Steklov Spectral Asymptotics for Polygons M. Levitin 1 L. Parnovski 2 - PowerPoint PPT Presentation

Background The Polygon Case Steklov Spectral Asymptotics for Polygons M. Levitin 1 L. Parnovski 2 I. Polterovich 3 D. Sher 4 1 University of Reading 2 University College London 3 Université de Montréal 4 DePaul University Miniconference on Sharp Eigenvalue Estimates for Partial Differential Operators, 2020 Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background The Polygon Case Outline Background 1 Definitions Spectral Asymptotics The Polygon Case 2 Matched Asymptotic Expansions Main Result Quantum Graphs Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background Definitions The Polygon Case Spectral Asymptotics The Problem Assume (Ω n , g ) is a smooth compact manifold with smooth boundary. The Steklov eigenvalue problem is � ∆ u = 0 in Ω; ∂ u ∂ n = λ u on ∂ Ω . Discrete spectrum: 0 = λ 1 ≤ λ 2 ≤ . . . , with corresponding eigenfunctions { u m } . Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background Definitions The Polygon Case Spectral Asymptotics The Problem Assume (Ω n , g ) is a smooth compact manifold with smooth boundary. The Steklov eigenvalue problem is � ∆ u = 0 in Ω; ∂ u ∂ n = λ u on ∂ Ω . Discrete spectrum: 0 = λ 1 ≤ λ 2 ≤ . . . , with corresponding eigenfunctions { u m } . Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background Definitions The Polygon Case Spectral Asymptotics Example Let D be the unit disk. Then: Steklov spectrum is { 0 , 1 , 1 , 2 , 2 , 3 , 3 . . . } Eigenfunctions are 1, then r n e ± in θ , n ∈ N . Scaling: if D L is a disk of circumference L , Steklov spectrum is { 0 , 2 π L , 2 π L , 4 π L , 4 π L , 6 π L , 6 π L . . . } . Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background Definitions The Polygon Case Spectral Asymptotics Example Let D be the unit disk. Then: Steklov spectrum is { 0 , 1 , 1 , 2 , 2 , 3 , 3 . . . } Eigenfunctions are 1, then r n e ± in θ , n ∈ N . Scaling: if D L is a disk of circumference L , Steklov spectrum is { 0 , 2 π L , 2 π L , 4 π L , 4 π L , 6 π L , 6 π L . . . } . Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background Definitions The Polygon Case Spectral Asymptotics Operator Formulation Define the Dirichlet-to-Neumann map D : C ∞ ( ∂ Ω) → C ∞ ( ∂ Ω) by: Take f ∈ C ∞ ( ∂ Ω) ; Let u be the harmonic extension of f to Ω ; Set D f := ∂ u ∂ n . Spectrum of D is Steklov spectrum { λ m } . Eigenfunctions of D are { u m | ∂ Ω } . Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background Definitions The Polygon Case Spectral Asymptotics Operator Formulation Define the Dirichlet-to-Neumann map D : C ∞ ( ∂ Ω) → C ∞ ( ∂ Ω) by: Take f ∈ C ∞ ( ∂ Ω) ; Let u be the harmonic extension of f to Ω ; Set D f := ∂ u ∂ n . Spectrum of D is Steklov spectrum { λ m } . Eigenfunctions of D are { u m | ∂ Ω } . Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background Definitions The Polygon Case Spectral Asymptotics Spectral Asymptotics In the setting with ∂ Ω smooth, D is a first-order elliptic pseudodifferential operator on ∂ Ω . This has some consequences: A Weyl law for the eigenvalues: N ( λ m ≤ λ ) = c n · Vol ( ∂ Ω) · λ n − 1 + O ( λ n − 2 ) . Boundary locality (based on Lee-Uhlmann ’88). If Ω and � Ω are isometric near the boundary then | λ m (Ω) − λ m ( � Ω) | = O ( m −∞ ) . Eigenfunction decay. If K is compactly contained in the interior of Ω , then for any k , � u m � C k ( K ) = O ( m −∞ ) . Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background Definitions The Polygon Case Spectral Asymptotics Spectral Asymptotics In the setting with ∂ Ω smooth, D is a first-order elliptic pseudodifferential operator on ∂ Ω . This has some consequences: A Weyl law for the eigenvalues: N ( λ m ≤ λ ) = c n · Vol ( ∂ Ω) · λ n − 1 + O ( λ n − 2 ) . Boundary locality (based on Lee-Uhlmann ’88). If Ω and � Ω are isometric near the boundary then | λ m (Ω) − λ m ( � Ω) | = O ( m −∞ ) . Eigenfunction decay. If K is compactly contained in the interior of Ω , then for any k , � u m � C k ( K ) = O ( m −∞ ) . Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background Definitions The Polygon Case Spectral Asymptotics Spectral Asymptotics In the setting with ∂ Ω smooth, D is a first-order elliptic pseudodifferential operator on ∂ Ω . This has some consequences: A Weyl law for the eigenvalues: N ( λ m ≤ λ ) = c n · Vol ( ∂ Ω) · λ n − 1 + O ( λ n − 2 ) . Boundary locality (based on Lee-Uhlmann ’88). If Ω and � Ω are isometric near the boundary then | λ m (Ω) − λ m ( � Ω) | = O ( m −∞ ) . Eigenfunction decay. If K is compactly contained in the interior of Ω , then for any k , � u m � C k ( K ) = O ( m −∞ ) . Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background Definitions The Polygon Case Spectral Asymptotics Spectral Asymptotics In the setting with ∂ Ω smooth, D is a first-order elliptic pseudodifferential operator on ∂ Ω . This has some consequences: A Weyl law for the eigenvalues: N ( λ m ≤ λ ) = c n · Vol ( ∂ Ω) · λ n − 1 + O ( λ n − 2 ) . Boundary locality (based on Lee-Uhlmann ’88). If Ω and � Ω are isometric near the boundary then | λ m (Ω) − λ m ( � Ω) | = O ( m −∞ ) . Eigenfunction decay. If K is compactly contained in the interior of Ω , then for any k , � u m � C k ( K ) = O ( m −∞ ) . Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background Definitions The Polygon Case Spectral Asymptotics Spectral Asymptotics, 2D Case Theorem (Rozenblium ’86; Edward ’93) Suppose Ω is a compact surface with smooth, connected boundary of length L. Then | λ m (Ω) − λ m ( D L ) | = O ( m −∞ ) . Massive improvement over the usual Weyl law! Extended to the case of disconnected boundary by Girouard-Parnovski-Polterovich-S., 2014. Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background Definitions The Polygon Case Spectral Asymptotics Spectral Asymptotics, 2D Case Theorem (Rozenblium ’86; Edward ’93) Suppose Ω is a compact surface with smooth, connected boundary of length L. Then | λ m (Ω) − λ m ( D L ) | = O ( m −∞ ) . Massive improvement over the usual Weyl law! Extended to the case of disconnected boundary by Girouard-Parnovski-Polterovich-S., 2014. Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background Definitions The Polygon Case Spectral Asymptotics Spectral Asymptotics, 2D Case Theorem (Rozenblium ’86; Edward ’93) Suppose Ω is a compact surface with smooth, connected boundary of length L. Then | λ m (Ω) − λ m ( D L ) | = O ( m −∞ ) . Massive improvement over the usual Weyl law! Extended to the case of disconnected boundary by Girouard-Parnovski-Polterovich-S., 2014. Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background Definitions The Polygon Case Spectral Asymptotics What’s So Special About 2D? A physics-style explananation: matched asymptotic expansions . Model eigenfunction: on lower half-plane ( y < 0), consider e y cos( x + χ ) . Harmonic, satisfies Steklov (i.e. Robin) boundary condition with λ = 1. Take a coordinate patch near ∂ Ω . Let y point out of Ω and x point along ∂ Ω . An eigenfunction with eigenvalue σ “should look like" e σ y cos( σ ( x + χ )) . Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background Definitions The Polygon Case Spectral Asymptotics What’s So Special About 2D? A physics-style explananation: matched asymptotic expansions . Model eigenfunction: on lower half-plane ( y < 0), consider e y cos( x + χ ) . Harmonic, satisfies Steklov (i.e. Robin) boundary condition with λ = 1. Take a coordinate patch near ∂ Ω . Let y point out of Ω and x point along ∂ Ω . An eigenfunction with eigenvalue σ “should look like" e σ y cos( σ ( x + χ )) . Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

Background Definitions The Polygon Case Spectral Asymptotics What’s So Special About 2D? A physics-style explananation: matched asymptotic expansions . Model eigenfunction: on lower half-plane ( y < 0), consider e y cos( x + χ ) . Harmonic, satisfies Steklov (i.e. Robin) boundary condition with λ = 1. Take a coordinate patch near ∂ Ω . Let y point out of Ω and x point along ∂ Ω . An eigenfunction with eigenvalue σ “should look like" e σ y cos( σ ( x + χ )) . Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons