(|α|) if we take B(x, r) as max-norm balls in Ω. Hence (3) might be interpretated as an “asymptotic” assumption on the volume growth. Under certain assumptions on the Ricci-curvature of the manifold it has been shown in [14] that the spectrum of uniformly elliptic operators depends on p. We use the same method of proof to treat domains which look like comets or stars, i.e. which have several horn-shaped outlets. Let Ω0 be a domain in R2 , the “basis” to which we will attach the horns or rather “rays”. Let (Rn )n∈J be a countable (finite or infinite) collection of mutually disjoint “rays” Rn , each one congruent to a “horn” Hn := {(x, y) : x > 0, |y| < ϕn (x)/2} by a mapping Sn , i.e. Rn = Sn (Hn ), where ϕn : [0, ∞) → (0, ∞) satisfies (2). We assume that Rn ∩ Ω0 = ∅ for n ∈ J, and use the following notation: For each 0 < L ≤ ∞ we define RnL := 0 Sn ({(x, y) : 0 ≤ x < L, |y| ≤ ϕn (x)/2}) and Rn := L>0 RnL (hence, e.g., Rn∞ = Rn0 ∪ Rn ). We require Rn0 ⊂ ∂Ω0 so that we can attach the rays
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Rn∞ to the basis Ω0 , and define Ω := Ω0 ∪ n∈J Rn∞ . Hence Ω resembles a star or a comet depending on the directions of the rays Rn . We shall require in addition that the (in some sense) “finite parts” of Ω behave well with respect to the embedding H 1 → L2 . To be precise, we assume for any finite subset F ⊂ J and any finite sequence L := (Ln )n∈F of positive real numbers, H 1 (ΩL ) → L2 (Ω) is compact the embedding L n where ΩL := Ω0 ∪ n∈F Rn . (4) Hence ΩL is the union of the basis Ω0 with a finite collection of truncated rays. If the index set J is infinite then J = N without loss of generality, and we require the following uniformity in (2) lim ϕn ∞ + ϕn ∞ = 0
n→∞
(5)
in order to be able to approximate by growing finite collections of rays. We are then able to do a similar reduction of the problem of determining the essential spectrum, this time to an lp -direct sum of weighted Lp -spaces on (0, ∞). For finite J this leads to Theorem 3. Assume in addition to the above that J is finite and (α(n))n∈J is such that ϕn /ϕn − α(n) ∈ c0 (L2 ) for all n ∈ J. Then
σess (∆Ω ∂Qp,α(n) . p)= n∈J
For λ ∈
σess (∆Ω p)
we have
ind(λ − ∆Ω p ) = εp · #{n : λ ∈ int (Qp,α(n) )} where εp = 1 if 1 < p < 2 and εp = −1 if 2 < p < ∞. If J is not finite then the description of σess (∆Ω p ) is in general not as simple (cf. Remark 17), but it is clear how to obtain Neumann Laplacians with “maximal” Lp -spectrum: As mentioned above ∆Ω 2 is the generator of a submarkovian semigroup, i.e. we have consistent positive contraction semigroups (Tp (t)) on Lp (Ω), 1 < p < ∞. A result of Liskevich and Semenov ([12]) states that (Tp (t)), 1 < p < ∞, is a holomorphic semigroup in Lp (Ω) with angle θp ∈ [0, π/2] where cos θp = |1−2/p|. Thus σ(∆Ω p)⊂ Sp always, where Sp is the closed sector with vertex 0 and half opening angle π/2−θp , symmetric to the negative real axis. Hence the Neumann Laplacian in the following theorem has maximal spectrum in Lp . Theorem 4. There is a planar domain Ω of finite measure such that Ω σ(∆Ω p ) = σess (∆p ) = Sp .
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Remark 5. J. Voigt showed in [15] that the angle of holomorphy θp is optimal for the class of generators A of submarkovian semigroups by constructing an example for which σ(Ap ) = Sp . The operator is y(d/dx)x2 (d/dx) in Lp ((0, ∞)2 ). In [10] the author constructed a uniformly elliptic, purely second order operator A in divergence form with C ∞ -coefficients on a planar domain satisfying σ(Ap ) = Sp which showed that θp is optimal for uniformly elliptic operators in Euclidian space. Theorem 4 shows that θp is even optimal for the class of Neumann Laplacians in Euclidian space, and that the Lp -spectrum of Neumann Laplacians is in general not smaller than predicted by the theory of generators of submarkovian semigroups. The paper is organized as follows. In Sect. 2 we reduce the problem of determining the essential spectrum in Lp of Neumann Laplacians on horns to a problem in a weighted Lp -space over (0, ∞). This is basic for all our proofs. In Sect. 3 we use the result of Sect. 2 to prove Theorem 1 by reducing to a constant coefficient operator on L2 (0, ∞). The keystone is a perturbation argument (Lemma 13). In Sect. 4 we prove a reduction result for comets and stars (Proposition 16) that relies on estimates obtained in Sect. 2, and we then use the result of Sect. 3 to prove Theorem 3. In Sect. 5 we prove Theorem 4. We close this work with some remarks on our method of proof in Sect. 6. The author wants to thank J. Voigt for a helpful discussion on an early version of this work and for the encouragement to treat comets and stars. 2 Neumann Laplacians on horns In this section we use ideas from [4] and reduce the problem of determining the essential spectrum of ∆Ω p to problems for operators in Lp -spaces over (0, ∞). Throughout this section 1 < p < ∞. We shall use the following notations. The space Lp ((0, ∞), dx) where dx means Lebesgue measure is denoted by Lp (dx) and the weighted Lp space Lp ((0, ∞), ϕ(x) dx) is denoted by Lp (ϕ dx). We define the isometric injection Jp : Lp (ϕ dx) → Lp (Ω) by (Jp u)(x, y) := u(x). Then its adjoint Jp : Lp (Ω) → Lp (ϕ dx) is given by (Jp g)(x)
1 = ϕ(x)
ϕ2 (x)
ϕ1 (x)
g(x, y) dy.
We also define Pp := I − Jp Jp : Lp (Ω) → Lp (Ω) where I denotes the identity. Note that Jp , Jp and Pp are consistent on all Lp , 1 ≤ p < ∞.
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Occasionally we shall drop the index p and just write J, J and P . Observe that JJ is an orthogonal projection in L2 (Ω). We define the form ¯bϕ in L2 (ϕ dx) by ∞ ¯bϕ (u, v) := u (x)v (x)ϕ(x) dx (6) 0
with form domain Vϕ := {u ∈ L2 (ϕ dx) : u ∈ L2 (ϕ dx), u(0) = 0}. The associated operator is then Bϕ u :=
1 ϕ (ϕu ) = u + u ϕ ϕ
with boundary condition u(0) = 0. By Beurling-Deny the operator Bϕ is the generator of a submarkovian semigroup. For 1 < p < ∞, we denote its generator in Lp (ϕ dx) by Bϕ,p or simply by Bp if ϕ is clear from the context. We now do the announced reduction to a problem in Lp (ϕ dx). Here we shall need the following Lp -versions of results in [4]. The difference is that we reduce only to Lp (ϕ dx) whereas in [4], for p = 2, the reduction is done to L2 (dx) via the canonical isometry L2 (dx) → L2 (ϕ dx) and requires more smoothness of ϕ. For the convenience of the reader and since we shall later refer to the precise estimates obtained in the proof we give full details here. Proposition 6. (i) The operator Kp := R(1, ∆Ω p )Jp − Jp R(1, Bp ) : Lp (ϕ dx) → Lp (Ω) is compact. 1/2 P and P R(1, ∆Ω )1/2 are compact in (ii) The operators R(1, ∆Ω p p p) p Lp (Ω). Proof. (i): Clearly, Kp is bounded for all 1 ≤ p < ∞. By [2, Theorem 1.6.1] it rests to show compactness for p = 2. We shall drop the index 2 in the proof. We have K = R(1, ∆Ω )J − JR(1, B) = R(1, ∆Ω )(JB − ∆Ω J)R(1, B), and use arguments of the proof of [4, Prop. 2.1]. For u ∈ Cc∞ (0, ∞) and g ∈ H 1 (Ω) =: V we have u ∈ D(Bϕ ), Ju ∈ 1 H (Ω) = V , and ∆Ω Ju ∈ V (here ∆Ω : V → V means the operator corresponding to the form n ¯ in the Gelfand triple V → L2 (Ω) → V ). We obtain |(g, (JBϕ − ∆Ω J)u)|
Lp -spectral properties of the Neumann Laplacian on horns, comets and stars
189
ϕ2 ϕ2 u (ϕ−1 g¯x dy − (ϕ−1 g¯ dy) )ϕ dx| ϕ1 ϕ1 0 ∞ ϕ2 ϕ ϕ2 ϕ1 ≤ − g¯(·, ϕ1 ) ϕ dx |u | − 2 g¯ dy + g¯(·, ϕ2 ) ϕ ϕ1 ϕ ϕ 0 2 ∞ ϕj ϕ2 ≤ |u | 2 g¯(·, y) − g¯(·, ϕj ) dy ϕ dx ϕ ϕ1 0
=|
∞
j=1
≤
2 j=1
=
2
∞
0
|u
||ϕj |
ϕ2
ϕ1
|gy | dy dx
(J(|u ||ϕj |), |gy |)
j=1
≤ u (|ϕ1 | + |ϕ2 |)L2 (ϕ dx) (1 − ∆Ω )1/2 gL2 (Ω) , where we used t 2 ∇g ∇g + |g|2 d(x, y) = ((1 − ∆Ω )1/2 g, (1 − ∆Ω )1/2 g) gy L2 (Ω) ≤ Ω
(7)
in the last step. Hence (1 − ∆Ω )−1/2 (JBϕ − ∆Ω J)uL2 (Ω) ≤ (|ϕ1 | + |ϕ2 |) · u L2 (ϕ dx) , (8) and, since ϕ1/2 ∈ L∞ and Cc∞ (0, ∞) is a form-core, the map can be continuously extended Vϕ → L2 (Ω). Choose a smooth function χ such that χ = 1 on (−∞, −1) and χ = 0 on (0, ∞) and let χn := χ(· − n). Then Kn := R(1, ∆Ω )(JB − ∆Ω J)χn R(1, B) is compact since we can factorize χn R(1, B) : L2 (ϕ dx) → Vϕ through a restriction of the compact embedding W22 (0, n) → W21 (0, n). Using (8) we estimate the norm of (K − Kn )u by ess sup (|ϕ1 (x)| + |ϕ2 (x)|) · ((1 − χn )R(1, B)u) L2 (ϕ dx) x≥n−1
and the last norm by (R(1, B)u) L2 (ϕ dx) +χ ∞ R(1, B)uL2 (ϕ dx) ≤ (1+χ )uL2 (ϕ dx) where we use an argument similar to (7) in the last step. Then (2) gives K − Kn → 0 as n → ∞. Hence K is compact. (ii): Note that the operators are adjoints of each other in L2 (Ω). By the following they are compact in L2 (Ω). Since they are bounded in all Lp (Ω), 1 ≤ p < ∞, interpolation (see [2, Theorem 1.6.1]) gives the result.
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We want to show first that the embedding H 1 (Ω) ∩ ker J → L2 (Ω) is compact (which has implicitely been shown in the proof of [4, Theorem 3.1]). Let I ⊂ (0, ∞) be an interval and Ω I := {(x, y) ∈ Ω : x ∈ I} = {(x, y) : x ∈ I, ϕ1 (x) < y < ϕ2 (x)} for the moment. If I is finite then the embedding H 1 (Ω I ) → L2 (Ω) is compact since the boundary of the bounded domain Ω I is Lipschitz. In the general case we obtain for u ∈ H 1 (Ω I ) ∩ ker J by partial integration ϕ2 ϕ2 y |u|2 = uu dydx = − u dη uy dydx. ΩI
I
ϕ1
I
ϕ1
ϕ1
By Cauchy-Schwarz and the elementary inequality ab ≤ (a2 + b2 )/2 this implies ϕ2 ϕ2 2 |u| d(x, y) ≤ ( |u|2 dη)1/2 ϕ1/2 |uy | dydx ΩI I ϕ1 ϕ1
ϕ2 ϕ2 1 2 2 ϕ |u| dy + ϕ |uy | dydx ≤ 2 I ϕ1 ϕ1 1 ≤ · ess sup ϕ · (u22 + ∇u22 ). 2 I By (2) the norms of the embeddings H 1 (Ω (n,∞) ) ∩ ker J → L2 (Ω) tend to 0 as n → ∞. A standard argument using cut-off functions shows that the embedding H 1 (Ω) ∩ ker J → L2 (Ω) is compact. Now im P = ker J and P leaves invariant H 1 (Ω). Indeed, letting g ∈ 1 H (Ω) we have ∂y JJ g = 0 and using the arguments in the proof of (i) we have ϕ2 −1 ∂x JJ g = ∂x (ϕ g dy) =ϕ
−1
ϕ1
ϕ2
ϕ1
gx dy +
2
j−1
(−1)
j=1
ϕj ϕ2
ϕ2
ϕ1
y
ϕj
gy (·, η) dydx.
This implies by JJ ≤ 1 that 2 ∂x JJ g2 ≤ JJ gx 2 + ( ϕj ∞ )JJ |gy |2 j=1
≤ max(1,
2
ϕj ∞ ) (gx 2 + gy 2 ) .
j=1
Hence JJ leaves H 1 (Ω) invariant, and the same holds for P = I − JJ .
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This means that we can factorize P (1 − ∆Ω )−1/2 through the compact embedding H 1 (Ω) ∩ ker J → L2 (Ω), and (ii) is proved. We are now ready to prove Proposition 7. We have σess (Bp ) = σess (∆Ω p ). Proof. The operators P and JJ = I − P are projections in Lp (Ω) for 1 < p < ∞. Let R := R(1, ∆Ω p ). We have the direct sum decomposition Lp (Ω) = ker P ⊕ im P and λ − R takes the form
λ − (I − P )R (I − P )RP : ker P ⊕ im P → ker P ⊕ im P. P R(I − P ) λ − PR By Proposition 6(ii) the operators (I − P )RP , P R(I − P ) and P R are compact. Thus λ − R is a compact perturbation of the operator
λ − (I − P )R 0 . 0 λ Hence λ − R ∈ F(Lp (Ω) if and only if λ − (I − P )R ∈ F(ker P ). On ker P we have (λ − (I − P )R) = (I − P )(λJ − RJ)J = J(λ − R(1, Bp ) − J Kp )J which implies by the compactness of Kp that λ − (I − P )R ∈ F(im P ) if and only if λ − R(1, Bp ) ∈ F(Lp (ϕ dx)) since J and J are isometries Lp (dx) ↔ ker P = im J. Hence σess (∆Ω p ) = σess (Bp ) by the spectral mapping theorem for the essential spectrum (see [5], Theorem IX.2.3(iii)). 3 Proof of Theorem 1 We fix p ∈ (1, ∞), take (6) and denote the corresponding operator by Bp when considered in Lp (ϕ dx). By Proposition 7 we only have to determine σess (Bp ). By symmetry it is sufficient to treat p ∈ (1, 2]. In several steps the task will be reduced to a constant coefficient operator in Lp (dx). The logic of proof, however, requires to do the steps backward. Hence we start in Lp (dx) with the constant coefficient operator
1 2 1 2 u +α u Ap,α u := u + α 1 − − p p2 p with Dirichlet boundary condition u(0) = 0. For α = 0, the operator (0,∞) A0 := Ap,0 is independent of p, and we have σess (A0 ) = σess (∆D ) =
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(−∞, 0]. For α < 0, the operator Ap,α is similar to the operator α2 Ap,−1 via the scaling isomorphism f → f (|α|(·)). Hence the spectrum scales with a factor α2 and we only have to calculate it for α = −1 which we shall drop in notation from now on, i.e. Ap := Ap,−1 . Clearly, A2 = A0 − 1/4, hence σess (A2 ) = (−∞, −1/4]. Lemma 8. If 1 < p < 2 then σess (Ap ) = ∂Qp . For λ ∈ Qp we have ind(λ − Ap ) = 0, and for λ ∈ int (Qp ) we have ind(λ − Ap ) = 1. Proof. If λ ∈ C \ {−1/4} then Ap u − λu = 0 has two linearly independent −1 −1 solutions, given by exp(µ± (·)) where µ± = 2 − p ± λ + 1/4, and we take the root such that its real part is ≥ 0. For λ ∈ Qp we have Reµ+ > 0 and Reµ− < 0 (recall 1 < p < 2), and the resolvent R(λ, Ap ) is an integral operator with kernel 1 µ− x (e−µ− y − e−µ+ y ) , if 0 ≤ y ≤ x µ+ −µ− e kλ (x, y) = . (9) 1 µ+ x − eµ− x )e−µ+ y , if y > x µ+ −µ− (e Noticing that 0 > −Reµ+ > Reµ− , we obtain the following estimate |kλ (x, y)| ≤
2 e−|Reµ+ ||x−y| =: gλ (x − y) |µ+ − µ− |
(x, y > 0). (10)
By gλ 1 < ∞ we get λ ∈ ρ(Ap ). For λ ∈ int (Qp ) we have Reµ± < 0 and 1 (eµ+ x − eµ− x ) (x ≥ 0) vλ (x) := µ+ − µ− defines an eigenfunction vλ ∈ Lp (dx) of Ap to the eigenvalue λ. For λ = −1/4 (which is equivalent to µ+ = µ− ) we let v−1/4 (x) := xeµ+ x which is an eigenfunction. In any case, the eigenspace has dimension 1, and for g ∈ Lp (dx) the function h := vλ ∗ g is a solution of (Ap − λ)h = g. Hence ind(λ − Ap ) = 1. Since the index is constant on connected components this finishes the proof of the lemma. Remark 9. For later purposes we note that, for λ ∈ int (Qp ), the operator R(λ) given by R(λ) := vλ ∗ g defines a continuous linear right inverse for Ap − λ, and that we have the estimate |vλ (x)| ≤ gλ (x)
(x ≥ 0)
(11)
where gλ is as in (10) (but now Reµ+ < 0). We also define R(λ) := R(λ, Ap ) for λ ∈ Qp . Recall that Qp,0 := (−∞, 0], and, for α = 0, Qp,α = α2 Qp . The scaling argument above then gives
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Corollary 10. If 1 < p ≤ 2 then σess (Ap,α ) = ∂Qp,α . For λ ∈ Qp,α we have ind(λ − Ap,α ) = 0, and for λ ∈ int (Qp,α ) we have ind(λ − Ap,α ) = 1. The weight ϕ contains a factor eα(·) and a multiplicative rest. We deal with the rest first. Let ψ(x) := − log ϕ(x) + αx. Then −ψ = ϕ /ϕ − α ∈ c0 (L2 ), and ϕ = e−ψ eα(·) . We consider the operator Ap,α in Lp (e−ψ dx), i.e. we define Ap,α,ψ to be the operator in Lp (e−ψ dx) whose resolvents for large λ are consistent with R(λ, Ap ). The following lemma shows in particular that this is possible. Lemma 11. For λ ∈ ∂Qp,α the operator R(λ) acts as a bounded operator in Lp (e−ψ dx). Proof. We show first that the weight ρ := e−ψ satisfies the following subexponential growth condition: for all ε > 0 exists Cε > 0 such that ρ(x) ≤ Cε eε|x−y| ρ(y)
(x, y > 0).
(12)
Clearly (12) is equivalent to the following: For all ε > 0 exists cε ∈ R such that |ψ(x) − ψ(y)| ≤ ε|x − y| + cε (x, y > 0). (13) Now ψ is absolutely continuous. If 0 < x < y and j, k ∈ N0 such that j ≤ x < j + 1 and k − 1 < y ≤ k then y k |ψ(x) − ψ(y)| ≤ |ψ (ξ)| dξ ≤ |ψ | dξ x
≤
k−1
j
ψ 1[ν,ν+1] L2
(14)
ν=j
≤ (y − x + 2)
max
ν=j,...,k−1
ψ 1[ν,ν+1] 2 .
(15)
By ψ ∈ c0 (L2 ) there is, for given ε > 0, an integer j(ε) such that ψ 1[ν,ν+1] 2 ≤ ε for ν ≥ j(ε). For y > x ≥ j(ε) we get by (15) that |ψ(y) − ψ(x)| ≤ ε|y − x| + 2ε. For j(ε) ≥ y > x we have by (14) that j(ε) |ψ(y) − ψ(x)| ≤ ν=0 ψ 1[ν,ν+1] 2 =: C(ε), and for y > j(ε) > x we thus have |ψ(y) − ψ(x)| ≤ |ψ(y) − ψ(j(ε))| + |ψ(j(ε)) − ψ(x)| ≤ ε|y − j(ε)| + 2ε + C(ε) ≤ ε|y − x| + 2ε + C(ε) which proves (13) with cε := 2ε + C(ε). Putting together (10), (11), and (12) we see that ρ1/p R(λ)ρ−1/p acts as a bounded operator in Lp (dx). Since f → ρ−1/p f is an isometry Lp (dx) → Lp (ρ dx) the claim is proved.
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It is clear that R(λ), λ ∈ / Qp,α , is the resolvent of a closed linear operator Ap,α,ψ in Lp (e−ψ dx). By the arguments in the proof of Lemma 8 we get Corollary 12. If 1 < p ≤ 2 then σess (Ap,α,ψ ) = ∂Qp,α . For λ ∈ Qp,α we have ind(λ−Ap,α,ψ ) = 0, and for λ ∈ int (Qp,α ) we have ind(λ−Ap,α,ψ ) = 1. In the next step we perturb Ap,α,ψ by the operator
ϕ α ϕ α −α u − − α u = −ψ u + ψ u. u → ϕ p ϕ p In the last step we will then pass from Lp (e−ψ dx) to Lp (ϕ dx) via the canonical isometry. Lemma 13. The operator Cp,α,ψ , given by Cp,α,ψ u := −ψ u + (α/p)ψ u, is a relatively compact perturbation of Ap,α,ψ for 1 < p ≤ 2. Proof. Dropping all indices we denote Cp,α,ψ simply by C and we denote the resolvent R(1, Ap,α,ψ ) by R. We have to show that T := CR is compact. For n ∈ N we let Tn := Cχn R where χn := χ(· − n) and χ is smooth with χ = 1 on (−∞, −1) and χ = 0 on (0, ∞). Then χn R : Lp (e−ψ dx) → Wp2 (0, n) is continuous, the embedding Wp2 (0, n) → Wp1 (0, n) is compact, and C : Wp1 (0, n) → Lp (e−ψ dx) is continuous (recall that ψ = α − ϕ /ϕ ∈ L∞,loc since ϕ is Lipschitz continuous). Hence each Tn is compact. We want to estimate the norm of T − Tn as an operator in Lp (e−ψ dx) which is the same as the norm of e−ψ/p C(1 − χn )Reψ/p as an operator in Lp (dx) since f → eψ/p f is an isometry Lp (dx) → Lp (e−ψ dx). For a function v we have α C(1 − χn )v = −ψ ((1 − χn )v) + ψ (1 − χn )v p α = ψ χn v − (1 − χn )(ψ v − ψ v). p Thus we see that we have to estimate ψ χn Sp→p and (1 − χn )ψ Sp→p for S ∈ {e−ψ/p Reψ/p , e−ψ/p (d/dx)Re−ψ/p } where · p→p means the norm for operators Lp (dx) → Lp (dx). By (9) R is given by a kernel k satisfying
|k(x, y)| + |∂x k(x, y)| ≤ c e−δ |x−y|
(x, y ≥ 0).
By (12) this means that S is given by a kernel m satisfying |m(x, y)| ≤ ce−δ|x−y|
(x, y ≥ 0).
(16)
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Let p−1 = 2−1 + r−1 . By (16) and Young’s inequality we get Sp→r < ∞ which by H¨older and ψ ∈ c0 (L2 ) implies ψ χn Sup ≤ c(χ)ψ 1(n−1,n) 2 Sur
≤ c(χ)ψ 1(n−1,n) 2 Sp→r up −→ 0
(n → ∞)
uniformly in up ≤ 1. We now use an argument similar to the one that proved [13, Prop. 3.2]. For j, k ∈ N0 and u ∈ Lp (dx) with support in [k, k + 1] we get by exp(−δ|x − y|) ≤ exp(2δ − δ|j − k|) for x ∈ (j, j + 1), y ∈ (k, k + 1), and (16) that 1(j,j+1) Sur ≤ 1(j,j+1) Su∞ ≤ ce2δ e−δ|j−k| u1 ≤ ce2δ e−δ|j−k| up . Hence, using H¨older again and denoting βn := supj≥n ψ 1(j,j+1) 2 , we obtain (1 − χn )ψ Supp = (1 − χn )ψ 1(j,j+1) Supp j≥n−1
≤ c(χ)
j≥n−1
≤ c(χ)βn−1
ψ 1(j,j+1) p2 1(j,j+1) Supr
j≥0
≤ c(χ)βn−1 cp e2δp
k≥0
j≥0
p 1(j,j+1) S(1(k,k+1) u)r
p e−δ|j−k| 1(k,k+1) up .
k≥0
Since, by the discrete Young inequality, (xk )k≥0 → ( k≥0 exp(−δ|j − k|)xk )j≥0 is a bounded operator in lp with norm, say, Mδ < ∞, we conclude that (1 − χn )ψ Spp→p ≤ c(χ)βn−1 cp e2δp Mδp which, by ψ ∈ c0 (L2 ) and the definition of βn , tends to 0 as n → ∞. Hence we have shown that T − Tn → 0 in the norm of operators in Lp (e−ψ dx), and T is compact. We define Bp,α := Ap,α,ψ + Cp,α,ψ . Then
2 ϕ u + − α u Bp,α u = u + α 1 − p ϕ
α ϕ 1 1 2 u− −α u +α − p2 p p ϕ
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with boundary condition u(0) = 0 in Lp (e−ψ dx). This operator is similar to the operator Bp in Lp (ϕ dx) via the isometry Lp (e−ψ dx) → Lp (ϕ dx), f → exp(−(α/p)(·))f . Hence Corollary 12 and Lemma 13 give Corollary 14. If 1 < p ≤ 2 then σess (Bp ) = ∂Qp,α . For λ ∈ Qp,α we have ind(λ − Bp ) = 0, and for λ ∈ int (Qp,α ) we have ind(λ − Bp ) = 1. which together with Proposition 7 proves Theorem 1 if we recall that Ω Ω Ω σess (∆Ω p ) = σess (∆p ) and ind(λ − ∆p ) = −ind(λ − ∆p ) for p > 2. 4 Neumann Laplacians on comets and stars Following the strategy of Sect. 2 we reduce the problem of determining the essential spectrum of ∆Ω 2 in Lp (Ω), but now to an lp -direct sum of weighted Lp -spaces on (0, ∞) corresponding to the family of rays (Rn )n∈J . To this purpose let Xp := lp − Lp (ϕn dx) n∈J
for 1 < p < ∞, i.e. Xp is the space of all sequences (un )n∈J such that un ∈ Lp (ϕn dx), n ∈ J, and the following norm 1/p 1/p ∞ p (un )Xp := un Lp (ϕn dx) = |un |p ϕn dx n∈J
n∈J
0
is finite. Let Jp : Xp → Lp (Ω) be the isometric injection given by un (x) , (ξ, η) = Sn (x, y) ∈ Rn . Jp (un )(ξ, η) := 0 elsewhere Define the operator B in X2 by the form ¯b((un ), (vn )) :=
n∈J
0
∞
un vn ϕn dx
with form domain V := l2 − n∈J Vn where Vn := {u ∈ L2 (ϕn dx) : u ∈ L2 (ϕn dx), u(0) = 0}, i.e. V is the space of all sequences (un ) such that un ∈ Vn , n ∈ J, and 1/2 un 2L2 (ϕn dx) + un 2L2 (ϕn dx) < ∞. (un )V := n∈J
The operator B is given by B(un ) = (un +
ϕn u ) =: (Bn un ) ϕn n
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with boundary conditions un (0) = 0, n ∈ J. By Beurling-Deny, B is the generator of a submarkovian semigroup and we hence generators Bp for 1 < p < ∞. Observe that the direct sum have ∞ D := Cc (0, ∞) is a form core and a subset of each D(Bp ), and that Jp maps D into V . We want to proceed as in Sect. 2 and prove the following analogue of Proposition 6 where Pp means – as before – the map I − Jp Jp : Lp (Ω) → Lp (Ω). Proposition 15. (i) The operator Kp := R(1, ∆Ω p )Jp −Jp R(1, Bp ) : Xp → Lp (Ω) is compact. 1/2 P and P R(1, ∆Ω )1/2 are compact in (ii) The operators R(1, ∆Ω p p p) p Lp (Ω). Proof. (i) For (un ) ∈ D one can show as in the proof of Proposition 6 (i) that (1 − ∆Ω )−1/2 (JB − ∆Ω J)(un )L2 (Ω) ≤ (un |ϕn |)X2 .
(17)
Since D is a form core, this means that (1 − ∆Ω )−1/2 (JB − ∆Ω J) : V → L2 (Ω) is bounded. The case of finite J being simpler we prove the case J = N. As before we take a smooth function χ such that χ = 1 on (−∞, −1) and χ = 0 on (0, ∞). For each m ∈ N, we define the operator χ ¯m on X2 by χ(· − m) n ≤ m . (18) χ ¯m ((un )) := (χm,n un ) where χm,n := χ n>m Then, for each m ∈ N, the operator Km := R(1, ∆Ω )1/2 (JB − ∆Ω J)χ ¯m R(1, B) is compact since we can factorize through 1a restriction of the compact embedding n≤m W22 (0, m) → n≤m W2 (0, m). Using (17) we estimate the norm of (K − Km )(un ) by max( sup ess sup |ϕn (x)|, sup ϕ ∞ )·((1−χm,n )R(1, Bn )un ) )X2 n≤m
x≥m−1
n>m
and the last norm as before by (1 + χ ∞ ) · (un )X2 . By (5) we have limm K − Km = 0, hence K is compact. (ii) If we have shown that the embedding H 1 (Ω) ∩ ker J → L2 (Ω) is compact we can finish the proof just as the proof of Proposition 6 (ii). We fix further notation. For any open interval I := (a, b) ⊂ (0, ∞) we define RnI := Sn ({(x, y) : x ∈ I, |y| < ϕn (x)/2}). For a family I := (In )n∈J of such intervals we define
RnIn . Ω I := n∈J
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Then – by the same proof as before – we obtain for g ∈ H 1 (Ω I ) ∩ ker J the estimate 1 (19) g2L2 (Ω) ≤ sup(sup ϕn )g2H 1 (Ω I ) . 2 n In We now define, for any m ∈ N, the families L(m) := (Lm,n ) and I(m) := (Im,n ) by m, n ≤ m (m, ∞), n ≤ m and Im,n := . Lm,n := 0, n > m (0, ∞), n > m Then each embedding H 1 (ΩL(m) ) → L2 (Ω), m ∈ N, is compact by assumption (4). By (19) and (5) we obtain that the norms of the embeddings H 1 (Ω I(m) ) ∩ ker J → L2 (Ω) tend to 0 as m → ∞. Using a standard cutoff function argument we conclude that the embedding H 1 (Ω) ∩ ker J → L2 (Ω) is compact. As mentioned before the rest of the proof of (ii) is the same as in Sect. 2. The following is the analogue of Proposition 7 and is proved as before. Proposition 16. For all 1 < p < ∞, we have σess (∆Ω p ) = σess (Bp ). It is clear that σess (Bp ) ⊃ n∈J σess (Bn,p ) and that equality holds if J is finite. By the results of Sect. 3 we hence have proved Theorem 3. Remark 17. If J is not finite then σess (Bp ) also contains accumulation points of the discrete spectra of the operators Bn,p , n ∈ J. Moreover it also contains points λ ∈ n∈J\F ρ(Bn,p ), where F ⊂ J is finite, for which (R(λ, Bn,p ))n∈J\F is not uniformly bounded. The latter do not exist if p = 2 but for p = 2 this seems not to be clear. 5 Neumann Laplacians with maximal Lp -spectrum In this section we use the results of Sections 3 and 4 to prove Theorem 4. We take an enumeration (α(n)) of the positive rational numbers and define β(n) := 2−n min(1, α(n), α(n)−1 ), and ϕn (x) := β(n)e−α(n)x . Then the ϕn satisfy (5), and we define Hn := {(x, y) : x > 0, |y| ≤ ϕn (x)/2}, n ∈ N. We take a sequence of rays (Rn ) with Rn congruent to Hn and a basis Ω0 of finite measure satisfying the assumptions in Sect. 1. We define Ω := Ω0 ∪ n Rn∞ . Then |Ω| ≤ |Ω0 | + n 2−n < ∞. By Proposition 16 and the result of Liskevich/Semenov we then have
Ω Sp ⊃ σ(∆Ω ) ⊃ σ (∆ ) ⊃ σ (B ) = ∂Qp,α(n) = Sp \ {0} ess ess n,p p p n
n
where the last equality follows from the fact that the set ∂Sp coincides with the union of two tangents starting from 0 and touching the parabola ∂Qp (cf. the similar argument in [15]). Since the (essential) spectrum is closed Ω we conclude that σ(∆Ω p ) = σess (∆p ) = Sp , and Theorem 4 is proved.
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6 Concluding remarks Remark 18. In [6] the Neumann Laplacian on generalized ridged domains, which include horn-shaped regions as a special case, has been investigated and a criterion has been given for the compactness of the resolvent in L2 (Ω). In the special case of horn-shaped regions it reads (cf. [4], p.106)
x ∞ −1 lim ϕ(ξ) dξ ϕ(ξ) dξ = 0. (20) x→∞
0
x
By [2], Theorem 1.6.3, the resolvent is then compact in Lp (Ω) for 1 < p < ∞, and the spectrum σ(∆Ω p ), which consists entirely of eigenvalues of finite multiplicity, and the eigenfunctions are independent of p. Observe that, for α < 0 and ϕ1/2 (x) := ±eαx , we have ϕ /ϕ = α and the limit in (20) equals α−2 . Hence it is natural to ask if σess (∆Ω p ) = ∂Qp,1/µ whenever the limit in (20) equals µ ∈ (0, ∞). It might be also be interesting to consider generalized ridged domains instead of horns. By the results of [6] we know 0 ∈ σess (∆Ω 2 ) if the limit in (20) is ∞. Remark 19. If ϕ1/2 (x) = ± exp(−x2 )/2 then ∆Ω 2 has compact resolvent in L2 (Ω) thus in all Lp (Ω), 1 < p < ∞. Hence σess (∆Ω p ) = σess (Bp ) = ∅ for 1 < p < ∞ where Bp u = u + ϕ /ϕu = u − 2xu in Lp (ϕ dx) with boundary condition u(0) = 0. The situation for p = 1 is different: ˜ 1 ) = {Re z ≥ 0} and that {Rez > 0} Section 4.3 in [2] shows that σ(H ˜ 1 u = −(1/2)u + xu consists of eigenvalues of multiplicity two where H 2 in L1 (exp(−x ) dx) which implies that {Re z < 0} consists of eigenvalues of B1 and hence {Re z = 0} ⊂ σess (B1 ). It is remarkable that in this example as well as for the Ω in Theorem 4 (see Sect. 5) there is no uniform volume growth estimate |B(x, r)| ≤ m(r)|B(x, 1)| with m(r) < ∞ for r large. Remark 20. Roughly speaking, the assumption ϕ /ϕ−α ∈ c0 (L2 ) in Theorem 1 means that ϕ (x)/ϕ(x) tends to a limit as x → ∞. If ϕ /ϕ is periodic, a reduction similar to that of Sect. 3 can be done to an operator Ap in L2 (dx) which has periodic coefficients, but it seems not to be so easy to compute the spectrum of Ap . Remark 21. The condition (12) was used in [9] where it was shown to be sufficient to guarantee σ(Ap,ρ ) = σ(A2 ) if A2 is the generator of a semigroup (T2 (t)) in L2 (dx) which satisfies Gaussian bounds and Ap,ρ denotes the generator of the semigroup (Tp,ρ (t)) in Lp (ρ dx) which is consistent with (T2 (t)). This is not sufficient to prove the assertions on σess and the index in Corollary 12 and we therefore have given a direct proof here. It
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would be interesting to know whether independence of the essential spectrum and the index hold in the general situation of Gaussian bounds if we assume (12) for the weight. Remark 22. The proof of Lemma 6 would of course have been easier if we had assumed the stronger condition that ϕ /ϕ − α belongs to c0 (L∞ ) instead of c0 (L2 ). In any case, the close relation to the problem of stability of the essential spectrum of differential operators under perturbation of their coefficients (see, e.g., [11]) is obvious. Remark 23. Proposition 16 is also interesting for p = 2 in the case that σess (Bn,p ) = ∅ for all n ∈ J (then σ(Bn,p ) does not depend on p ∈ (1, ∞)). If J is infinite then we conclude that σess (∆Ω 2 ) consists exactly of the accumulation points of the eigenvalues of the operators Bn,2 . This is very close in spirit to the decoupling result for comb-like domains in [7] which was used to give a solution to the inverse problem for the essential spectrum Is there, for each closed subset S ⊂ [0, ∞), a domain Ω such that σess (−∆Ω 2 ) = S? If there are functions ϕn such that λ2 (ϕn ) → ∞ whereas λ1 (ϕn ) can be arbitrarily arranged (where λj (ϕn ) denotes the j-th eigenvalue of −Bn,2 ) then one would have another solution of this problem which would be (together with its method of proof) different from the solution in [7]. References 1. W. Arendt, Gaussian estimates and interpolation of the spectrum in Lp , Differential and Integral Equations 7, 1153–1168 (1994) 2. E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge 1989 3. E.B. Davies, Lp spectral independence and L1 analyticity, J. London Math. Soc. 52, 177–184 (1995) 4. E.B. Davies, B. Simon, Spectral properties of the Neumann Laplacian of horns, Geom. Funct. Anal. 2, 105–117 (1992) 5. D.E. Edmunds, W.D. Evans, Spectral Theory and Differential Operators, Oxford University Press, Oxford 1987 6. W.D. Evans, D.J. Harris, Sobolev embeddings for generalized ridged domains, Proc. Lond. Math. Soc. 54, 141–175 (1987) 7. R. Hempel, L.A. Seco, B. Simon, The essential spectrum of Neumann Laplacians on some bounded singular domains, J. Funct. Anal. 102, 448–483 (1991) 8. R. Hempel, J. Voigt, The spectrum of a Schr¨odinger operator on Lp (Rν ) is pindependent, Comm. Math. Phys. 104, 243–250 (1986) 9. P.C. Kunstmann, Heat kernel estimates and Lp -spectral independence of elliptic operators, Bull. London Math. Soc. 31, 345–353 (1999)
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10. P.C. Kunstmann, On uniformly elliptic operators with maximal Lp -spectrum in planar domains, Arch. Math. 76, 377–384 (2001) 11. V.A. Liskevich, H. Vogt, On Lp spectra and essential spectra of second order elliptic operators, Proc. London Math. Soc. 80, 590–610 (2000) 12. V.A. Liskevich, Yu.A. Semenov Some problems on Markov semigroups, in Demuth, Michael (ed.) et al., Schroedinger operators, Markov semigroups, wavelet analysis, operator algebras, Berlin, Akademie Verlag. Math. Top. 11, 163–217 (1996) 13. G. Schreieck, J. Voigt, Stability of the Lp -spectrum of generalized Schr¨odinger operators with form small negative part of the potential, in “Functional Analysis” Proc. Essen 1991, Marcel-Dekker, New York 1994 14. K.-Th. Sturm, On the Lp -spectrum of uniformly elliptic operators on Riemannian manifolds, J. Funct. Anal. 118, 442–453 (1993) 15. J. Voigt, The sector of holomorphy for symmetric submarkovian semigroups, in Dierolf, Susanne (ed.) et al., Functional analysis, Berlin, de Gruyter. 449–453 (1996)
