Semi-classical edge states for the Robin Laplacian
aa r X i v : . [ m a t h . SP ] F e b SEMI-CLASSICAL EDGE STATES FOR THE ROBINLAPLACIAN
B. HELFFER AND A. KACHMAR
Abstract.
Motivated by the study of high energy Steklov eigen-functions, we examine the semi-classical Robin Laplacian. In thetwo dimensional situation, we determine an effective operator de-scribing the asymptotic distribution of the negative eigenvalues,and we prove that the corresponding eigenfunctions decay awayfrom the boundary, for all dimensions. Introduction
Motivation: Generalized Steklov eigenfunctions.
Let us consider an open bounded set Ω ⊂ R n with a smooth con-nected boundary Γ. Let − ∆ D be the Dirichlet Laplace operator onΩ with spectrum σ ( − ∆ D ). We fix a constant w ∈ R \ σ ( − ∆ D ). Forevery function ψ ∈ H / (Γ), we assign the unique function u = u w,ψ asfollows(1.1) − ∆ u = w u on M and u = ψ on Γ . The operator(1.2) ψ ∈ H / (Γ) Λ( w ) ψ := ∂u w,ψ ∂ν ∈ H − / (Γ)is the Dirichlet to Neumann (DN) operator. Here ν denotes the unitoutward normal vector of Γ.The DN operator is a boundary pseudo-differential operator of or-der 1. Its spectrum consists of a non-decreasing sequence of eigenvalues( µ m ( w )) m ≥ counting multiplicities, known as the (generalized) Stekloveigenvalues . More precisely, σ (Λ( w )) = σ s , Date : February 16, 2021. The Steklov eigenvalues correspond to the case where w = 0. where σ s is the Steklov spectrum defined as the set of real numbers µ such that a non-trivial solution u exists for the following Robin problem(1.3) − ∆ u = w u on Ω , u ∈ H (Ω) and ∂u∂ν = µ u on Γ . The study of the localization of the normalized solutions u µ of (1.3) inthe limit µ → + ∞ is connected with the semi-classical Robin Lapla-cian studied in [16].Let us formulate the Steklov problem in the framework of [16]. Weintroduce the semi-classical parameter h = µ − and denote by u h anon-trivial solution of (1.3); the eigenfunction u h satisfies(1.4) − ∆ u h = w u h in Ω ∂u h ∂ν = h − / u on Γ . We introduce the self-adjoint operator T h with domain D ( T h ) as follows T h = − h ∆ , D ( T h ) = { u ∈ H (Ω) : ∂u∂ν = h − / u on Γ } . Then (1.4) can be rewritten in the form(1.5) T h u h = w h u h , u h ∈ D ( T h ) \ { } , w h := h w . By [16], in the planar situation n = 2, if w h < u h decays exponentially as follows: Given M ∈ (0 , and α ∈ (0 , √ M ) , there exist h , C > such that (1.6) Z Ω (cid:0) | u h | + h |∇ u h | (cid:1) exp (cid:18) α d ( x, Γ) h / (cid:19) dx ≤ C k u h k L (Ω) , for h ∈ (0 , h ] and w h < − M h . Here d ( · , Γ) is the normal distance to the boundary(1.7) d ( x, Γ) = inf {| x − y | : y ∈ Γ } ( x ∈ R n ) . This decay is a consequence of Agmon type estimates. If we note thatthe ground state energy of the operator T h satisfies λ ( T h ) = − h + o ( h )as h → + , the theorem applies with α <
1. This decay result can beeasily extended to the n -dimensional situation [23] from which we candeduce pointwise estimates on u h (see Theorem 2.1).Examining the case of the annulus, Ω = { x ∈ R : r < | x | < } ,we observe that the constant α and the distance function d ( x, Γ) in(1.6) are non-optimal. The example of the annulus suggest the optimaldecay rate is achieved with α ≈ d Γ thatdepends on the curvature of the boundary (see [10, Sec. 1.1.3] and [6]). This amounts to the study of the Steklov eigenpair ( u µ m , µ m ) as m → + ∞ . EMI-CLASSICAL EDGE STATES FOR THE ROBIN LAPLACIAN 3
Returning to the problem in (1.3), we see that a consequence of(1.6) is that the Steklov eigenfunction decays away from the boundaryprovided the Steklov eigenvalue λ satisfies w ≤ − M λ and λ ≫ | w | ≥ M λ ≫ w . Thisquestion is motivated by the paper by Galkowski-Toth [10] (who alsorefer to Hislop-Lutzer [18] and Polterovich-Sher-Toth [24]) and by thePHD thesis of G. Gendron [11] discussing for special manifolds withboundary the correspondence between the spectrum of the Steklov andthe metric given on the manifold. In the first contribution, it is assumedthat w = 0, and the above decay is obtained with α = 1, but underthe condition that the boundary is analytic. Although not writtenexplicitly, the computations by G. Gendron can also lead to the sameresult (but for a particular case). This has been developed in the recentwork [6].In the semi-classical framework, we will study the spectral proper-ties of the eigenvalues of the Robin Laplacian T h below the energylevel h λ D (Ω), where λ D (Ω) is the ground state energy of the DirichletLaplacian. We obtain a boundary effective operator that describes theasymptotic distribution of the eigenvalues in the semi-classical limit(see Theorem 5.1 below). The corresponding eigenfunctions (whichcan be viewed as interior Steklov eigenfunctions in the sense of [18]and [10]) are expected to be localized near the domain’s boundary(thereby called edge states in the literature). We confirm this propertyin Theorem 1.1 below, which is valid for any dimension n ≥ Decay of eigenfunctions.
Using the boundary pseudo-differentialcalculus (as in [18]), we obtain that all eigenfunctions corresponding tonon-positive eigenvalues of the Robin Laplacian T h decay away from theboundary, uniformly with respect to the non-positive eigenvalues. Thisextends the result of [18] up to the boundary, and presents a weakerversion of the result of [10] but valid for the non-zero modes of T h . Theorem 1.1.
Let λ D (Ω) the principal eigenvalue of the DirichletLaplacian − ∆ on Ω .For any p ∈ N and ǫ < λ D (Ω) , there exist positive constants C p,ǫ , h p,ǫ and such that if ( h, u h , w ) is a solution of (1.8)(1.8) − ∆ u h = w u h in Ω ∂u h ∂ν = h − / u h on Γ , B. HELFFER AND A. KACHMAR with h ∈ (0 , h p,ǫ ] , w ≤ ǫ , and k u h k L ( ∂ Ω) = 1 then it satisfies (1.9) | u h ( x ) | ≤ C p,ǫ h (cid:0) d ( x, Γ) (cid:1) ! p , ∀ x ∈ Ω , where d ( x ) = d ( x, Γ) is the distance to the boundary introduced in (1.7) . One could hope in the case of an analytic boundary to have by usingan analytic pseudo-differential calculus a control of the constant C p,ǫ in (1.9) with respect to p leading to an estimate of the following form(1.10) | u h ( x ) | ≤ C d ( x ) − n h − exp (cid:18) − C d ( x ) h − (cid:19) , ∀ x s.t. d ( x ) ≤ C , for some constants C , C , C >
0, which could be difficult to determineexplicitly. We will discuss this in Subsection 3.5. Note that, for w = 0,(3.25) is established with C = 1 − η and η arbitrarly small in [10]by using analytic microlocal methods. This was improving the non-optimal exponential bound of [24] in the 2D case.In the case of an analytic boundary, based on the analysis in [10],we are able to improve improve the decay in Theorem 1.1 for w = 0. Theorem 1.2.
Assume that Γ , the boundary of Ω , is analytic. Forany ζ < λ D (Ω) and η > , there exist positive constants ε, C, h suchthat if ( h, u h , w ) is a solution of (1.8) with h ∈ (0 , h ] , w ≤ ζ , and k u h k L ( ∂ Ω) = 1 , then the following estimate holds, (1.11) ∀ x ∈ Ω , | u h ( x ) | ≤ C h − n + exp (cid:18) − (1 − η ) inf( d ( x, Γ) , ε ) h / (cid:19) . At the moment, it is unclear if the analytic assumptions are im-portant for the validity of the estimates (and more accurate estimatesdiscussed around (3.25)). Note that in the C ∞ case, the microlocalapproach proposed in [10] could at most give an information modulo O ( h ∞ ) leading perhaps to (1.9) with ǫ = ˆ Ch | log h | , for some ˆ C > α , as the eigenvalue w approaches0. However, Theorem 1 of Galkowski-Toth [10] and Theorem 1.1 aboveshow that all eigenfunctions decay with a constant exponential profileunder the analytic boundary hypothesis. It would then be interestingto extend these estimates to the case of a C ∞ - boundary. EMI-CLASSICAL EDGE STATES FOR THE ROBIN LAPLACIAN 5
Positive indications will be given in the 2 dimensional case that wewill discuss in the next section. Let us denote by(1.12) L = | Γ | , where | Γ | is the length of the boundary Γ. Assuming Γ is connected,we will encounter quasi-modes normalized in L (Γ) and having thefollowing profile u h ≈ (2 L ) − / exp (cid:0) − h − / d ( x, Γ) (cid:1) e ikπs/L with k ∈ Z . Such quasi-modes appear also in Polterovich-Sher-Toth’spaper [24] for the eigenvalue w = 0, where it is proved, in the case ofan analytic boundary, that they are close to the actual zero-modes ofthe operator T h . In the case where Γ is not connected [24], we stillencounter the foregoing quasi-modes on each connected component ofΓ and their linear combinations.1.3. Asymptotic distribution of eigenvalues.
There is a one-to-one correspondence between the negative eigenvalues of T h and theSteklov eigenvalues below the energy level h − / (see [5] and [2, Lem. 1]in a slightly different context). The correspondence being not explicit,it does not yield a precise description of the eigenvalues of the operator T h , based on the existing eigenvalue asymptotics for the Steklov eigen-values, but it does allow to deduce the asymptotics for the countingfunction of the operator T h from that of the DN operator Λ(0). Ourresult on the Robin eigenvalues (Theorem 1.3) is new and within ourapproach we can quantify the correspondence between the Robin andSteklov eigenvalues, and also to derive Weyl laws for the operator T h (and consequently for the DN operator) in Theorem 1.4.Let us consider the case n = 2 for the sake of simplicity and assumethat Ω is simply connected. We denote by (cid:0) λ n ( T h ) (cid:1) n ≥ the sequenceof min-max eigenvalues of the operator T h . We will determine theasymptotic behavior of λ n ( T h ) in the regime h → + thereby describingthe distribution of all the negative eigenvalues of T h .For all n ≥ L introduced in (1.12), we introduce the eigenvalues λ F n ( L ) = π k L for n ∈ { k, k + 1 } & k ∈ N , which correspond to the Fourier modes e ± iπks/L on R / L Z . Theorem 1.3.
Let λ N (Ω) denote the second eigenvalue of the Neu-mann Laplacian − ∆ on Ω and consider a positive constant ǫ < λ N (Ω) . B. HELFFER AND A. KACHMAR
Assume furthermore that Ω is simply connected. Then, there exist pos-itive constants C and h such that, for all h ∈ (0 , h ] , the followingestimates hold, (cid:12)(cid:12) λ n ( T h ) + h − h λ F n ( L ) (cid:12)(cid:12) ≤ Ch / (cid:0) h / λ F n ( L ) (cid:1) , provided that λ n ( T h ) < ǫh . The proof of Theorem 1.3 follows by deriving an effective operatorapproximating the operator T h . The precise statement is given in The-orem 5.1.The estimates of Theorem 1.3 are interesting when n ≫ h − / , sinceby [16, Thm. 2.1] and [17, Prop. 7.4], λ n ( T h ) ≤ − h − h / κ max + O ( h / )for n . h − / . In particular, the negative eigenvalues of T h satisfy λ k ( T h ) ∼ − h + π L − k h and λ k +1 ( T h ) − λ k ( T h ) = O ( h / )provided k ≫ h − / . This is consistent with the results in [7, 12, 27] and[24, Sec. 3.1] dealing with the spectrum of the DN operator Λ(0), whoseprincipal symbol coincides with √− ∆ Γ , the square root of the Laplace-Beltrami operator on Γ. In fact, the Steklov eigenvalues ( µ n ) n ≥ of Λ(0)satisfy the following asymptotics [27](1.13) µ k +1 = µ k + O ( k −∞ ) = πL k + O ( k −∞ ) ( k → + ∞ ) . So, we get the following correspondence between the negative Robineigenvalues { λ n ( T h ) < } and the Steklov eigenvalues { µ n < h − / } : µ n ∼ h − p h + λ n ( T h ) for h − / ≪ k n ≤ Lπ h − / + O ( h / ) . As a direct consequence of Theorem 1.3, we obtain a Weyl law extend-ing earlier results [17, 19, 20].
Theorem 1.4.
Assume that Ω is simply connected. Let ǫ ∈ [0 , λ N (Ω)) .For all h > and λ ∈ R , we denote by N ( T h , λ ) := tr (cid:16) ( −∞ ,λ ) (cid:0) T h (cid:1)(cid:17) . Then we have the following asymptotics as h → + , N ( T h , ǫh ) = | Γ | π h − / + O ( h − / ) , Furthermore, N ( T h , λh ) = | Γ | π √ λ h − / + O ( h − / ) , holds for all λ ∈ ( − , . EMI-CLASSICAL EDGE STATES FOR THE ROBIN LAPLACIAN 7
The asymptotics of N ( T h , λh ) and N ( T h , ǫh ) hold uniformly withrespect to λ ∈ ( − ,
0) and ǫ ∈ [0 , λ N (Ω)) respectively. Noting that N ( T h ,
0) = N (Λ(0) , h − / ) , we recover the leading order term for the existing results on the DNoperator (see [13, Eq. (2.1.4)])(1.14) N (Λ(0) , h − / ) = | Γ | π h − / + O (1) . The asymptotics in (1.14) continues to hold for the generalized DNoperator Λ( ǫ ) introduced in (1.2) if ǫ < λ D (Ω) is fixed (or in a compactinterval of ( −∞ , λ D (Ω)). Moreover, N (Λ( ǫ ) , h − / ) = N ( T h , ǫh ), hencewe get N ( T h , ǫh ) = | Γ | π h − / + O (1)which gives a more accurate estimate of the remainder than the oneappearing in Theorem 1.4. Organization of the paper. – In Sec. 2, we show how we can extract pointwise bounds on theeigenfunctions from the Agmon decay estimates.– In Sec. 3, we use a pseudo-differential calculus to prove Theo-rems 1.1 and 1.2.– In Sec. 4 we analyze 1D operators that we use later in Sec. 5 toderive an effective operator for the Robin Laplacian and proveTheorem 5.1.2.
Pointwise bounds via Agmon estimates
Using the elliptic and Agmon estimates, we can derive pointwisebounds on the low-energy eigenfunctions of the semi-classical RobinLaplacian operator T h . This was standard in the case of Dirichlet casebut because the Robin condition includes the parameter inside theboundary condition, we feel that it is useful to give the details in thisnew case. Theorem 2.1.
Given M ∈ (0 , and α ∈ (0 , √ M ) , there exist positiveconstants ε , h , C > such that, if h ∈ (0 , h ) and u h is a solution of − ∆ u h = w u h in Ω ∂u h ∂ν = h − / u h on Γ B. HELFFER AND A. KACHMAR with w < − M h − and k u h k L (Γ) = 1 , then (2.1) | u h ( x ) | ≤ Ch − n − exp − α min (cid:0) d ( x, Γ) , ε (cid:1) h / ! ( x ∈ Ω) . Proof.
For all ε >
0, we introduce the tubular neighborhood of theboundary,(2.2) Ω ε := { x ∈ Ω , d ( x, Γ) < ε } . Choose ε > x d ( x, Γ) is smooth on Ω ε . Weextend this function to a smooth function ˜ t on Ω as follows˜ t ( x ) = ( d ( x, Γ) if x ∈ Ω ε ε if x ∈ Ω \ Ω ε and ε ≤ ˜ t ( x ) ≤ ε if x ∈ Ω ε \ Ω ε . We introduce the function v h ( x ) = u h ( x ) exp (cid:0) α ˜ t ( x ) h / (cid:1) . We select α ∈ (0 , √ M ) and h > h ∈ (0 , h ), (1.6) holds, which inturn yields k v h k H (Ω) + h − / k v h k L (Ω) ≤ C h − / k u h k L (Ω) . The function v h satisfies the non-homogeneous Neumann problem:∆ v h = f h in Ω and ∂v h ∂ν = g h on Γ , where f h ( x ) = (cid:0) αh − / ∆˜ t − α h − |∇ ˜ t | − w (cid:1) v h + 2 αh − / ∇ ˜ t · ∇ v h , and g h ( x ) = ( α + 1) h − / v h ( x ) . By the elliptic estimates for the Neumann non homogeneous problem,we get k v h k H (Ω) ≤ C (cid:0) k f h k L (Ω) + k v h k L (Ω) + k g h k H / (Γ) (cid:1) ≤ ˜ C h − k u h k L (Ω) . In the cases n = 2 , n ≥
4, we pick the smallest integer k ∗ > n and we iterate the previous estimate so that k v h k H k ∗ (Ω) ≤ C ∗ (cid:0) k f h k H k ∗− (Ω) + k v h k H k ∗− (Ω) + k g h k H k ∗− / (Γ) (cid:1) ≤ ˜ C ∗ h − k ∗ / k u h k L (Ω) . We use Sobolev embedding of H k ∗ (Ω) in C (Ω) and that k ∗ ≤ n + 1.To finish the proof, we note that due to our normalization of u h / Γ , thenorm of u h in Ω satisfies k u h k L (Ω) = O ( h / ) since − M h − k u h k L (Ω) ≥ − w k u h k L (Ω) = k∇ u h k L (Ω) − h − / k u h k L (Γ) . EMI-CLASSICAL EDGE STATES FOR THE ROBIN LAPLACIAN 9 (cid:3) Boundary pseudo-differential calculus and decay ofeigenfunctions
Decay in the interior.
Here we discuss (and improve) the weakerresult of [18] leading to the conjecture proved by [10].
Theorem 3.1 (Hislop-Lutzer [18]) . For any p ∈ N , any K ⊂ Ω compact, there exists C K,p > and h K,p > such that if ( h, u h ) is a solution of (3.1)(3.1) − ∆ u h = 0 in Ω ∂u h ∂ν = h − / u h on Γ , with h ∈ (0 , h K,p ] and || u h || L ( ∂ Ω) = 1 , then it satisfies (3.2) | u h ( x ) | ≤ C K,p h p/ , ∀ x ∈ K. Note that our Thorem 1.1 extends the result of Theorem 3.1 up tothe boundary. The idea is to use the properties of the Poisson kernel ofthe operator − ∆ − w up to the boundary, while in [18], the propertiesof the Poisson kernel were used in the interior of the domain.3.2. Proof of Theorem 1.1 for w = 0 . The proof of [18] ( w = 0) is based on the classical Green-RepresentationFormula for u h (see [9, Ch. 2, Sec. 2.2.4] for the basic theory)(3.3) u h ( x ) = Z ∂ Ω u h ( · ) P ( x, · ) dσ , where P ( x, · ) is the Poisson kernel defined as follows(3.4) P ( x, · ) = − ∂ ν G ( x, · )where the distribution G ( x, y ) ∈ D ′ (Ω × Ω) is, given x ∈ Ω, the solutionof the inhomogeneous Dirichlet problem(3.5) − ∆ y G ( x, · ) = δ x , G ( x, y ) = 0 for y ∈ ∂ Ω . The properties of G (which is called the Green function) are rather wellknown in the case of a smooth boundary (see Theorem 2.3 in [18]) butfor the proof of the conjecture, we will need a more precise informationfor the Poisson kernel for y ∈ ∂ Ω and x close to ∂ Ω). This is done, atleast for w = 0 in [8] (see also [22]). The proof is based on the connection with the Dirichlet to Neumannoperator Λ( w ). Indeed, u h/∂ Ω is an eigenfunction of Λ( w ) associatedwith the eigenvalue h − . We can then write(3.6) u h ( x ) = h p ( P ◦ Λ( w ) p )( u h | ∂ Ω ) . For w = 0, (3.6) reads as follows,(3.7) u h ( x ) = h p Z ∂ Ω u h ( y ) · (Λ(0) p ( y, D y ) P ( x, y )) dσ . For x ∈ K , it is then easy to get the result obtained in [18], i.e. theinterior decay estimate of Theorem 3.1. As for the estimate of Theo-rem 1.1 up to the boundary, we recall the estimate given by M. Englisin [8]. Theorem 3.2.
Let Ω be a bounded domain in R n with smooth bound-ary. Then the Poisson kernel P ( x, y ) admits the following decomposi-tion (3.8) P ( x, y ) = c n d ( x ) | x − y | n (cid:20) F ( y, | x − y | , x − y | x − y | ) + H ( x, y ) | x − y | n log | x − y | (cid:21) , where • d ∈ C ∞ ( ¯Ω) , d > on Ω , • d ( x ) = d ( x, ∂ Ω) for x near ∂ Ω , • c n = Γ( n / π n / • F ∈ C ∞ ( ∂ Ω × ¯ R + × S n − ) , F ( y, , ω ) = 1 for y ∈ ∂ Ω , ω ∈ S n − , • H ∈ C ∞ ( ¯Ω × ∂ Ω) . This implies in particular the weak version mentioned by Krantz [22]which reads, for n ≥ | ∂ αy P ( x, y ) | ≤ C α d ( x ) | x − y | − n −| α | , ∀ y ∈ ∂ Ω , x ∈ Ω . This last estimate directly implies(3.10) | ∂ αy P ( x, y ) | ≤ C α d ( x ) − n −| α | , ∀ y ∈ ∂ Ω , x ∈ Ω . Coming back to (3.7), we can write for p even (if we do not want touse the complete Boutet de Monvel calculus)(3.11) u h ( x ) = h p Z ∂ Ω u h ( y ) · (Λ(0) p · ( − ∆ y ) − p/ ) (( − ∆ y ) p/ ) P ( x, y ) dσ = h p Z ∂ Ω (Λ(0) p · ( − ∆ y ) − p/ ) u h ( y ) · (( − ∆ y ) p/ ) P ( x, y ) dσ . EMI-CLASSICAL EDGE STATES FOR THE ROBIN LAPLACIAN 11
We now observe that (Λ(0) p · ( − ∆ y ) − p/ ) is a boundary pseudodiffer-ential operator of degree 0 (with constant principal symbol) and using(3.10) we obtain, for any p ≥ | u h ( x ) | ≤ C p h p d ( x ) − n − p . This proves Theorem 1.1 for w = 0.3.3. Proof of Theorem 1.1 for w ∈ [ − π , λ D (Ω)) . Now assume that − π ≤ w < λ D (Ω) and w = 0. The proof issimilar to the case w = 0 but we should replace the Green function G by G w and the ND operator Λ(0) by Λ( w ). There is no problem ofdefinition if w is not an eigenvalue of the Dirichlet Laplacian. To avoidto analyze if the proof written for w = 0 goes on, we can use a weakertheorem which holds for general potential operators (or Poisson likeoperators). See [8, Thm. 8, p. 18] with n ≥ d = p and observe that | x − ζ | − ≤ d ( x ) − . The aforementioned result of [8] reads as follows: Theorem 3.3 (Englis [8]) . If K is a potential operator in K d (Ω) , where Ω ⊂ R n is a bounded domain with smooth boundary (or Ω = R n + ), thenthe Schwartz kernel k K satisfies, if d ∈ Z , d > − n (3.13) k K ( x, y ) = | x − y | − n − d F ( y, | x − y | , x − y | x − y | )+ H ( x, y ) log | x − y | , where F and H have the same property as in the previous theorem. In our application, we use that the Poisson operator (associated with( − ∆ − w )) is a potential operator P ( w ) if w is not an eigenvalue ofthe Dirichlet Laplacian. We also use the property that the Dirichlet toNeumann operator Λ( w ) is a boundary pseudo-differential operator ofdegree 1 with elliptic principal symbol.We apply Theorem 3.3 to K = ( P ◦ Λ( w ) p ) and use (3.7). Everythingdepends continuously of w in the interval I := [ − π , λ D (Ω)) and thecontrol is uniform in any compact interval in I . This is clear for thecomputation (symbolic calculus) of an approximate Poisson operator P app ( w ) modulo regularizing operators R reg ( w ) and r reg ( w ) withoutadditional assumptions. One gets( − ∆ − w ) P app ( w ) = R reg ( w ) , γ ◦ P app ( w ) = I + r reg ( w ) . For eliminating the remainder, we use the resolvent and this is therethat the assumption that w is not in the spectrum of the DirichletLaplacian is used. More precisely, we first eliminate r ( w ) by usingsimply an extension operator ǫ from C ∞ ( ∂ Ω) into C ∞ (Ω). We note that ǫ ◦ r reg ( w ) is regularizing.Then, we compute( − ∆ − w )( P app ( w ) − ǫ ◦ r reg ( w )) = R reg ( w ) − ( − ∆ − w ) ǫ ◦ r reg ( w )) := ˆ R reg ( w ) . Finally, we get for the Poisson kernel P ( w ) = P app ( w ) − ǫ ◦ r reg ( w ) − ( − ∆ − w ) − ˆ R reg ( w ) . At this stage we get (3.12) from (3.6) in the case where w = 0 is fixedin the interval [ − π , λ D (Ω)), the estimate being uniform in w for anycompact subinterval. We have the same result for any compact intervalin the resolvent set of the Dirichlet Laplacian in Ω. The choice of − π is only motivated by the next step.3.4. Proof of Theorem 1.1 for w < − π . The problem here is thatwe loose in the previous approach the control of the uniformity withrespect to w in the estimates of the Poisson kernel P ( w ). Actually,since h − w ∈ σ ( T h ), w = w ( h ) may approach −∞ in the semi-classicallimit, by Theorem 5.1.Pick the unique integer k ≥ kπ ≤ √− w < ( k + 1) π and set a = kπ √− w . Then,(3.14) w + k π a = 0 , k ∈ N , ≤ a ≤ . We introduce the a weighted Laplace operator − ∆ ˆΩ ,a in the cylinderˆΩ := Ω × T , where T = [0 ,
1) is the 1d torus. That is(3.15) ∆ ˆΩ ,a = X i =1 ∂ ∂x i + 1 a ∂ ∂θ where ( x , x ) denote the coordinates in Ω and θ denotes the coordinatein T = [0 , x = ( x, θ ) of ˆΩ. Weintroduce the following function(3.16) v h (ˆ x ) = e ikπθs u h ( x ) (cid:0) ˆ x = ( x, θ ) (cid:1) which satisfies(3.17) − ∆ ˆΩ ,a v h = 0 on ˆΩ , ν ˆΓ · ∇ ˆ x v h = h − / v h on ˆΓ . EMI-CLASSICAL EDGE STATES FOR THE ROBIN LAPLACIAN 13
Here ˆΓ = ( ∂ Ω) × T is the boundary of ˆΩ, and ν ˆΓ its unit outward nor-mal vector; we have ν ˆΓ = ( ν , ν ,
0) where ν = ( ν , ν ) is the outwardunit normal vector of Γ = ∂ Ω. .We can introduce the DN operator of ˆΩ, Λ ˆΩ ,a (0), defined on H / (ˆΓ)as in (1.2) (with ˆΩ , ˆΓ replacing Ω , Γ and − ∆ ˆΩ ,a replacing − ∆). Con-sequently, the function v h in (3.17) is an eigenfunction of Λ ˆΩ ,a (0) witheigenvalue h − / . We will use the Poisson kernel P ˆΩ ,a corresponding to − ∆ ˆΩ ,a . Using Theorem 3.3 for the domain ˆΩ and the operator − ∆ ˆΩ ,a ,we get the following Poisson kernel estimates (as in (3.9)–(3.10))(3.18) | ∂ α ˆ x P ˆΩ ,a (ˆ x, ˆ y ) | ≤ C α d (ˆ x, ˆΓ) − n −| α | (ˆ x ∈ ˆΩ , ˆ y ∈ ˆΓ)where the constant C α can be chosen independently of a ∈ [ , x = ( x, θ ) ∈ ˆΩ, v h (ˆ x ) = Z ˆΓ v h (ˆ y ) P ˆΩ ,a (ˆ x, ˆ y ) dσ ˆΓ (ˆ y )= h p/ Z ˆΓ Λ ˆΩ ,a (0) p v h (ˆ y ) P ˆΩ ,a (ˆ x, ˆ y ) dσ ˆΓ (ˆ y ) . Using the Poisson kernel estimate in (3.18) and the pseudodifferentialcalculus as in (3.11), we get, for any positive even integer p , any a ∈ [ , C p and h p > h ∈ (0 , h p ], | v h (ˆ x ) | ≤ C p h p/ d (ˆ x, ˆΓ) − n − p . Since | v h (ˆ x ) | = | u h ( x ) | by (3.16) and d (ˆ x, ˆΓ) = d ( x, ∂ Ω) = d ( x ) forˆ x = ( x, θ ) ∈ ˆΩ, this implies(3.19) | u h ( x ) | ≤ C p h p/ d ( x ) − n − p , ∀ x ∈ Ω , as stated in Theorem 1.1 for w ≤ − π .3.5. Analytic case.
We now consider the case when ∂ Ω is analyticand handle the case where w < λ D (Ω).3.5.1. Using analytic pseudodifferential calculus.
At a heuristic level,one could hope from the Boutet de Monvel analytic pseudodifferentialcalculus [3] that we will get an estimate in the form(3.20) | u h ( x ) | ≤ C p +1 p ! h p d ( x ) − n − p . A first step could be the following (to our knowledge unproved) result:If A is an analytic pseudo-differential operator on ∂ Ω (or more generallya compact analytic manifold) of degree 1 and u is an analytic functionon ∂ Ω, then A p u satisfies | ( A p u )( y ) | ≤ C p +1 p ! . This kind of estimate (with additional control with respect to the dis-tance of x to ∂ Ω) should be applied to the distribution kernel of thePoisson operator of − ∆ − w .Assuming that the estimate (3.20) is true we can try to optimizeover p . Using Stirling Formula, we get the simpler(3.21) | u h ( x ) | ≤ C p +1 p p +1 h p d ( x ) − n − p . Optimizing over p will give an estimate of the form (1.10).It seems difficult by this approach to have the optimal result ofGalkowski-Toth [10], i.e. to have a control of the constant C appearingin (1.10).We also refer the reader to an interesting discussion at the end of [8](Subsection 7.4) and to [24].3.5.2. Using Galkowski-Toth.
In this section, we prove Theorem 1.2.To keep tracking the uniformity with respect to w of the estimates, weintroduce a fixed positive constant 0 < ζ < λ D (Ω) and assume that w varies as follows, −∞ < w ≤ ǫ .We recall Theorem 1 of Galkowski-Toth [10]: Theorem 3.4 ([10]) . For all δ > and α ∈ N n , there exist ε > , C and h such that, for h ∈ (0 , h ] , any solution u h of (3.22) − ∆ u h = 0 in Ω ∂u h ∂ν = h − / u h on Γ k u h k L ( ∂ Ω) = 1 , satisfies the following estimate in { d ( x, Γ) < ε } , (3.23) | ∂ αx u h ( x ) | ≤ C h − n + − | α | exp − d ( x, Γ) (cid:0) C Ω − δ ) d ( x, Γ) (cid:1) h / ! . Here C Ω = − + inf ( x ′ ,ξ ′ ) ∈ S ∗ Γ Q ( x ′ , ξ ′ ) , Q is the symbol of the secondfundamental form of the boundary Γ . It results from Theorem 3.4 the following weaker estimate. Thereexist constants ε, C, ˆ C such that, for d ( x, Γ) < ε , we have(3.24) | ∂ αx u h ( x ) | ≤ C h − n + − | α | exp − d ( x, Γ) (cid:0) − ˆ Cd ( x, Γ) (cid:1) h / ! . Looking at the proof, Theorem 3.4 can be generalized in two differentways:
EMI-CLASSICAL EDGE STATES FOR THE ROBIN LAPLACIAN 15 • When replacing − ∆ by − ∆ − w , the constants in the estimatescan be controlled uniformly with respect to w in any compactinterval of ( −∞ , λ D (Ω)). • When replacing − ∆ by div( c ∇ ) with c ∈ R n a constant vectorwith positive components, the constants in the estimates canalso be controlled uniformly with respect to | c | when it variesin a compact interval in (0 , + ∞ ).In the two aforementioned situations, (3.23) continues to hold, whichalso yields that, for all η >
0, there exist positive constants ε, C, h suchthat, for h ∈ (0 , h ], any solution u h of (3.22) satisfies the followingestimate in { d ( x, Γ) < ε } ,(3.25) | u h ( x ) | ≤ C h − n + exp (cid:18) − (1 − η ) d ( x, Γ) h / (cid:19) . Note that we just keep (3.24) which is the weaker version of (3.23) forsimplification. In the procedure of addition of one variable describedbelow, we can not keep the additional information related to the cur-vature of Γ, but we can always write the following estimate (which alsoleads to (3.25)):
There exist positive constants C, ˆ C, h such that, for all h ∈ (0 , h ] , (3.26) | u h ( x ) | ≤ C h − n + exp (cid:18) − d ˆ C ( x ) h / (cid:19) , with d ˆ C ( x ) = ( d ( x, Γ) − ˆ Cd ( x, Γ) )1 { y, d ( y, Γ) <
12 ˆ C } ( x )+ 12 ˆ C (cid:16) − { y, d ( y, Γ) <
12 ˆ C } ( x ) (cid:17) . We proceed with the proof of Theorem 1.2. We start with the case w < − π and apply the Galkowski-Toth estimate (3.25) for the solution v h of (3.17). We get | v h (ˆ x ) | ≤ C α h − (cid:0) n + (cid:1) exp − (1 − η ) d (ˆ x, ˆΓ) h / ! , in a tubular neighborhood ˆΩ ε = { x ∈ ˆΩ , dist(ˆ x, ˆΓ) < ε } . Note thatthe second fundamental form of ˆΩ vanishes so the estimate does notdisplay the effect of the curvature of Ω as in (3.22). Remarking that | v h (ˆ x ) | = | u h ( x ) | and d (ˆ x, ˆΓ) = d ( x, Γ) = d ( x ) forˆ x = ( x, θ ) ∈ ˆΩ, we get | u h ( x ) | ≤ Ch − (cid:0) n + (cid:1) exp (cid:18) − (1 − η ) d ( x, Γ) h / (cid:19) , in Ω ε . To get the interior estimate | ˆ v h (ˆ x ) | ≤ ˆ C exp( − ˆ c h − / ) in Ω \ ˆΩ ε , we use the maximum principle, for the operator − ∆ ˆΩ ,a and the solutionˆ v h , in Ω \ ˆΩ ε (see [24, Lem. 3.2.9] for the details of the argument). Thisfinishes the proof of (3.25) for w < − π .We move now to the case where − π ≤ w ≤ ζ . We use the estimate(3.25) for the solution of − ∆ u h = wu h and get | u h ( x ) | ≤ Ch − n + exp (cid:18) − (1 − η ) d ( x, Γ) h / (cid:19) in Ω ε . If moreover w ≤
0, we use the maximum principle, as in [10, 24] toget the interior estimates. Notice that we use the maximum principlefor the operator − ∆ − w with w ≤ .If 0 < w ≤ ζ < λ D (Ω), we apply the maximum principle to thefunction f h defined by u h = f h u D , where u D is the normalized positiveground state of the Dirichlet Laplacian on Ω. The function f h satisfies − u D ) div (cid:16) ( u D ) ∇ f h (cid:17) + cf h = 0 with c := λ D (Ω) − w > . One dimensional Robin Laplacians
We revisit one dimensional model operators appearing in [16].4.1.
On the half line.
We start with the self-adjoint operator in L ( R + ) defined by(4.1) H = − ∂ τ on the domain(4.2) Dom ( H ) = { u ∈ H ( R + ) : u ′ (0) = − u (0) } . The quadratic form associated with this operator is V ∋ u Z + ∞ | u ′ ( τ ) | dτ − | u (0) | , with form domain V = H (0 , + ∞ ) . see for example Stampacchia [28, Thm. 3.8] for the maximum principle for − ∆ − w when w ≤ EMI-CLASSICAL EDGE STATES FOR THE ROBIN LAPLACIAN 17
The spectrum of this operator is (see [16]) σ ( H ) = {− } ∪ [0 , + ∞ )and the eigenvalue − L -normalized positive eigenfunction(4.3) u ( τ ) = √ − τ ) . On an interval.
Let us consider T ≥ L (0 , T ) and defined by(4.4) H T,D = − ∂ τ , with domain,(4.5) Dom ( H T,D ) = { u ∈ H (0 , T ) : u ′ (0) = − u (0) and u ( T ) = 0 } . This operator is associated with the quadratic form V T,D ∋ u Z T | u ′ ( τ ) | dτ − | u (0) | , with V T,D = { v ∈ H (0 , T ) | v ( T ) = 0 } .The spectrum of the operator H T,D is purely discrete. We denoteby (cid:0) λ Dn ( T ) (cid:1) n ≥ the sequence of min-max eigenvalues and by ( u T,Dn ) n ≥ some associated L (0 , T ) Hilbert basis of eigenfunctions. We can lo-calize the spectrum in the large T limit [16, Lem. 4.1 and Rem. 4.3]and [19, Lem. A.2]. Lemma 4.1. As T → + ∞ , it holds (4.6) λ T,D ( T ) = − (cid:0) o (1) (cid:1) exp (cid:0) − T (cid:1) , and the eigenfunction u T,D satisfies (4.7) (cid:13)(cid:13) e τ (cid:0) u T,D − u (cid:1)(cid:13)(cid:13) W , ∞ (0 ,T ) = O (cid:0) T (cid:1) , where u is the eigenfunction in (4.3) .Furthermore, for all T > and n ≥ , we have (cid:18) (2 n − π T (cid:19) < λ Dn ( T ) < (cid:18) ( n − πT (cid:19) . Also we consider the Neumann realization at the boundary t = T ,(4.8) H T,N = − ∂ τ , with domain,(4.9) Dom ( H T,N ) = { u ∈ H (0 , T ) : u ′ (0) = − u (0) and u ′ ( T ) = 0 } . The spectrum of the operator H { T } is purely discrete, consisting ofthe sequence of min-max eigenvalues (cid:16) λ n (cid:16) H T,N (cid:17)(cid:17) n ≥ . We denote by( u T,Nn ) n ≥ the corresponding Hilbert basis of eigenfunctions. We canlocalize the spectrum in the large T limit. Lemma 4.2. As T → + ∞ , it holds (4.10) λ N ( T ) = − (cid:0) o (1) (cid:1) exp (cid:0) − T (cid:1) and (4.11) (cid:13)(cid:13) e τ (cid:0) u T,N − u (cid:1)(cid:13)(cid:13) W , ∞ (0 ,T ) = O (cid:0) T (cid:1) , where u is the eigenfunction in (4.3) .Furthermore, for all T > and n ≥ , we have (cid:18) (2 n − π T (cid:19) < λ Nn ( T ) < (cid:18) ( n − πT (cid:19) . Proof.
The proof is similar to that of Lemma 4.1 but we give the mainpoints for the convenience of the reader. Let λ ≤ H T,N with an eigenfunction f . Solvingthe ODE f ′′ = λf with the boundary conditions f ′ (0) = − f (0) and f ′ ( T ) = 0 yields that the only possible value of λ is λ = − (cid:0) o (1) (cid:1) exp (cid:0) − T (cid:1) , which corresponds to the first eigenvalue (see [16]). The correspondingnormalized eigenfunction is u T,N ( τ ) = A T (cid:16) e − τ + e − αT e τ (cid:17) with A T = √ O ( T e − T ) and α = 1 − o (1)) e − T , so that e − αT e τ = O ( e τ − T ) = o ( e − τ ) . We then have the following uniform estimate, | u T,N ( τ ) − A T e − τ | = O (cid:0) e τ − T (cid:1) which also yields (cid:12)(cid:12) e τ (cid:0) u T,N ( τ ) − √ e − τ (cid:1)(cid:12)(cid:12) = O ( T ) . Although not needed here, note that we have the much more accurateapproximation | A − T u T,N ( τ ) − ( e − τ + e τ − T ) | = O ( T e − T ) . EMI-CLASSICAL EDGE STATES FOR THE ROBIN LAPLACIAN 19
Now we study the positive eigenvalues. Let ℓ > λ = ℓ be anon-negative eigenvalue of the operator H T,N with an eigenfunction u ,which will have the form u ( τ ) = A cos( ℓτ ) + B sin( ℓτ ) , for some constants A, B ∈ R that depend on T , with A = − Bℓ ,sin( ℓT ) = 0, and cot( ℓT ) = − ℓ , to respect the boundary conditions.The positive fixed points of the π/T -periodic function x
7→ − cot( xT )must belong to the intervals I k := ( π T , πT ) + kπT , k = 0 , , · · · . In eachinterval I k , there exists a unique fixed point ℓ k because the function g ( x ) = cot( xT ) + x satisfies g ′ ( x ) = − T (1 + cot ( xT )) + 1 < T >
1. For each k = 0 , , , · · · , the fixed point ℓ k ∈ I k is equal to q λ Nk +2 ( T ). (cid:3) On a weighted space.
Now we consider operators with weightterms, which can be viewed as perturbations of the operators studiedpreviously on the interval (0 , T ) with Dirichlet or Neumann realizationsat the endpoint t = T .In the sequel, ρ ∈ ( , ) and M > T = T h := h ρ − . We pick h = h ( ρ, M ) > h ∈ (0 , h ] and β ∈ [ − M, M ], we have < − h / βτ < τ ∈ (0 , T ).Consider the differential operator H h,β = − (1 − h / βτ ) − ddτ (cid:16) (1 − h / βτ ) ddτ (cid:17) = − d dτ + βh / (1 − h / βτ ) − ddτ . We work in the Hilbert space(4.12) X h,β = L (cid:0) (0 , T h ); (1 − h / βτ ) dτ (cid:1) , with inner product and norm defined by(4.13) h u, v i h,β = Z T u ( τ ) v ( τ ) (1 − h / βτ ) dτ, k u k h,β = h u, u i / h,β . Consider the two self-adjoint realizations of H h,β in X h,β , H Nh,β and H Dh,β , with domains(4.14) D Nh = { u ∈ H (0 , T ) , u ′ (0) = − u (0) & u ′ ( T ) = 0 } and D Dh = { u ∈ H (0 , T ) , u ′ (0) = − u (0) & u ( T ) = 0 } . We denote the sequences of min-max eigenvalues by (cid:0) λ Nn,h ( β ) (cid:1) n ≥ and (cid:0) λ Dn,h ( β ) (cid:1) n ≥ respectively.By the min-max principle, we can localize the foregoing eigenvaluesas follows(4.15) (cid:12)(cid:12) λ n,h ( β ) − λ T, n,h (cid:12)(cid:12) ≤ (cid:0) λ T, n,h (cid:1) h ρ uniformly with respect to β ∈ [ − M, M ] and h ∈ (0 , h ]. Here ∈{ N, D } and λ T, n,h are the eigenvalues of the operators introduced in(4.4) and (4.8) with T = h ρ − ≫
1. We deduce then that there exists h > h ∈ (0 , h ](4.16) λ ,h ( β ) ≥ π h − ρ − O ( h ρ ) ≥ π h − ρ > , since ρ ∈ ( , ). The first eigenvalue λ .h was analyzed in [16, Prop. 4.5]and [21, Lem. 2.5] for the Dirichlet case ( D ). The same analysisapplies for the Neumann case ( N ). we have(4.17) (cid:12)(cid:12)(cid:12) λ .h ( β ) − (cid:0) − − βh / − β h (cid:1)(cid:12)(cid:12)(cid:12) ≤ C ( | β | + 1) h , uniformly with respect to β ∈ [ − M, M ] and h ∈ (0 , h ].For the convenience of the reader, we present the outline of the proofof (4.17). The idea is to look for a formal eigenpair of the form u app h,β = v + h / v + hv and µ app h,β = µ + µ h / + µ h . We expand (cid:0) H h,β − µ app h,β (cid:1) u app h,β ( τ ) as L + h / L + hL + h / r β ( τ ) with L = (cid:16) − d dτ − µ (cid:17) v , L = (cid:16) − d dτ − µ (cid:17) v + (cid:16) β ddτ − µ (cid:17) v ,L = (cid:16) − d dτ − µ (cid:17) v + (cid:16) β ddτ − µ (cid:17) v + (cid:16) β ddτ − µ (cid:17) v | r β ( τ ) | ≤ C ( | β | + 1)( τ + 1) X i =1 | v i ( τ ) | We choose the pairs ( v i , µ i ) so that the coefficients L , L , L vanish[21, Lem. 2.5]. Eventually we get the approximate eigenfunction µ app h,β := − − βh / − β h , and the following quasi-mode(4.18) u app h,β ( τ ) := (cid:16) β h (cid:16) τ − (cid:17)(cid:17) u ( τ ) , EMI-CLASSICAL EDGE STATES FOR THE ROBIN LAPLACIAN 21 where u the eigenfunction in (4.3). The following estimate holds, forall τ ∈ (0 , T ),(4.19) (cid:12)(cid:12)(cid:12)(cid:16) H h,β − µ app h,β (cid:17) u app h,β ( τ ) (cid:12)(cid:12)(cid:12) ≤ Ch ( | β | + 1)( τ + 1) | u ( τ ) | uniformly with respect to β ∈ [ − M, M ] and τ ∈ (0 , T ).We introduce the following quasi-mode (it belongs to D ( H h,β ))(4.20) v h ( t ) = c h χ ( T − τ ) u app h,β ( τ ) , where χ ∈ C ∞ c ( R ) satisfies 0 ≤ χ ≤
1, supp χ ⊂ ( − , χ / [ − ,
12 ] = 1,and where c h is selected so that k v h k h,β = 1. By the exponential decayof u (see (4.3)), the constant c h and the quasi-mode v h satisfy c h = 1 + O ( h / )and(4.21) (cid:12)(cid:12)(cid:12)(cid:16) H h,β − µ app h,β (cid:17) v h ( τ ) (cid:12)(cid:12)(cid:12) ≤ ˜ Ch ( | β | + 1)( τ + 1) | u ( τ ) | . The spectral theorem and (4.16) yield the estimate in (4.17).We will need the following lemma on the ‘energy’ of functions or-thogonal to the quasi-mode v h in the space X h,β introduced in (4.12). Lemma 4.3.
Then there exist positive constants m, h such that, if h ∈ (0 , h ] and g h ∈ H (0 , T ) is orthogonal to v h in X h,β , then k g ′ h k h,β − | g h (0) | ≥ mh − ρ k g h k h,β . Proof.
Let u gs h,β ∈ D Nh be the normalized (in X h,β ) ground state of theoperator H Nh,β :(4.22) H h,β u gs h,β = λ N .h ( β ) u gs h,β . By the min-max principle, (4.15) and Lemma 4.2, if f belongs to theform domain of H h,β and satisfies h f , u gs h,β i h,β = 0, then(4.23) hH h,β f , f i h,β ≥ λ N .h ( β ) k f k h,β ≥ (cid:0) λ N .h ( β ) + c h − ρ (cid:1) k f k , where c is a positive constant.Now consider a function g h ∈ H (0 , T ) such that h g h , v h i h,β = 0. Wedecompose v h and g h as follows(4.24) v h = α h u gs h,β + f h and g h = γ h u gs h + e h , with(4.25) α h = h v h , u gs h,β i h,β , γ h = h g h , u gs h i h,β and h f h , u gs h,β i h,β = h e h , u gs h,β i h,β = 0 . We infer from (4.17), (4.21) and (4.22) that (cid:13)(cid:13)(cid:0) H h,β − λ N .h ( β ) (cid:1) f h (cid:13)(cid:13) h,β = O (cid:0) h (cid:1) . Consequently q h,β ( f h ) := h (cid:0) H h,β − λ N .h ( β ) (cid:1) f h , f h i h,β = O (cid:0) h (cid:1) k f h k h,β , and by (4.23), q h,β ( f h ) ≥ c h − ρ k f h k . Eventually we get that (cid:0) − | α h | (cid:1) / = k f h k h,β = O ( h +2 ρ ) , where α h is introduced in (4.24).We return to the function g h in (4.24). Since e h ⊥ u gs h,β , we get by(4.23),(4.26) k g ′ h k h,β − | g h (0) | = q h,β ( e h ) ≥ c h − ρ k e h k . Since h g h , v h i h,β = 0, we get from (4.24), γ h α h + h e h , f h i h,β = 0which yields that | γ h | ≤ | α h | k e h kk f h k = O ( h +2 ρ ) k e h k h,β and consequently k g h k h,β = | γ h | + k e h k h,β = (cid:0) O ( h ρ ) (cid:1) k e h k h,β . Inserting this into (4.26), we finish the proof of Lemma 4.3. (cid:3) The effective operator
The operator near the boundary.
Assume that Ω is simply connected, hence Γ consists of a single con-nected component. In the case of a multiply connected domain, withΓ having a finite number of connected components, we can do the con-structions below in each connected component of Γ.We introduce the coordinates ( s, t ) valid in a tubular neighborhood ofthe boundary, Ω ε := { x ∈ Ω , dist( x, ∂ Ω) < ε } , and defined as follows: t ( x ) = dist( x, Γ) measures the transversal distance to Γ − ∂ Ω, and s ( x ) ∈ [ − L, L ) measures the (arc-length) tangential distance along Γ,with 2 L = | Γ | is the length of the boundary. More precisely, we denoteby [ − L, L [ ∋ s M ( s ) the arc-length parameterization of Γ orientedcounter-clock wise and consider the transformationΦ : ( s, t ) M ( s ) − tν ( s ) EMI-CLASSICAL EDGE STATES FOR THE ROBIN LAPLACIAN 23 where ν ( s ) is the unit outward normal of ∂ Ω.The L -norm of u in Ω ε is k u k L (Ω ε ) = Z L − L Z ε | u ( s, t ) | a ( s, t ) dtds and the operator T h is expressed as follows T h = − a − ∂ t ( a∂ t ) + a − ∂ s ( a − ∂ s )where a ( s, t ) = 1 − tκ ( s )and κ ( s ) is the curvature of Γ at the point M ( s ).For every c ∈ R , let L ch denote the operator (on R / L Z ) L ch = − ( h / + ch / ) d ds − κ ( s ) − h / (cid:0) κ ( s ) (cid:1) + ch / , with domain D = { u ∈ H (] − L, L [) : u ( − L ) = u ( L ) & u ′ ( − L ) = u ( L ) } . For a self-adjoint semi-bounded operator P , we denote by ( λ n ( P )) n ≥ the sequence of min-max eigenvalues. For all h > ǫ ∈ R , weintroduce the following subset of N (5.1) I ǫh = { k ≥ λ k ( T h ) < ǫh } . Theorem 5.1.
Given ≤ ǫ < λ N (Ω) , there exist positive constants c , h , such that, for all h ∈ (0 , h ] and n ∈ I ǫh , (5.2) h / min (cid:0) λ n ( L − c h ) , h − / (cid:1) ≤ λ n ( T h ) + h ≤ h / min (cid:0) λ n ( L c h ) , ǫh / (cid:1) . In particular, for λ n ( T h ) < , we have, (5.3) h / λ n ( L − c h ) ≤ λ n ( T h ) + h ≤ h / min (cid:0) λ n ( L c h ) , (cid:1) . Remarks 5.2.1.
Note that for ǫ < Let λ D (Ω) be the first eigenvalue of the Dirichlet Laplacian on Ω.It follows from [25, 29] that λ N (Ω) < λ D (Ω) (see also [1, Eq. (2.2)]).The upper bound in (5.2) actually holds for ǫ < λ D (Ω). A comparison similar to the one in Theorem 5.1 has been proved in[17] when n ∈ I − ǫh := { k ≥ λ k ( T h ) < − ǫh } with 0 < ǫ < − (cid:0) h / + hb ( s ) (cid:1) d ds − κ ( s ) , with b ( s ) = O (1) uniformly w.r.t. s . Our result extends that in [17] allthe way up to ǫ = 0, but with a worse remainder term for the coefficientof d ds , in order to consider all the non-positive eigenvalues. Note that for the realization of − ∂ s on R / L Z , the spectrum is { π L − ( n − , n ≥ } with the first eigenvalue being simple and the others being of multi-plicity 2, hence λ ( − ∂ s ) = 0and λ k ( − ∂ s ) = λ k +1 ( − ∂ s ) = π L − k , k = 1 , , · · · . Theorem 5.1 then yields the existence of c > h > h ∈ (0 , h ] and n ∈ { k, k + 1 } with λ n ( T h ) < ǫh , we have h − / λ n ( T h ) ≤ − h − / + π k L (cid:0) c h / (cid:1) h / − κ min + M + h / + c h / , and h − / λ n ( T h ) ≥ − h − / + π k L (cid:0) − c h / (cid:1) h / − κ max + M − h / − c h / , where κ min = min s ∈ [ − L,L ) κ ( s ) , κ max = max s ∈ [ − L,L ) κ ( s ) ,M − = − max s ∈ [ − L,L ) | κ ( s ) | , M + = − min s ∈ [ − L,L ) | κ ( s ) | . These estimates yield Theorem 1.3. For a positive integer k = k ( h ) ≫ h − / satisfying k ≤ (1 + c h / ) − (cid:16) h / κ max + 12 hκ − c h / (cid:17) h − , we get λ k ( T h ) ∼ − h + π L (2 k − h & λ k +1 ( T h ) − λ k ( T h ) = O ( h / ) . Decomposition of L (Ω) . Let ρ ∈ ( , ) and consider the do-main Ω h ρ defined by (2.2).We decompose the Hilbert space L (Ω) as L (Ω h ρ ) ⊕ L (Ω \ Ω h ρ ). Wewill decompose further the space L (Ω h ρ ) by considering the orthogonalprojection on the function u tran h ( s, t ) = c h h − / χ (cid:0) h − ρ t (cid:1) u app h,κ ( s ) ( h − / t )where u app h,κ ( s ) is the function defined by (4.18), and χ ∈ C ∞ c ( R ) satisfies0 ≤ χ ≤
1, supp χ ⊂ ( − ,
1) and χ / [ − ,
12 ] = 1. The coefficient c h isdetermined by k u tran h k L (Ω) = 1 and satisfies c h = 1 + O ( h ∞ ). EMI-CLASSICAL EDGE STATES FOR THE ROBIN LAPLACIAN 25
Note that u tran h ∈ D ( T h ) and by (4.19) (cid:16) − a − ∂ t ( a∂ t ) − λ h ( s ) (cid:17) u tran h ( s, t ) = O ( h +2 ρ ) , with(5.4) λ h ( s ) = − h − h / κ ( s ) − h κ ( s ) . We introduce the projections in the space L (Ω h ρ ),Π s ψ = h ψ, u tran h i u tran h and Π ⊥ s ψ = ψ / Ω hρ − Π s ψ , and the isometry(5.5) ψ ∋ L (Ω h ρ ) (Π s ψ, Π ⊥ s ψ ) ∈ V h ⊕ W h where V h = L (cid:0) [ − L, L ) (cid:1) ⊗ { span( u tran h ) } = { v ∈ L (Ω h ρ ) , ∃ k ∈ L ([ − L, L )) , v ( s, t ) = k ( s ) u tran h ( s, t ) } and W h = { v ∈ L (Ω h ρ ) , Z h ρ v ( s, t ) u tran h ( s, t ) (1 − tκ ( s )) dt = 0 } . Using (5.6) and the decomposition of L (Ω) as L (Ω h ρ ) ⊕ L (Ω \ Ω h ρ ),we construct the following isometry(5.6) ψ ∋ L (Ω) χ ψ := (Π s ψ, Π ⊥ s ψ, ψ / Ω \ Ω hρ ) ∈ V h ⊕ W h ⊕ L (Ω \ Ω h ρ )Note that k ψ k L (Ω) = k χ ψ k = Z L − L | k ψ ( s ) | ds + Z Ω hρ | Π ⊥ s ψ | dx + Z Ω \ Ω hρ | ψ | dx , where(5.7) k ψ ( s ) := h ψ, u tran h i = Z h ρ ψ ( s, t ) u tran h ( s, t ) (1 − tκ ( s )) dt . Decomposition of the quadratic form.
We examine the qua-dratic form(5.8) q Ω h ( ψ ) := h Z Ω |∇ ψ | dx − h / Z ∂ Ω | ψ | ds ( x )= q Ω hρ h ( ψ ) + q int h,ρ ( ψ )where q int h,ρ ( ψ ) = Z Ω \ Ω hρ |∇ ψ | dx . Working in the ( s, t ) coordinates, we express the quadratic form q Ω hρ h ( ψ )as follows q Ω hρ h ( ψ ) = h Z L − L (cid:18)Z h ρ (cid:16) | ∂ t ψ | + a − | ∂ s ψ | (cid:17) adt − h − / | ψ ( s, t = 0) | (cid:19) . Freezing the s -variable, the Π s is an orthogonal projection in the weightedHilbert space L (cid:0) (0 , h ρ ); a ( s, t ) dt (cid:1) ; consequently, q tran h ( ψ ) := h Z h ρ | ∂ t ψ | (1 − tκ ( s )) dt − h / | ψ ( s, t = 0) | = q tran h (Π s ψ ) + q tran h (Π ⊥ s ψ )and Z h ρ | ∂ s ψ | (1 − tκ ( s )) dt = Z h ρ (cid:16) | Π s ∂ s ψ | + | Π ⊥ s ∂ s ψ | (cid:17) (1 − tκ ( s )) dt We have (see (5.4))(5.9) q tran h (Π s ψ ) = (cid:0) λ h ( s ) + O ( h +2 ρ ) (cid:1) Z h ρ | Π s ψ | (1 − tκ ( s )) dt = (cid:0) λ h ( s ) + O ( h +2 ρ ) (cid:1) | k ψ ( s ) | , (5.10) q tran h (Π ⊥ s ψ ) & h − ρ , and, setting K = 8 k κ k ∞ ,1 + 2 t K ≤ a − ≤ t K . Therefore, we end up with the following upper bound of the quadraticform(5.11) q Ω hρ h ( ψ ) ≤ h Z L − L (cid:18)(cid:0) λ h ( s ) + O ( h +2 ρ ) (cid:1) | k ψ ( s ) | + Z h ρ (1 + K t ) | Π s ∂ s ψ | adt (cid:19) ds + h Z L − L Z h ρ (cid:16) | ∂ t Π ⊥ s ψ | + (1 + K t ) | Π ⊥ s ∂ s ψ | (cid:17) adtds − h / Z L − L | Π ⊥ s ψ ( s, t = 0) | ds . EMI-CLASSICAL EDGE STATES FOR THE ROBIN LAPLACIAN 27
The same argument yields the following lower bound(5.12) q Ω hρ h ( ψ ) ≥ h Z L − L (cid:18)(cid:0) λ h ( s ) + O ( h +2 ρ ) (cid:1) | k ψ ( s ) | + Z h ρ (1 − K t ) | Π s ∂ s ψ | adt (cid:19) ds + h Z L − L Z h ρ (cid:16) | ∂ t Π ⊥ s ψ | + (1 − K t ) | Π ⊥ s ∂ s ψ | (cid:17) adtds − h / Z L − L | Π ⊥ s ψ ( s, t = 0) | ds . Let us now handle the term | Π s ∂ s ψ | . Let us introduce u = ∂ s ψ . It iseasy to check the following identities, ∂ s Π s ψ = ∂ s (cid:16) k ψ ( s ) u tran h (cid:17) = k ′ ψ ( s ) u tran h + k ψ ( s ) ∂ s u tran h = (cid:0) k u − κ ′ ( s ) k tψ + h ψ, ∂ s u tran h i (cid:1) u tran h + k ψ ( s ) ∂ s u tran h . Therefore,Π s ∂ s ψ = k ′ ψ ( s ) u tran h + k ψ ( s ) ∂ s u tran h + (cid:0) κ ′ ( s ) k tψ ( s ) − h ψ, ∂ s u tran h i (cid:1) u tran h | {z } := w ψ ( s,t ) . Note that if we perform the change of variable, t = h / τ , we can write Z h ρ t | u tran h ( s, t ) | dt = O ( h / )uniformly with respect to s . In a similar manner, we can check that Z h ρ | ∂ s u tran h | dt = O ( h ) . Consequently, if we introduce the norms N ± ( f ) = (cid:18)Z L − L Z h ρ (1 ± K t ) | f | adtds (cid:19) / , we get that N ± ( k ψ u tran h ) = (cid:0) O ( h / ) (cid:1) Z L − L | k ′ ψ ( s ) | ds and N ± ( w ψ ) = O ( h / ) Z L − L Z h ρ | ψ | dsdt = O (cid:0) h / k ψ k L (Ω hρ ) (cid:1) . Armed with the foregoing estimates, and Cauchy’s inequality, we write,for all η ∈ (0 , N ± (Π s ∂ s ψ ) ≥ (1 − η ) N ± ( k ψ u tran h ) − η − N ± ( w ψ ) and N ± (Π s ∂ s ψ ) ≤ (1 + η ) N ± ( k ψ u tran h ) + (1 + η − ) N ± ( w ψ ) . Choosing η = h / , we eventually get estimates for the energy of Π s ∂ s ψ as follows(5.13) (cid:12)(cid:12)(cid:12)(cid:12)Z L − L Z h ρ Z h ρ (1 ± K t ) | Π s ∂ s ψ | adtds − Z L − L | k ′ ψ ( s ) | ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ M h / (cid:16) Z L − L | k ′ ψ ( s ) | ds + k ψ k L (Ω hρ ) (cid:17) = M h / (cid:16) Z L − L | k ′ ψ ( s ) | ds + k Π s ψ k L (Ω hρ ) | {z } = R [ − L,L ) | k ψ ( s ) | ds + k Π ⊥ s ψ k L (Ω hρ ) (cid:17) for h ∈ (0 , h ], where M, h are positive constants.5.4. Comparison of eigenvalues.
Upper bounds.
Consider the self-adjoint operator T K h h in L (Ω),defined by the quadratic form(5.14) K h ∋ u h Z Ω |∇ ψ | dx − h / Z ∂ Ω | u | ds ( x )where the form domain K h consists of functions with zero trace on theboundary of Ω \ Ω h ρ , i.e. K h = { v ∈ H (Ω) : v = 0 for dist( x, Γ) = h ρ } By the min-max principle and comparison of the form domains, for all n ≥ λ n ( T h ) ≤ λ n ( T K h h ) . For all ψ ∈ K h , we investigate the quadratic form q Ω h ( ψ ) = h Z Ω |∇ ψ | dx − h / Z ∂ Ω | u | ds ( x ) ≤ q h ( χ ψ )where q h ( χ ψ ) := q h, ( k ψ ) + q h, ( f ψ ) + q h, ( u ψ )and χ ψ = ( k ψ u tran h , f ψ , u ψ ) . EMI-CLASSICAL EDGE STATES FOR THE ROBIN LAPLACIAN 29
The quadratic forms q h,i are defined as follows q h, ( k ψ ) = Z L − L (cid:16)(cid:0) λ h ( s ) + O ( h +2 ρ ) (cid:1) | k ψ ( s ) | + (cid:0) h + O ( h ) (cid:1) | k ′ ψ ( s ) | (cid:17) ds , q h, ( f ψ ) = h Z L − L (cid:18)Z h ρ (cid:16) | ∂ t f ψ | + O ( h ) | f ψ | (cid:17) adt − h − | f ψ ( s, | (cid:19) ds q h, ( u ψ ) = h Z Ω hρ |∇ u ψ | dx ≥ h λ D (Ω) k u ψ k L (Ω hρ ) . For all i ∈ { , , } , let L h,i be the operator defined by the quadraticform q h,i . By the min-max principle, λ n ( T K h h ) ≤ λ n (cid:0) ⊕ i =1 L h,i (cid:1) . We insert this into (5.15) and choose ρ = ∈ ( , ). Note that λ n ( L h, ) > n ≥ f ψ ⊥ u tran h in L (cid:0) (0 , h ρ ); (1 − tκ ( s )) dt ), we get by Lemma 4.3 andour choice of ρ = that λ n ( L h, ) & h − ρ = h / > n ≥ ∀ n ∈ I h , λ n ( T h ) ≤ − h + h / min (cid:16) λ n ( L + h ) , (cid:17) , where, for some constant c + > L + h is the operator acting on L (cid:0) [ − L, L ) (cid:1) as follows,(5.17) L + h = − h / (1 + c + h / ) d ds − κ ( s ) − h / κ ( s ) + c + h / . If we consider the eigenvalues of T h below the energy level ǫh , with ǫ < λ D (Ω), we still get λ n ( T h ) ≤ − h + h / min (cid:16) λ n ( L + h ) , ǫh (cid:17) Lower bounds.
For all ψ ∈ H (Ω), we write the lower bound q Ω h ( ψ ) = h Z Ω |∇ ψ | dx − h / Z ∂ Ω | u | ds ( x ) ≥ p h ( χ ψ )where p h ( χ ψ ) := p h, ( k ψ ) + p h, ( f ψ ) + p h, ( u ψ )and χ ψ = ( k ψ u tran h , f ψ , u ψ ) . The quadratic forms p h,i are defined as follows p h, ( k ψ ) = Z L − L (cid:16)(cid:0) λ h ( s ) − O ( h +2 ρ ) (cid:1) | k ψ ( s ) | + (cid:0) h − O ( h ) (cid:1) | k ′ ψ ( s ) | (cid:17) ds , p h, ( f ψ ) = h Z L − L (cid:18)Z h ρ (cid:16) | ∂ t f ψ | − O ( h ) | f ψ | (cid:17) adt − h − | f ψ ( s, | (cid:19) ds p h, ( u ψ ) = h Z Ω hρ |∇ u ψ | dx ≥ h λ N (Ω h ρ ) k u ψ k L (Ω hρ ) = 0 . For all i ∈ { , , } , let l h,i be the operator defined by the quadraticform p h,i . By the min-max principle, λ n ( T h ) ≥ λ n (cid:0) ⊕ i =1 l h,i (cid:1) , with λ ( l h, ) ≥ λ n ( l h, ) > n ≥
2. We choose ρ = and observe that, by Lemma 4.3, λ n ( l h, ) & h / > n ≥ c − > L − h the operator (acting on L (cid:0) [ − L, L ) (cid:1) )(5.18) L − h = − h / (1 − c − h / ) d ds − κ ( s ) − h / κ ( s ) − c − h / , we get(5.19) ∀ n ∈ I h , λ n ( T h ) ≥ − h + h / λ n ( L − h ) . When dealing with the eigenvalues of T h below ǫh , with ǫ < λ N (Ω),we still get λ n ( T h ) ≥ − h + h / min (cid:0) λ n ( L − h ) , h − / (cid:1) , becausemin (cid:0) − h + h / λ n ( L − h ) , (cid:1) = − h + min (cid:0) h / λ n ( L − h ) , h (cid:1) . Remark 5.3.
Consider ǫ ∈ (0 , λ N (Ω)). Since λ ( l h, ) = 0 is a simpleeigenvalue, the min-max principle allows us to extend (5.19) as follows.Set N ∗ ( h ) = max { n ≥ , − h + h / λ n ( L − h ) < } . Then, for h smallenough, we have ∀ n ∈ I ǫh ∩ [2 + N ∗ ( h ) , + ∞ ) , λ n ( T h ) ≥ − h + h / λ n ( L − h ) . Acknowledgements.
The authors would like to thank Gerd Grubb,Thierry Daud´e and Fran¸cois Nicoleau for helpful discussions. The firstauthor was inspired by the very interesting talks proposed at the sem-inar “Spectral geometry in the clouds” organized by A. Girouard andJ. Lagac´e and initially due to this terrible COVID period. The sec-ond author is supported by the Lebanese University within the project “Analytical and numerical aspects of the Ginzburg Landau model” . EMI-CLASSICAL EDGE STATES FOR THE ROBIN LAPLACIAN 31
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Email address : [email protected] (A. Kachmar)
Lebanese University, Department of Mathematics, Ha-dath, Lebanon.
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