Eigenvalues of Robin Laplacians on infinite sectors and application to polygons

Transcription

1 Eigenvalues of Robin Laplacians on infinite sectors and application to polygons Magda Khalile (joint work with Konstantin Pankrashkin) Université Paris Sud /25 Robin Laplacians on infinite sectors 1 /

2 Robin eigenvalue problem Let Ω R d be an open set with a sufficiently regular boundary. We consider the eigenvalue problem : ψ = ( d j=1 ) 2 xj 2 ψ = Eψ on Ω, ψ ν = γψ on Ω, where ν is the outward unit normal of Ω, γ > 0 and E is a discrete eigenvalue. 2/25 Robin Laplacians on infinite sectors 2 /

3 Robin eigenvalue problem Let Ω R d be an open set with a sufficiently regular boundary. We consider the eigenvalue problem : ψ = ( d j=1 ) 2 xj 2 ψ = Eψ on Ω, ψ ν = γψ on Ω, where ν is the outward unit normal of Ω, γ > 0 and E is a discrete eigenvalue. More precisely, we study the spectral problem for the self-ajdoint operator T γ Ω on L 2 (Ω) associated with the sesquilinear form : t γ Ω (ψ, ψ) = ψ 2 dx γ ψ 2 dσ, ψ H 1 (Ω). Ω Ω 2/25 Robin Laplacians on infinite sectors 2 /

4 Smooth domains Main goal : Study of E n (T γ Ω ) as γ +. - Change of variables : E n (T γ Ω ) = γ2 E n (TγΩ 1 ). - Link with the study of superconductors. [Lacey-Ockendon-Sabina, 1998 ; Lou-Zhu,2004 ; Levitin-Parnovski 2008, Bruneau-Popoff,2016 ;...] 3/25 Robin Laplacians on infinite sectors 3 /

5 Smooth domains Main goal : Study of E n (T γ Ω ) as γ +. - Change of variables : E n (T γ Ω ) = γ2 E n (TγΩ 1 ). - Link with the study of superconductors. [Lacey-Ockendon-Sabina, 1998 ; Lou-Zhu,2004 ; Levitin-Parnovski 2008, Bruneau-Popoff,2016 ;...] Theorem [Daners-Kennedy, 2010] If Ω is C 1, for each fixed n N, E n (T γ Ω ) = γ2 + o(γ 2 ), γ +. Theorem [Exner-Minakov-Parnovski, 2014 ; Pankrashkin-Popoff, 2015] If Ω is C 3, for each fixed n N, E n (T γ Ω ) = γ2 (d 1)H max (Ω)γ + O(γ 2 3 ), γ +, where H max (Ω) is the maximum of the mean curvature of Ω. 3/25 Robin Laplacians on infinite sectors 3 /

6 What happens on non-smooth domains? Theorem [Levitin-Parnovski, 2008 ; Bruneau-Popoff, 2016] If Ω is a corner domain (Lipschitz, piecewise smooth boundary + little more), E 1 (T γ Ω ) = Cγ2 + o(γ 2 ), γ +, where C 1 depends only on the tangent cones of Ω. 4/25 Robin Laplacians on infinite sectors 4 /

7 What happens on non-smooth domains? Theorem [Levitin-Parnovski, 2008 ; Bruneau-Popoff, 2016] If Ω is a corner domain (Lipschitz, piecewise smooth boundary + little more), E 1 (T γ Ω ) = Cγ2 + o(γ 2 ), γ +, where C 1 depends only on the tangent cones of Ω. If Ω R 2 is a curvilinear polygon, can we obtain a more detailed eigenvalue asymptotics? In this case, the tangent cones are the infinite sectors. 4/25 Robin Laplacians on infinite sectors 4 /

8 What happens on non-smooth domains? Theorem [Levitin-Parnovski, 2008 ; Bruneau-Popoff, 2016] If Ω is a corner domain (Lipschitz, piecewise smooth boundary + little more), E 1 (T γ Ω ) = Cγ2 + o(γ 2 ), γ +, where C 1 depends only on the tangent cones of Ω. If Ω R 2 is a curvilinear polygon, can we obtain a more detailed eigenvalue asymptotics? In this case, the tangent cones are the infinite sectors. Theorem [Pankrashkin,2013] If Ω R 2 has a piecewise smooth boundary which admits non-convex corners then, E 1 (T γ Ω ) = γ2 κ max γ + O(γ 2 3 ), γ +. i.e : the non convex corners do not contribute in the asymptotics. Role of convex corners? /25 Robin Laplacians on infinite sectors 4 /

9 Robin Laplacian on infinite sectors α (0, π), U α := { x R 2 : arg (x 1 + ix 2 ) < α }. /25 Robin Laplacians on infinite sectors 5 /

10 Robin Laplacian on infinite sectors α (0, π), U α := { x R 2 : arg (x 1 + ix 2 ) < α }. T γ α = Robin Laplacian on L 2 (U α ), γ > 0 : T γ α ψ = ψ on U α, ψ ν = γψ on U α. 5/25 Robin Laplacians on infinite sectors 5 /

11 Robin Laplacian on infinite sectors α (0, π), U α := { x R 2 : arg (x 1 + ix 2 ) < α }. T γ α = Robin Laplacian on L 2 (U α ), γ > 0 : T γ α ψ = ψ on U α, ψ ν = γψ on U α. Behavior of the eigenvalues of T γ α with respect to α? 5/25 Robin Laplacians on infinite sectors 5 /

12 Robin Laplacian on infinite sectors α (0, π), U α := { x R 2 : arg (x 1 + ix 2 ) < α }. T γ α = Robin Laplacian on L 2 (U α ), γ > 0 : T γ α ψ = ψ on U α, ψ ν = γψ on U α. Behavior of the eigenvalues of T γ α with respect to α? U α is invariant by dilations : E n (T γ α ) = γ 2 E n (T 1 α). In the following : T 1 α := T α. 5/25 Robin Laplacians on infinite sectors 5 /

13 Some known results Proposition [Levitin-Parnovski, 2008] For all α (0, π), spec ess (T α ) = [ 1, + ). 6/25 Robin Laplacians on infinite sectors 6 /

14 Some known results Proposition [Levitin-Parnovski, 2008] For all α (0, π), spec ess (T α ) = [ 1, + ). If α π 2, inf spec(t α) = 1 = spec(t α ) = [ 1, + ). 6/25 Robin Laplacians on infinite sectors 6 /

15 Some known results Proposition [Levitin-Parnovski, 2008] For all α (0, π), spec ess (T α ) = [ 1, + ). If α π 2, inf spec(t α) = 1 = spec(t α ) = [ 1, + ). If α (0, π 2 ), E 1 (T α ) = 1 sin 2 (α) < 1, ϕ 1,α(x 1, x 2 ) = exp( x 1 sin(α) ). 6/25 Robin Laplacians on infinite sectors 6 /

16 Some known results Proposition [Levitin-Parnovski, 2008] For all α (0, π), spec ess (T α ) = [ 1, + ). If α π 2, inf spec(t α) = 1 = spec(t α ) = [ 1, + ). If α (0, π 2 ), E 1 (T α ) = 1 sin 2 (α) < 1, ϕ 1,α(x 1, x 2 ) = exp( x 1 sin(α) ). Main questions for α (0, π 2 ), 6/25 Robin Laplacians on infinite sectors 6 /

17 Some known results Proposition [Levitin-Parnovski, 2008] For all α (0, π), spec ess (T α ) = [ 1, + ). If α π 2, inf spec(t α) = 1 = spec(t α ) = [ 1, + ). If α (0, π 2 ), E 1 (T α ) = 1 sin 2 (α) < 1, ϕ 1,α(x 1, x 2 ) = exp( x 1 sin(α) ). Main questions for α (0, π 2 ), Is spec disc (T γ α ) finite or infinite? 6/25 Robin Laplacians on infinite sectors 6 /

18 Some known results Proposition [Levitin-Parnovski, 2008] For all α (0, π), spec ess (T α ) = [ 1, + ). If α π 2, inf spec(t α) = 1 = spec(t α ) = [ 1, + ). If α (0, π 2 ), E 1 (T α ) = 1 sin 2 (α) < 1, ϕ 1,α(x 1, x 2 ) = exp( x 1 sin(α) ). Main questions for α (0, π 2 ), Is spec disc (T γ α ) finite or infinite? What is the behavior (regularity, monotonicity) of the eigenvalues with respect to α? 6/25 Robin Laplacians on infinite sectors 6 /

19 Some known results Proposition [Levitin-Parnovski, 2008] For all α (0, π), spec ess (T α ) = [ 1, + ). If α π 2, inf spec(t α) = 1 = spec(t α ) = [ 1, + ). If α (0, π 2 ), E 1 (T α ) = 1 sin 2 (α) < 1, ϕ 1,α(x 1, x 2 ) = exp( x 1 sin(α) ). Main questions for α (0, π 2 ), Is spec disc (T γ α ) finite or infinite? What is the behavior (regularity, monotonicity) of the eigenvalues with respect to α? What is their behavior as α 0? 6/25 Robin Laplacians on infinite sectors 6 /

20 Some known results Proposition [Levitin-Parnovski, 2008] For all α (0, π), spec ess (T α ) = [ 1, + ). If α π 2, inf spec(t α) = 1 = spec(t α ) = [ 1, + ). If α (0, π 2 ), E 1 (T α ) = 1 sin 2 (α) < 1, ϕ 1,α(x 1, x 2 ) = exp( x 1 sin(α) ). Main questions for α (0, π 2 ), Is spec disc (T γ α ) finite or infinite? What is the behavior (regularity, monotonicity) of the eigenvalues with respect to α? What is their behavior as α 0? What are the properties of the associated eigenfunctions? 6/25 Robin Laplacians on infinite sectors 6 /

21 Finiteness of the spectrum and monotonicity Theorem The discrete spectrum of T α is finite for all α (0, π 2 ). /25 Robin Laplacians on infinite sectors 7 /

22 Finiteness of the spectrum and monotonicity Theorem The discrete spectrum of T α is finite for all α (0, π 2 ). -The result fails in dimension 3 (cones can have infinite discrete spectrum). -Proof based on the idea of A. Morame et F. Truc (2005) : reduction to a one-dimensional Bargman-type estimate. /25 Robin Laplacians on infinite sectors 7 /

23 Finiteness of the spectrum and monotonicity Theorem The discrete spectrum of T α is finite for all α (0, π 2 ). -The result fails in dimension 3 (cones can have infinite discrete spectrum). -Proof based on the idea of A. Morame et F. Truc (2005) : reduction to a one-dimensional Bargman-type estimate. Notation : N α = #{n N : E n (T α ) < 1}. /25 Robin Laplacians on infinite sectors 7 /

24 Finiteness of the spectrum and monotonicity Theorem The discrete spectrum of T α is finite for all α (0, π 2 ). -The result fails in dimension 3 (cones can have infinite discrete spectrum). -Proof based on the idea of A. Morame et F. Truc (2005) : reduction to a one-dimensional Bargman-type estimate. Notation : N α = #{n N : E n (T α ) < 1}. Proposition The eigenvalues of T α are non-decreasing and continuous with respect to α. (0, π/2) α N α is decreasing. For all α π/6, N α = 1. 7/25 Robin Laplacians on infinite sectors 7 /

25 Asymptotic behavior as the angle becomes small Proposition There exists κ > 0 such that N α κ/α as α 0. In particular, N α +, α 0. 8/25 Robin Laplacians on infinite sectors 8 /

26 Asymptotic behavior as the angle becomes small Proposition There exists κ > 0 such that N α κ/α as α 0. In particular, N α +, α 0. Theorem : First order asymptotics For each n N : 1 E n (T α ) = (2n 1) 2 + O(1), α 0. α2 The constant can t be improve : E 1 (T α ) = 1 α o(1), α /25 Robin Laplacians on infinite sectors 8 /

27 Ideas of the proof of the first order asymptotics To avoid the singularity near the origin we introduce a dense subspace of H 1 (U α ) : } F := {u C (U α ) R 1, R 2 > 0 : u = 0 for x < R 1, and x > R 2. Polar coordinates : U : L 2 (U α, dx) L 2 (V α, drdθ) u r 1 2 u(r cos(θ), r sin(θ)), 9/25 Robin Laplacians on infinite sectors 9 /

28 Ideas of the proof of the first order asymptotics To avoid the singularity near the origin we introduce a dense subspace of H 1 (U α ) : } F := {u C (U α ) R 1, R 2 > 0 : u = 0 for x < R 1, and x > R 2. Polar coordinates : U : L 2 (U α, dx) L 2 (V α, drdθ) u r 1 2 u(r cos(θ), r sin(θ)), G := U(F) = {v C (V α ) R 1, R 2 > 0 : u(r, θ) = 0 for r < R 1 and r > R 2 }. 9/25 Robin Laplacians on infinite sectors 9 /

29 The unitarily equivalent operator is Q α associated to where : q α (v, v) := t α (U (v), U (v)), v G, q α (v, v) = v r 2 1 v 2 V α 4 r 2 drdθ { 1 α + R r 2 v θ 2 dθ r v(r, α) 2 r v(r, α) }dr. 2 + α 10/25 Robin Laplacians on infinite sectors 10

30 The unitarily equivalent operator is Q α associated to where : q α (v, v) := t α (U (v), U (v)), v G, q α (v, v) = v r 2 1 v 2 V α 4 r 2 drdθ { 1 α + R r 2 v θ 2 dθ r v(r, α) 2 r v(r, α) }dr. 2 + α Robin Laplacian B α,r acting on L 2 ( α, α), r R + : B α,r u = u sur ( α, α) ±u (±α) = ru(±α). First eigenvalue : E 1 (α, r) associated to the eigenfunction φ α. 10/25 Robin Laplacians on infinite sectors 10

31 Reduction of the dimension : we apply q α on functions of the form v(r, θ) = f (r)φ α (r, θ) : { q α (v, v) = f (r) 2 1 R + 4r 2 f (r) 2 1 } αr f (r) 2 dr + K α (r) f (r) 2 dr. R + We define the operator H a acting on L 2 (R + ) by (H a )(v) = ( d 2 dr 2 1 4r 2 1 ) v(r), v Cc (R + ), ar and Ha its Friedrichs extension. Then, spec ess (Ha ) = [0, + ) and its discrete eigenvalues are : 1 E n (a) = (2n 1) 2 a 2, n N. 11/25 Robin Laplacians on infinite sectors 11

32 Orthogonal projections : Πv(r, θ) := f (r)φ α (r, θ), f (r) := Pv(r, θ) := v(r, θ) Πv(r, θ). For all α (0, 1) : ( ) H (1 α 2 )I α(1 α 2 ) 0 I 0 0 α α v(r, θ)φ(r, θ)dθ and ( ) ( ) Πv M Q Pv α I H α 0 I 0 0 M R +, I is the unitary operator satisfying I(Πv, Pv) = (f, Pv). We conclude with the min-max principle. ( ) Πv + M, Pv 12/25 Robin Laplacians on infinite sectors 12

33 Theorem : Complete asymptotic expansion For each n N, there exists λ j,n R, j N {0}, such that for all N N {0} : E n (T α ) = 1 α 2 N j=0 λ j,n α 2j + O(α 2N ), α 0, with λ 0,n = 1 (2n 1) 2. Proof : standard perturbation theory, each eigenvalue is simple as α 0. 13/25 Robin Laplacians on infinite sectors 13

34 Theorem : Complete asymptotic expansion For each n N, there exists λ j,n R, j N {0}, such that for all N N {0} : E n (T α ) = 1 α 2 N j=0 λ j,n α 2j + O(α 2N ), α 0, with λ 0,n = 1 (2n 1) 2. Proof : standard perturbation theory, each eigenvalue is simple as α 0. Theorem : An Agmon-type estimate for the eigenfunctions Let E be a discrete eigenvalue of T α and V be an associated eigenfunction. Then, for all ɛ (0, 1), ( V 2 + V 2) e 2(1 ɛ) 1 E x dx < +. U α 13/25 Robin Laplacians on infinite sectors 13

35 Application to Robin Laplacians on polygons V := {vertices of Ω}, 14/25 Robin Laplacians on infinite sectors 14

36 Application to Robin Laplacians on polygons V := {vertices of Ω}, α v := half aperture at v V, 14/25 Robin Laplacians on infinite sectors 14

37 Application to Robin Laplacians on polygons V := {vertices of Ω}, α v := half aperture at v V, Q γ := Robin Laplacian on L 2 (Ω), 14/25 Robin Laplacians on infinite sectors 14

38 Application to Robin Laplacians on polygons V := {vertices of Ω}, α v := half aperture at v V, Q γ := Robin Laplacian on L 2 (Ω), spec ess (Q γ ) =, spec disc (Q γ ) = {(E n (Q γ )) n N }. 14/25 Robin Laplacians on infinite sectors 14

39 Application to Robin Laplacians on polygons V := {vertices of Ω}, α v := half aperture at v V, Q γ := Robin Laplacian on L 2 (Ω), spec ess (Q γ ) =, spec disc (Q γ ) = {(E n (Q γ )) n N }. Behavior of E n (Q γ ) as γ +? Proposition [Levitin-Parnovski,2008 ; Bruneau-Popoff,2016] γ 2 E 1 (Q γ ) = sin 2 (min v V α v ) + o(γ2 ), γ +. 4/25 Robin Laplacians on infinite sectors 14

40 Model operator We define T the Laplacian acting on v V L2 (U αv ) and defined by : T = v V T αv. 15/25 Robin Laplacians on infinite sectors 15

41 Model operator We define T the Laplacian acting on v V L2 (U αv ) and defined by : T = v V T αv. Then, spec(t ) = v V spec(t α v ), spec ess (T ) = [ 1, + ), N := #{n N, E n (T ) < 1} = N αv < +, v V E 1 (T 1 ) = sin 2 (min v V α v ). 15/25 Robin Laplacians on infinite sectors 15

42 Asymptotics of the first eigenvalues of Q γ Theorem For all n N, E n (Q γ ) = γ 2 E n (T ) + O(e cγ ), γ +. 16/25 Robin Laplacians on infinite sectors 16

43 Asymptotics of the first eigenvalues of Q γ Theorem For all n N, E n (Q γ ) = γ 2 E n (T ) + O(e cγ ), γ +. Ideas of the proof [Bonnaillie-Noël-Dauge, 2006] : - Construction of quasi-modes : for v V, let ψn γ,v be a normalized eigenfunction of Tα γ v and χ v a smooth radial cut-off function such that supp χ v B(v, r). We define φ γ,v n := ψ γ,v n χ v. 16/25 Robin Laplacians on infinite sectors 16

44 Asymptotics of the first eigenvalues of Q γ Theorem For all n N, E n (Q γ ) = γ 2 E n (T ) + O(e cγ ), γ +. Ideas of the proof [Bonnaillie-Noël-Dauge, 2006] : - Construction of quasi-modes : for v V, let ψn γ,v be a normalized eigenfunction of Tα γ v and χ v a smooth radial cut-off function such that supp χ v B(v, r). We define φ γ,v n := ψ γ,v n χ v. - φ γ,v n D(Q γ ) and Q γ φ γ,v n γ 2 E n (T αv ) 2 φ γ,v n 2 = O(e cγ ), γ +. 16/25 Robin Laplacians on infinite sectors 16

45 Spectral theorem implies dist(e n (T γ α v ), spec(q γ )) = O(e cγ ), γ +. It s not enough... 17/25 Robin Laplacians on infinite sectors 17

46 Spectral theorem implies dist(e n (T γ α v ), spec(q γ )) = O(e cγ ), γ +. It s not enough... Notations : - Λ := {λ 1 < λ 2 <... < λ K } = eigenvalues of T without multiplicity, 17/25 Robin Laplacians on infinite sectors 17

47 Spectral theorem implies dist(e n (T γ α v ), spec(q γ )) = O(e cγ ), γ +. It s not enough... Notations : - Λ := {λ 1 < λ 2 <... < λ K } = eigenvalues of T without multiplicity, - S l := {(n, v) : v V, 1 n N αv : E n (T γ α v ) = γ 2 λ l }, 17/25 Robin Laplacians on infinite sectors 17

48 Spectral theorem implies dist(e n (T γ α v ), spec(q γ )) = O(e cγ ), γ +. It s not enough... Notations : - Λ := {λ 1 < λ 2 <... < λ K } = eigenvalues of T without multiplicity, - S l := {(n, v) : v V, 1 n N αv : E n (T γ α v ) = γ 2 λ l }, - κ l = #S l = multiplicity of λ l. 17/25 Robin Laplacians on infinite sectors 17

49 Spectral theorem implies dist(e n (T γ α v ), spec(q γ )) = O(e cγ ), γ +. It s not enough... Notations : - Λ := {λ 1 < λ 2 <... < λ K } = eigenvalues of T without multiplicity, - S l := {(n, v) : v V, 1 n N αv : E n (T γ α v ) = γ 2 λ l }, - κ l = #S l = multiplicity of λ l. Properties of quasi-modes For γ large enough, (φ γ,v n ) K (n,v) S l l=1 is linearly independent, 17/25 Robin Laplacians on infinite sectors 17

50 Spectral theorem implies dist(e n (T γ α v ), spec(q γ )) = O(e cγ ), γ +. It s not enough... Notations : - Λ := {λ 1 < λ 2 <... < λ K } = eigenvalues of T without multiplicity, - S l := {(n, v) : v V, 1 n N αv : E n (T γ α v ) = γ 2 λ l }, - κ l = #S l = multiplicity of λ l. Properties of quasi-modes For γ large enough, (φ γ,v n ) K (n,v) S l l=1 is linearly independent, q γ (φ γ,v n, φ γ,v n ) E n (Tα γ v ) γce cγ, 17/25 Robin Laplacians on infinite sectors 17

51 Spectral theorem implies dist(e n (T γ α v ), spec(q γ )) = O(e cγ ), γ +. It s not enough... Notations : - Λ := {λ 1 < λ 2 <... < λ K } = eigenvalues of T without multiplicity, - S l := {(n, v) : v V, 1 n N αv : E n (T γ α v ) = γ 2 λ l }, - κ l = #S l = multiplicity of λ l. Properties of quasi-modes For γ large enough, (φ γ,v n ) K (n,v) S l l=1 q γ (φ γ,v n, φ γ,v q γ (φ γ,v i is linearly independent, n ) E n (Tα γ v ) γce cγ,, φ γ,v j ) γce cγ, i j. 17/25 Robin Laplacians on infinite sectors 17

52 Spectral theorem implies dist(e n (T γ α v ), spec(q γ )) = O(e cγ ), γ +. It s not enough... Notations : - Λ := {λ 1 < λ 2 <... < λ K } = eigenvalues of T without multiplicity, - S l := {(n, v) : v V, 1 n N αv : E n (T γ α v ) = γ 2 λ l }, - κ l = #S l = multiplicity of λ l. Properties of quasi-modes For γ large enough, (φ γ,v n ) K (n,v) S l l=1 q γ (φ γ,v n, φ γ,v q γ (φ γ,v i is linearly independent, n ) E n (Tα γ v ) γce cγ,, φ γ,v j ) γce cγ, i j. Proof : Localization property of ψ γ,v n. 17/25 Robin Laplacians on infinite sectors 17

53 Lemma For all 1 l K and for γ large enough, E κ1+...+κ l (Q γ ) γ 2 λ l + Cγ 2 e cγ, E κ1+...+κ l +1(Q γ ) γ 2 λ l+1 C. Proof : Min-max principle + partition of unity. 18/25 Robin Laplacians on infinite sectors 18

54 Lemma For all 1 l K and for γ large enough, E κ1+...+κ l (Q γ ) γ 2 λ l + Cγ 2 e cγ, E κ1+...+κ l +1(Q γ ) γ 2 λ l+1 C. Proof : Min-max principle + partition of unity. Cluster of eigenvalues For 1 n κ 1, Cγ 4 3 En (Q γ ) γ 2 E n (T ) Cγ 2 e cγ. For κ 1 < n N, C E n (Q γ ) γ 2 E n (T ) Cγ 2 e cγ. It s not enough... 18/25 Robin Laplacians on infinite sectors 18

55 Spectral approximation Let A be a self-adjoint operator acting on a Hilbert space H and λ R. If there exists ψ 1,..., ψ n D(A) linearly independent and η > 0 such that then, (A λ)ψ j η ψ j, j = 1,..., n, dim Ran P A (λ Cη, λ + Cη) n, where P A (a, b)= spectral projection of A on (a, b), C > 0 depends on the Gramian matrix of (ψ j ) j. In particular, if spec ess (A) (λ Cη, λ + Cη) =, there exist at least n eigenvalues in (λ cη, λ + cη). 9/25 Robin Laplacians on infinite sectors 19

56 Spectral approximation Let A be a self-adjoint operator acting on a Hilbert space H and λ R. If there exists ψ 1,..., ψ n D(A) linearly independent and η > 0 such that then, (A λ)ψ j η ψ j, j = 1,..., n, dim Ran P A (λ Cη, λ + Cη) n, where P A (a, b)= spectral projection of A on (a, b), C > 0 depends on the Gramian matrix of (ψ j ) j. In particular, if spec ess (A) (λ Cη, λ + Cη) =, there exist at least n eigenvalues in (λ cη, λ + cη). In our case : - spec ess (Q γ ) =, - η = O(e cγ ). 19/25 Robin Laplacians on infinite sectors 19

57 Work in progress The asymptotics remains true for curvilinear polygons but the remainders are polynomials. 20/25 Robin Laplacians on infinite sectors 20

58 Work in progress The asymptotics remains true for curvilinear polygons but the remainders are polynomials. What are the differences? - Construction of test functions : φ γ,v n (x) := χ γ v (ψn γ,v f v (x)), for x near v. 20/25 Robin Laplacians on infinite sectors 20

59 Work in progress The asymptotics remains true for curvilinear polygons but the remainders are polynomials. What are the differences? - Construction of test functions : φ γ,v n (x) := χ γ v (ψn γ,v f v (x)), for x near v. - Pseudo quasi-modes only : φ γ,v n / D(Q γ ). 20/25 Robin Laplacians on infinite sectors 20

60 Work in progress The asymptotics remains true for curvilinear polygons but the remainders are polynomials. What are the differences? - Construction of test functions : φ γ,v n (x) := χ γ v (ψn γ,v f v (x)), for x near v. - Pseudo quasi-modes only : φ γ,v n / D(Q γ ). - Estimates on φ γ,v n are polynomials because of the change of variables. 20/25 Robin Laplacians on infinite sectors 20

61 Work in progress The asymptotics remains true for curvilinear polygons but the remainders are polynomials. What are the differences? - Construction of test functions : φ γ,v n (x) := χ γ v (ψn γ,v f v (x)), for x near v. - Pseudo quasi-modes only : φ γ,v n / D(Q γ ). - Estimates on φ γ,v n are polynomials because of the change of variables. Important remark Proof = E N +1 γ 2 κ max γ + O(γ 2 3 ), γ +. What happens for E N +j? 20/25 Robin Laplacians on infinite sectors 20

62 Weyl asymptotics We want to study N(Q γ, cγ 2 ) := #{n N, E n (Q γ ) < cγ 2 } as γ +. What are the interesting constants c R? 21/25 Robin Laplacians on infinite sectors 21

63 Weyl asymptotics We want to study N(Q γ, cγ 2 ) := #{n N, E n (Q γ ) < cγ 2 } as γ +. What are the interesting constants c R? Theorem [Helffer-Kachmar-Raymond, 2017] Let D R 2 be an open, bounded connected set such that D is C 4 smooth, and T γ be the Robin Laplacian acting on L 2 (D). Then, for all λ R, γ N(T γ D, γ2 + λγ) (κ(s) + λ)+ dσ, γ + π D and for all E ( 1, 0), N(T γ D, Eγ2 ) is the curvature of D. γ + D π γ E + 1, where D s κ(s) Remark. For E < 1, lim γ + N(T γ D, Eγ2 ) = 0. 21/25 Robin Laplacians on infinite sectors 21

64 Work in progress The Weyl formulae remain true for curvilinear polygons. - There is no contribution of the vertices in the asymptotics. - If Ω is a polygon with straight edges, N(Q γ, γ 2 ) lim = 0. γ + γ - Ideas of the proof : Upper bound : partition of unity adapted to truncated sectors : the truncated sectors do not contribute, the regular part gives the asymptotics. Lower bound : Dirichlet bracketing. 22/25 Robin Laplacians on infinite sectors 22

65 What comes next? Asymptotics of eigenvalues on circular cones as the angle goes to 0? What happens for the next eigenvalues, i.e : for j N, E N +j(q γ ) γ +? Can we adapt the proof in higher dimension? Study of polyhedra? 3/25 Robin Laplacians on infinite sectors 23

66 V. Bonnaillie-Noël, M. Dauge : Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners. Ann. Henri Poincaré, 7, 5, pp (2006). V. Bruneau, N. Popoff : On the negative spectrum of the Robin Laplacian in corner domains.anal. PDE 9, no.5, (2016). D. Daners, J. B Kennedy : On the asymptotic behaviour of the eigenvalues of a Robin problem. Differ. Integr. Eq.23 :7/8 (2010) M. Khalile, K. Pankrashkin : Eigenvalues of Robin Laplacians in infinite sectors. Preprint available at M. Levitin, L. Parnovski : On the principal eigenvalue of a Robin problem with a large parameter.math. Nachr. 281 (2008) B.J. McCartin :Laplacian eigenstructure on the equilateral triangle. Hikari, A. Morame, F. Truc : Remarks on the spectrum of the Neumann problem with magnetic field in the half-space. J. Math. Phys. 46 (2005) K. Pankrashkin, N. Popoff : Mean curvature bounds and eigenvalues of Robin Laplacians.Calc. Var. 54 (2015) /25 Robin Laplacians on infinite sectors 24

67 Thank you for your attention 25/25 Robin Laplacians on infinite sectors 25