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

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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 Universit de Montral 4 DePaul University Miniconference on

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SLIDE 1

Background The Polygon Case

Steklov Spectral Asymptotics for Polygons

  • M. Levitin1
  • L. Parnovski2
  • I. Polterovich3
  • D. Sher4

1University of Reading 2University College London 3Université de Montréal 4DePaul University

Miniconference on Sharp Eigenvalue Estimates for Partial Differential Operators, 2020

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 2

Background The Polygon Case

Outline

1

Background Definitions Spectral Asymptotics

2

The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 3

Background The Polygon Case Definitions Spectral Asymptotics

The Problem

Assume (Ωn, g) is a smooth compact manifold with smooth boundary. The Steklov eigenvalue problem is

  • ∆u = 0

in Ω;

∂u ∂n = λu

  • n ∂Ω.

Discrete spectrum: 0 = λ1 ≤ λ2 ≤ . . . , with corresponding eigenfunctions {um}.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 4

Background The Polygon Case Definitions Spectral Asymptotics

The Problem

Assume (Ωn, g) is a smooth compact manifold with smooth boundary. The Steklov eigenvalue problem is

  • ∆u = 0

in Ω;

∂u ∂n = λu

  • n ∂Ω.

Discrete spectrum: 0 = λ1 ≤ λ2 ≤ . . . , with corresponding eigenfunctions {um}.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 5

Background The Polygon Case Definitions 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 ne±inθ, n ∈ N. Scaling: if DL is a disk of circumference L, Steklov spectrum is {0, 2π L , 2π L , 4π L , 4π L , 6π L , 6π L . . .}.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 6

Background The Polygon Case Definitions 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 ne±inθ, n ∈ N. Scaling: if DL is a disk of circumference L, Steklov spectrum is {0, 2π L , 2π L , 4π L , 4π L , 6π L , 6π L . . .}.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 7

Background The Polygon Case Definitions 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 Df := ∂u

∂n.

Spectrum of D is Steklov spectrum {λm}. Eigenfunctions of D are {um|∂Ω}.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 8

Background The Polygon Case Definitions 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 Df := ∂u

∂n.

Spectrum of D is Steklov spectrum {λm}. Eigenfunctions of D are {um|∂Ω}.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 9

Background The Polygon Case Definitions 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(λm ≤ λ) = cn · Vol(∂Ω) · λn−1 + O(λn−2). Boundary locality (based on Lee-Uhlmann ’88). If Ω and Ω are isometric near the boundary then |λm(Ω) − λm( Ω)| = O(m−∞). Eigenfunction decay. If K is compactly contained in the interior of Ω, then for any k, umCk(K) = O(m−∞).

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 10

Background The Polygon Case Definitions 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(λm ≤ λ) = cn · Vol(∂Ω) · λn−1 + O(λn−2). Boundary locality (based on Lee-Uhlmann ’88). If Ω and Ω are isometric near the boundary then |λm(Ω) − λm( Ω)| = O(m−∞). Eigenfunction decay. If K is compactly contained in the interior of Ω, then for any k, umCk(K) = O(m−∞).

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 11

Background The Polygon Case Definitions 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(λm ≤ λ) = cn · Vol(∂Ω) · λn−1 + O(λn−2). Boundary locality (based on Lee-Uhlmann ’88). If Ω and Ω are isometric near the boundary then |λm(Ω) − λm( Ω)| = O(m−∞). Eigenfunction decay. If K is compactly contained in the interior of Ω, then for any k, umCk(K) = O(m−∞).

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 12

Background The Polygon Case Definitions 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(λm ≤ λ) = cn · Vol(∂Ω) · λn−1 + O(λn−2). Boundary locality (based on Lee-Uhlmann ’88). If Ω and Ω are isometric near the boundary then |λm(Ω) − λm( Ω)| = O(m−∞). Eigenfunction decay. If K is compactly contained in the interior of Ω, then for any k, umCk(K) = O(m−∞).

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 13

Background The Polygon Case Definitions 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 |λm(Ω) − λm(DL)| = 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

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SLIDE 14

Background The Polygon Case Definitions 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 |λm(Ω) − λm(DL)| = 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

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SLIDE 15

Background The Polygon Case Definitions 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 |λm(Ω) − λm(DL)| = 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

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Background The Polygon Case Definitions Spectral Asymptotics

What’s So Special About 2D?

A physics-style explananation: matched asymptotic expansions. Model eigenfunction: on lower half-plane (y < 0), consider ey 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" eσy cos(σ(x + χ)).

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 17

Background The Polygon Case Definitions Spectral Asymptotics

What’s So Special About 2D?

A physics-style explananation: matched asymptotic expansions. Model eigenfunction: on lower half-plane (y < 0), consider ey 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" eσy cos(σ(x + χ)).

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 18

Background The Polygon Case Definitions Spectral Asymptotics

What’s So Special About 2D?

A physics-style explananation: matched asymptotic expansions. Model eigenfunction: on lower half-plane (y < 0), consider ey 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" eσy cos(σ(x + χ)).

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 19

Background The Polygon Case Definitions Spectral Asymptotics

What’s So Special About 2D?

Take an overlapping, neighboring coordinate patch. Same

  • model. But phases must align (“matching").

Keep doing this. Go all the way around ∂Ω. When you come back to the first patch, the phases must match. This imposes a quantization condition on σ: σ = 2π L · n, n ∈ N. In fact, the true λ turn out to be close to these σ.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 20

Background The Polygon Case Definitions Spectral Asymptotics

What’s So Special About 2D?

Take an overlapping, neighboring coordinate patch. Same

  • model. But phases must align (“matching").

Keep doing this. Go all the way around ∂Ω. When you come back to the first patch, the phases must match. This imposes a quantization condition on σ: σ = 2π L · n, n ∈ N. In fact, the true λ turn out to be close to these σ.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 21

Background The Polygon Case Definitions Spectral Asymptotics

What’s So Special About 2D?

Take an overlapping, neighboring coordinate patch. Same

  • model. But phases must align (“matching").

Keep doing this. Go all the way around ∂Ω. When you come back to the first patch, the phases must match. This imposes a quantization condition on σ: σ = 2π L · n, n ∈ N. In fact, the true λ turn out to be close to these σ.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Curvilinear Polygons

Now we let Ω be a curvilinear polygon in R2, with all interior angles less than π:

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Previously Known Results

Steklov eigenvalue problem makes sense for Lipschitz domains, including Ω. Asymptotics are very weak (Agranovich ’06): λm = πm ℓ(∂Ω) + o(m). Our idea: apply matching asymptotic expansions method to this case, find an approximation for what the eigenvalues should be. . . and turn it all into a rigorous proof.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 24

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Previously Known Results

Steklov eigenvalue problem makes sense for Lipschitz domains, including Ω. Asymptotics are very weak (Agranovich ’06): λm = πm ℓ(∂Ω) + o(m). Our idea: apply matching asymptotic expansions method to this case, find an approximation for what the eigenvalues should be. . . and turn it all into a rigorous proof.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 25

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Local Models

Away from corners, same as before: zoom in to scale σ, see ey cos(x + χ). Near corners? Zoom in to scale σ, see (approximately) an infinite sector:

  • Levitin, Parnovski, Polterovich, Sher

Steklov Spectral Asymptotics for Polygons

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SLIDE 26

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Local Models

Away from corners, same as before: zoom in to scale σ, see ey cos(x + χ). Near corners? Zoom in to scale σ, see (approximately) an infinite sector:

  • Levitin, Parnovski, Polterovich, Sher

Steklov Spectral Asymptotics for Polygons

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SLIDE 27

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Local Models

Model: harmonic in sector, satisfies Robin conditions with parameter 1 along sides, oscillates with frequency 1 along sides, decays elsewhere. Explicitly constructed by Lewy, Peters (1950). Matching condition: if incoming behavior is cin cin

  • ·

eis e−is

  • ,
  • utgoing behavior is

cout cout

  • ·

eis e−is

  • , then:

cout cout

  • = A(α)

cin cin

  • ,

A(α) =

  • csc( π2

2α)

−i cot( π2

2α)

i cot( π2

2α)

csc( π2

2α)

  • .

If α = π/n, something different happens: total transmission (if n is odd), or total reflection (if n is even).

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 28

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Local Models

Model: harmonic in sector, satisfies Robin conditions with parameter 1 along sides, oscillates with frequency 1 along sides, decays elsewhere. Explicitly constructed by Lewy, Peters (1950). Matching condition: if incoming behavior is cin cin

  • ·

eis e−is

  • ,
  • utgoing behavior is

cout cout

  • ·

eis e−is

  • , then:

cout cout

  • = A(α)

cin cin

  • ,

A(α) =

  • csc( π2

2α)

−i cot( π2

2α)

i cot( π2

2α)

csc( π2

2α)

  • .

If α = π/n, something different happens: total transmission (if n is odd), or total reflection (if n is even).

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 29

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Local Models

Model: harmonic in sector, satisfies Robin conditions with parameter 1 along sides, oscillates with frequency 1 along sides, decays elsewhere. Explicitly constructed by Lewy, Peters (1950). Matching condition: if incoming behavior is cin cin

  • ·

eis e−is

  • ,
  • utgoing behavior is

cout cout

  • ·

eis e−is

  • , then:

cout cout

  • = A(α)

cin cin

  • ,

A(α) =

  • csc( π2

2α)

−i cot( π2

2α)

i cot( π2

2α)

csc( π2

2α)

  • .

If α = π/n, something different happens: total transmission (if n is odd), or total reflection (if n is even).

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

slide-30
SLIDE 30

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Local Models

Model: harmonic in sector, satisfies Robin conditions with parameter 1 along sides, oscillates with frequency 1 along sides, decays elsewhere. Explicitly constructed by Lewy, Peters (1950). Matching condition: if incoming behavior is cin cin

  • ·

eis e−is

  • ,
  • utgoing behavior is

cout cout

  • ·

eis e−is

  • , then:

cout cout

  • = A(α)

cin cin

  • ,

A(α) =

  • csc( π2

2α)

−i cot( π2

2α)

i cot( π2

2α)

csc( π2

2α)

  • .

If α = π/n, something different happens: total transmission (if n is odd), or total reflection (if n is even).

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 31

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Matching the Expansions

Start with a local model near a corner with incoming phase c := c c

  • and frequency/scale σ.

Outgoing phase from first corner: A(α1)c. Propagation along first side of length ℓ1 multiplies phase by B(σℓ1) := eiσℓ1 e−iσℓ1

  • .

Keep going. At the end the phase must be the same, so c = B(σℓn)A(αn) · · · B(σℓ1)A(α1)c. Quantization condition: σ is such that this matrix product has eigenvalue 1. Equivalently, Tr B(σℓn)A(αn) · · · B(σℓ1)A(α1) = 2. (1)

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 32

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Matching the Expansions

Start with a local model near a corner with incoming phase c := c c

  • and frequency/scale σ.

Outgoing phase from first corner: A(α1)c. Propagation along first side of length ℓ1 multiplies phase by B(σℓ1) := eiσℓ1 e−iσℓ1

  • .

Keep going. At the end the phase must be the same, so c = B(σℓn)A(αn) · · · B(σℓ1)A(α1)c. Quantization condition: σ is such that this matrix product has eigenvalue 1. Equivalently, Tr B(σℓn)A(αn) · · · B(σℓ1)A(α1) = 2. (1)

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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SLIDE 33

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Matching the Expansions

Start with a local model near a corner with incoming phase c := c c

  • and frequency/scale σ.

Outgoing phase from first corner: A(α1)c. Propagation along first side of length ℓ1 multiplies phase by B(σℓ1) := eiσℓ1 e−iσℓ1

  • .

Keep going. At the end the phase must be the same, so c = B(σℓn)A(αn) · · · B(σℓ1)A(α1)c. Quantization condition: σ is such that this matrix product has eigenvalue 1. Equivalently, Tr B(σℓn)A(αn) · · · B(σℓ1)A(α1) = 2. (1)

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

slide-34
SLIDE 34

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Matching the Expansions

Start with a local model near a corner with incoming phase c := c c

  • and frequency/scale σ.

Outgoing phase from first corner: A(α1)c. Propagation along first side of length ℓ1 multiplies phase by B(σℓ1) := eiσℓ1 e−iσℓ1

  • .

Keep going. At the end the phase must be the same, so c = B(σℓn)A(αn) · · · B(σℓ1)A(α1)c. Quantization condition: σ is such that this matrix product has eigenvalue 1. Equivalently, Tr B(σℓn)A(αn) · · · B(σℓ1)A(α1) = 2. (1)

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

slide-35
SLIDE 35

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Matching the Expansions

Start with a local model near a corner with incoming phase c := c c

  • and frequency/scale σ.

Outgoing phase from first corner: A(α1)c. Propagation along first side of length ℓ1 multiplies phase by B(σℓ1) := eiσℓ1 e−iσℓ1

  • .

Keep going. At the end the phase must be the same, so c = B(σℓn)A(αn) · · · B(σℓ1)A(α1)c. Quantization condition: σ is such that this matrix product has eigenvalue 1. Equivalently, Tr B(σℓn)A(αn) · · · B(σℓ1)A(α1) = 2. (1)

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

slide-36
SLIDE 36

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Matching the Expansions

Start with a local model near a corner with incoming phase c := c c

  • and frequency/scale σ.

Outgoing phase from first corner: A(α1)c. Propagation along first side of length ℓ1 multiplies phase by B(σℓ1) := eiσℓ1 e−iσℓ1

  • .

Keep going. At the end the phase must be the same, so c = B(σℓn)A(αn) · · · B(σℓ1)A(α1)c. Quantization condition: σ is such that this matrix product has eigenvalue 1. Equivalently, Tr B(σℓn)A(αn) · · · B(σℓ1)A(α1) = 2. (1)

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Main Theorem

Theorem (Levitin-Parnovski-Polterovich-S. ’19) Let Ω be a curvilinear polygon with angles α1, . . . , αn, side lengths ℓ1, . . . , ℓn, and all αi < π. Assume (for simplicity only) that no αi = π/n with n even. Denote the non-negative roots of (1) by {σm}∞

m=1. Then

|λm(Ω) − σm| = O(m−ε), where ε > 0 is explicit. If any αi = π/n with n even there is a replacement quantization condition. The quantization condition is for the zeroes of a generalized trigonometric polynomial in σ. It is easy to compute these numerically.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Main Theorem

Theorem (Levitin-Parnovski-Polterovich-S. ’19) Let Ω be a curvilinear polygon with angles α1, . . . , αn, side lengths ℓ1, . . . , ℓn, and all αi < π. Assume (for simplicity only) that no αi = π/n with n even. Denote the non-negative roots of (1) by {σm}∞

m=1. Then

|λm(Ω) − σm| = O(m−ε), where ε > 0 is explicit. If any αi = π/n with n even there is a replacement quantization condition. The quantization condition is for the zeroes of a generalized trigonometric polynomial in σ. It is easy to compute these numerically.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Main Theorem

Theorem (Levitin-Parnovski-Polterovich-S. ’19) Let Ω be a curvilinear polygon with angles α1, . . . , αn, side lengths ℓ1, . . . , ℓn, and all αi < π. Assume (for simplicity only) that no αi = π/n with n even. Denote the non-negative roots of (1) by {σm}∞

m=1. Then

|λm(Ω) − σm| = O(m−ε), where ε > 0 is explicit. If any αi = π/n with n even there is a replacement quantization condition. The quantization condition is for the zeroes of a generalized trigonometric polynomial in σ. It is easy to compute these numerically.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Quantum Graph Interpretation

The boundary of a curvilinear polygon is a metric graph G. Consider the eigenvalue problem for a quantum graph Laplacian on G: f ′′ = −νf, with vertex matching conditions at each vertex Vj: sin( π2 4αj )f|Vj+0 = cos( π2 4αj )f|Vj−0; cos( π2 4αj )f ′|Vj+0 = sin( π2 4αj )f ′|Vj−0. (2) This problem has eigenvalues νm, and counting with multiplicity, σm = √νm.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Quantum Graph Interpretation

The boundary of a curvilinear polygon is a metric graph G. Consider the eigenvalue problem for a quantum graph Laplacian on G: f ′′ = −νf, with vertex matching conditions at each vertex Vj: sin( π2 4αj )f|Vj+0 = cos( π2 4αj )f|Vj−0; cos( π2 4αj )f ′|Vj+0 = sin( π2 4αj )f ′|Vj−0. (2) This problem has eigenvalues νm, and counting with multiplicity, σm = √νm.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Quantum Graph Interpretation

The boundary of a curvilinear polygon is a metric graph G. Consider the eigenvalue problem for a quantum graph Laplacian on G: f ′′ = −νf, with vertex matching conditions at each vertex Vj: sin( π2 4αj )f|Vj+0 = cos( π2 4αj )f|Vj−0; cos( π2 4αj )f ′|Vj+0 = sin( π2 4αj )f ′|Vj−0. (2) This problem has eigenvalues νm, and counting with multiplicity, σm = √νm.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Quantum Graph Interpretation

Eigenfunctions of quantum graph are NOT same as boundary Steklov eigenfunctions – they are perturbed near the corners. The error does go to zero in the L2 sense but not in the H1 sense.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

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Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Quantum Graph Interpretation

Eigenfunctions of quantum graph are NOT same as boundary Steklov eigenfunctions – they are perturbed near the corners. The error does go to zero in the L2 sense but not in the H1 sense.

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

slide-45
SLIDE 45

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Brief Comments on Proofs

Despite the 1D interpretation, all proofs must involve two dimensions. Proofs proceed via explicit construction of quasimodes via intuition from matched asymptotic expansions. Build approximate eigenfunctions. Show that near each σm there is a true eigenvalue λj. Challenge #1: does this produce all λj? Challenge #2: how to get curvature all the way down to the vertices?

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

slide-46
SLIDE 46

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Brief Comments on Proofs

Despite the 1D interpretation, all proofs must involve two dimensions. Proofs proceed via explicit construction of quasimodes via intuition from matched asymptotic expansions. Build approximate eigenfunctions. Show that near each σm there is a true eigenvalue λj. Challenge #1: does this produce all λj? Challenge #2: how to get curvature all the way down to the vertices?

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

slide-47
SLIDE 47

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Brief Comments on Proofs

Despite the 1D interpretation, all proofs must involve two dimensions. Proofs proceed via explicit construction of quasimodes via intuition from matched asymptotic expansions. Build approximate eigenfunctions. Show that near each σm there is a true eigenvalue λj. Challenge #1: does this produce all λj? Challenge #2: how to get curvature all the way down to the vertices?

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

slide-48
SLIDE 48

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Brief Comments on Proofs

Despite the 1D interpretation, all proofs must involve two dimensions. Proofs proceed via explicit construction of quasimodes via intuition from matched asymptotic expansions. Build approximate eigenfunctions. Show that near each σm there is a true eigenvalue λj. Challenge #1: does this produce all λj? Challenge #2: how to get curvature all the way down to the vertices?

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons

slide-49
SLIDE 49

Background The Polygon Case Matched Asymptotic Expansions Main Result Quantum Graphs

Thank you for your time!

Levitin, Parnovski, Polterovich, Sher Steklov Spectral Asymptotics for Polygons