A coupling problem for entire functions
Jonathan Eckhardt
University of Vienna
[email protected]
May 23, 2017
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Let σ be a discrete set of nonzero reals such that
X 1
<∞
|λ|
λ∈σ
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Let σ be a discrete set of nonzero reals such that
Y
X 1
z
→
W (z) =
1−
<∞
λ
|λ|
λ∈σ
λ∈σ
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Let σ be a discrete set of nonzero reals such that
Y
X 1
z
→
W (z) =
1−
<∞
λ
|λ|
λ∈σ
λ∈σ
Given a sequence η ∈
R̂σ
(where R̂ = R ∪ {∞}), we consider:
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Let σ be a discrete set of nonzero reals such that
Y
X 1
z
→
W (z) =
1−
<∞
λ
|λ|
λ∈σ
λ∈σ
Given a sequence η ∈
R̂σ
(where R̂ = R ∪ {∞}), we consider:
Coupling problem
Find a pair of real entire functions (Φ− , Φ+ ) of exp. type zero with
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Let σ be a discrete set of nonzero reals such that
Y
X 1
z
→
W (z) =
1−
<∞
λ
|λ|
λ∈σ
λ∈σ
Given a sequence η ∈
R̂σ
(where R̂ = R ∪ {∞}), we consider:
Coupling problem
Find a pair of real entire functions (Φ− , Φ+ ) of exp. type zero with
(C) Coupling condition:
Φ− (λ) = η(λ)Φ+ (λ),
Jonathan Eckhardt
λ∈σ
A coupling problem for entire functions
A coupling problem for entire functions
Let σ be a discrete set of nonzero reals such that
Y
X 1
z
→
W (z) =
1−
<∞
λ
|λ|
λ∈σ
λ∈σ
Given a sequence η ∈
R̂σ
(where R̂ = R ∪ {∞}), we consider:
Coupling problem
Find a pair of real entire functions (Φ− , Φ+ ) of exp. type zero with
(C) Coupling condition:
Φ− (λ) = η(λ)Φ+ (λ),
(G) Growth and positivity condition:
zΦ− (z)Φ+ (z)
Im
≥ 0,
W (z)
Jonathan Eckhardt
λ∈σ
Im(z) > 0
A coupling problem for entire functions
A coupling problem for entire functions
Let σ be a discrete set of nonzero reals such that
Y
X 1
z
→
W (z) =
1−
<∞
λ
|λ|
λ∈σ
λ∈σ
Given a sequence η ∈
R̂σ
(where R̂ = R ∪ {∞}), we consider:
Coupling problem
Find a pair of real entire functions (Φ− , Φ+ ) of exp. type zero with
(C) Coupling condition:
Φ− (λ) = η(λ)Φ+ (λ),
(G) Growth and positivity condition:
zΦ− (z)Φ+ (z)
Im
≥ 0,
W (z)
λ∈σ
Im(z) > 0
(N) Normalization condition:
Φ− (0) = Φ+ (0) = 1
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
For a solution (Φ− , Φ+ ) of the coupling problem
z 7→
zΦ− (z)Φ+ (z)
W (z)
is a Herglotz–Nevanlinna function:
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
For a solution (Φ− , Φ+ ) of the coupling problem
z 7→
zΦ− (z)Φ+ (z)
W (z)
is a Herglotz–Nevanlinna function:
• Zeros of Φ− and Φ+ are real and interlace the set σ
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
For a solution (Φ− , Φ+ ) of the coupling problem
z 7→
zΦ− (z)Φ+ (z)
W (z)
is a Herglotz–Nevanlinna function:
• Zeros of Φ− and Φ+ are real and interlace the set σ
• The functions Φ− and Φ+ are of genus zero and satisfy
Y
|z|
|Φ± (z)| ≤
1+
|λ|
λ∈σ
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
For a solution (Φ− , Φ+ ) of the coupling problem
z 7→
zΦ− (z)Φ+ (z)
W (z)
is a Herglotz–Nevanlinna function:
• Zeros of Φ− and Φ+ are real and interlace the set σ
• The functions Φ− and Φ+ are of genus zero and satisfy
Y
|z|
|Φ± (z)| ≤
1+
|λ|
λ∈σ
• Residues at poles are necessarily negative:
η(λ)Φ+ (λ)2
≤0
λW 0 (λ)
η(λ)
≤0
λW 0 (λ)
for all those λ ∈ σ for which the coupling constant η(λ) is finite
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
For a solution (Φ− , Φ+ ) of the coupling problem
z 7→
zΦ− (z)Φ+ (z)
W (z)
is a Herglotz–Nevanlinna function:
• Zeros of Φ− and Φ+ are real and interlace the set σ
• The functions Φ− and Φ+ are of genus zero and satisfy
Y
|z|
|Φ± (z)| ≤
1+
|λ|
λ∈σ
• Residues at poles are necessarily negative:
η(λ)Φ+ (λ)2
≤0
λW 0 (λ)
η(λ)
≤0
λW 0 (λ)
for all those λ ∈ σ for which the coupling constant η(λ) is finite
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Definition
Coupling constants η ∈ R̂σ are called admissible
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Definition
Coupling constants η ∈ R̂σ are called admissible if the inequality
η(λ)
≤0
λW 0 (λ)
holds for all those λ ∈ σ for which η(λ) is finite
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Definition
Coupling constants η ∈ R̂σ are called admissible if the inequality
η(λ)
≤0
λW 0 (λ)
holds for all those λ ∈ σ for which η(λ) is finite
Theorem (Existence and Uniqueness)
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Definition
Coupling constants η ∈ R̂σ are called admissible if the inequality
η(λ)
≤0
λW 0 (λ)
holds for all those λ ∈ σ for which η(λ) is finite
Theorem (Existence and Uniqueness)
If the coupling constants η ∈ R̂σ are admissible, then the coupling
problem with data η has a unique solution
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Definition
Coupling constants η ∈ R̂σ are called admissible if the inequality
η(λ)
≤0
λW 0 (λ)
holds for all those λ ∈ σ for which η(λ) is finite
Theorem (Existence and Uniqueness)
If the coupling constants η ∈ R̂σ are admissible, then the coupling
problem with data η has a unique solution
Theorem (Stability)
The solution depends continuously on the data
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Example
Suppose that σ = {λ0 } for some nonzero λ0 ∈ R so that
W (z) = 1 −
Jonathan Eckhardt
z
λ0
A coupling problem for entire functions
A coupling problem for entire functions
Example
Suppose that σ = {λ0 } for some nonzero λ0 ∈ R so that
W (z) = 1 −
z
λ0
• Some η ∈ R̂σ is admissible ⇔ η(λ0 ) is infinite or non-negative
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Example
Suppose that σ = {λ0 } for some nonzero λ0 ∈ R so that
W (z) = 1 −
z
λ0
• Some η ∈ R̂σ is admissible ⇔ η(λ0 ) is infinite or non-negative
• In this case, the unique solution (Φ− , Φ+ ) is given by
1 − min 1, η(λ0 )∓1
Φ± (z) = 1 − z
λ0
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Example
Suppose that σ = {λ0 } for some nonzero λ0 ∈ R so that
W (z) = 1 −
z
λ0
• Some η ∈ R̂σ is admissible ⇔ η(λ0 ) is infinite or non-negative
• In this case, the unique solution (Φ− , Φ+ ) is given by
1 − min 1, η(λ0 )∓1
Φ± (z) = 1 − z
λ0
• When the coupling constant η(λ0 ) is finite and negative, the
coupling problem with data η has no solution at all
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
General data
Let η ∈ R̂σ be coupling constants
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
General data
Let η ∈ R̂σ be coupling constants and define η̃ ∈ R̂σ by
(
η(λ), λ ∈ σ\ρ,
η̃(λ) =
0,
λ ∈ ρ,
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
General data
Let η ∈ R̂σ be coupling constants and define η̃ ∈ R̂σ by
(
η(λ), λ ∈ σ\ρ,
η̃(λ) =
0,
λ ∈ ρ,
where the set ρ consists of all λ ∈ σ for which η(λ) is finite and
η(λ)
>0
λW 0 (λ)
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
General data
Let η ∈ R̂σ be coupling constants and define η̃ ∈ R̂σ by
(
η(λ), λ ∈ σ\ρ,
η̃(λ) =
0,
λ ∈ ρ,
where the set ρ consists of all λ ∈ σ for which η(λ) is finite and
η(λ)
>0
λW 0 (λ)
• The new coupling constants η̃ are admissible
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
General data
Let η ∈ R̂σ be coupling constants and define η̃ ∈ R̂σ by
(
η(λ), λ ∈ σ\ρ,
η̃(λ) =
0,
λ ∈ ρ,
where the set ρ consists of all λ ∈ σ for which η(λ) is finite and
η(λ)
>0
λW 0 (λ)
• The new coupling constants η̃ are admissible
• Unique solution (Φ− , Φ+ ) of the coupling problem with data η̃
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
General data
Let η ∈ R̂σ be coupling constants and define η̃ ∈ R̂σ by
(
η(λ), λ ∈ σ\ρ,
η̃(λ) =
0,
λ ∈ ρ,
where the set ρ consists of all λ ∈ σ for which η(λ) is finite and
η(λ)
>0
λW 0 (λ)
• The new coupling constants η̃ are admissible
• Unique solution (Φ− , Φ+ ) of the coupling problem with data η̃
• The coupling problem with data η is solvable if and only if the
function Φ+ vanishes on the set ρ
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
General data
Let η ∈ R̂σ be coupling constants and define η̃ ∈ R̂σ by
(
η(λ), λ ∈ σ\ρ,
η̃(λ) =
0,
λ ∈ ρ,
where the set ρ consists of all λ ∈ σ for which η(λ) is finite and
η(λ)
>0
λW 0 (λ)
• The new coupling constants η̃ are admissible
• Unique solution (Φ− , Φ+ ) of the coupling problem with data η̃
• The coupling problem with data η is solvable if and only if the
function Φ+ vanishes on the set ρ
• In this case, the solution of the coupling problem with data η is
unique and coincides with the solution with data η̃
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Relevance
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Relevance
1
Inverse spectral theory
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Relevance
1
Inverse spectral theory
(Indefinite) Krein strings, canonical systems
...with two singular endpoints but purely discrete spectrum
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Relevance
1
Inverse spectral theory
(Indefinite) Krein strings, canonical systems
...with two singular endpoints but purely discrete spectrum
2
Nonlinear wave equations
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Relevance
1
Inverse spectral theory
(Indefinite) Krein strings, canonical systems
...with two singular endpoints but purely discrete spectrum
2
Nonlinear wave equations (completely integrable systems)
Camassa–Holm equation, Hunter–Saxton equation
...when the underlying spectral problem is purely discrete
→ replacement of Riemann–Hilbert problems
Jonathan Eckhardt
A coupling problem for entire functions
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
with a positive Borel measure ω on (0, 1) satisfying
Z 1
(1 − x)x dω(x) < ∞
−f 00 = z ωf
0
Jonathan Eckhardt
A coupling problem for entire functions
(∗)
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
with a positive Borel measure ω on (0, 1) satisfying
Z 1
(1 − x)x dω(x) < ∞
−f 00 = z ωf
0
• Spectrum σ is positive with
X 1
<∞
|λ|
λ∈σ
Jonathan Eckhardt
A coupling problem for entire functions
(∗)
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
with a positive Borel measure ω on (0, 1) satisfying
Z 1
(1 − x)x dω(x) < ∞
−f 00 = z ωf
(∗)
0
• Spectrum σ is positive with
X 1
<∞
|λ|
λ∈σ
• Solutions: φ(z, x) ∼ x as x → 0 and ψ(z, x) ∼ 1 − x as x → 1
Jonathan Eckhardt
A coupling problem for entire functions
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
with a positive Borel measure ω on (0, 1) satisfying
Z 1
(1 − x)x dω(x) < ∞
−f 00 = z ωf
(∗)
0
• Spectrum σ is positive with
X 1
<∞
|λ|
λ∈σ
• Solutions: φ(z, x) ∼ x as x → 0 and ψ(z, x) ∼ 1 − x as x → 1
• Wronskian:
Y
z
0
0
ψ(z, x)φ (z, x) − ψ (z, x)φ(z, x) =
1−
= W (z)
λ
λ∈σ
Jonathan Eckhardt
A coupling problem for entire functions
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
with a positive Borel measure ω on (0, 1) satisfying
Z 1
(1 − x)x dω(x) < ∞
−f 00 = z ωf
(∗)
0
• Spectrum σ is positive with
X 1
<∞
|λ|
λ∈σ
• Solutions: φ(z, x) ∼ x as x → 0 and ψ(z, x) ∼ 1 − x as x → 1
• Wronskian:
Y
z
0
0
ψ(z, x)φ (z, x) − ψ (z, x)φ(z, x) =
1−
= W (z)
λ
λ∈σ
• Define norming constants γλ ∈ R+ for λ ∈ σ:
Z 1
γλ =
|φ0 (λ, x)|2 dx
0
Jonathan Eckhardt
A coupling problem for entire functions
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
with a positive Borel measure ω on (0, 1) satisfying
Z 1
(1 − x)x dω(x) < ∞
−f 00 = z ωf
(∗)
0
• Spectrum σ is positive with
X 1
<∞
|λ|
λ∈σ
• Solutions: φ(z, x) ∼ x as x → 0 and ψ(z, x) ∼ 1 − x as x → 1
• Wronskian:
Y
z
0
0
ψ(z, x)φ (z, x) − ψ (z, x)φ(z, x) =
1−
= W (z)
λ
λ∈σ
• Define norming constants γλ ∈ R+ for λ ∈ σ:
Z 1
X
γλ =
|φ0 (λ, x)|2 dx → µ =
γλ−1 δλ
0
λ∈σ
Jonathan Eckhardt
A coupling problem for entire functions
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
with a positive Borel measure ω on (0, 1) satisfying
Z 1
(1 − x)x dω(x) < ∞
−f 00 = z ωf
(∗)
0
• Spectrum σ is positive with
X 1
<∞
|λ|
λ∈σ
• Solutions: φ(z, x) ∼ x as x → 0 and ψ(z, x) ∼ 1 − x as x → 1
• Wronskian:
Y
z
0
0
ψ(z, x)φ (z, x) − ψ (z, x)φ(z, x) =
1−
= W (z)
λ
λ∈σ
• Define norming constants γλ ∈ R+ for λ ∈ σ:
Z 1
X
γλ =
|φ0 (λ, x)|2 dx → µ =
γλ−1 δλ
0
λ∈σ
How to recover ω from the eigenvalues and norming constants?
Jonathan Eckhardt
A coupling problem for entire functions
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
−f 00 = z ωf
with a positive Borel measure ω on (0, 1) satisfying (∗)
Jonathan Eckhardt
A coupling problem for entire functions
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
−f 00 = z ωf
with a positive Borel measure ω on (0, 1) satisfying (∗)
1 For every eigenvalue λ ∈ σ:
γλ
ψ(λ, · )
φ(λ, · ) = −
λW 0 (λ)
Jonathan Eckhardt
A coupling problem for entire functions
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
−f 00 = z ωf
with a positive Borel measure ω on (0, 1) satisfying (∗)
1 For every eigenvalue λ ∈ σ:
γλ
ψ(λ, · )
φ(λ, · ) = −
λW 0 (λ)
2
Green’s function is a Herglotz–Nevanlinna function:
zφ(z, x)ψ(z, x)
z 7→
W (z)
Jonathan Eckhardt
A coupling problem for entire functions
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
−f 00 = z ωf
with a positive Borel measure ω on (0, 1) satisfying (∗)
1 For every eigenvalue λ ∈ σ:
γλ
ψ(λ, · )
φ(λ, · ) = −
λW 0 (λ)
2
Green’s function is a Herglotz–Nevanlinna function:
zφ(z, x)ψ(z, x)
z 7→
W (z)
3
Normalization:
φ(0, x) = x,
Jonathan Eckhardt
ψ(0, x) = 1 − x
A coupling problem for entire functions
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
−f 00 = z ωf
with a positive Borel measure ω on (0, 1) satisfying (∗)
1 For every eigenvalue λ ∈ σ:
γλ
ψ(λ, · )
φ(λ, · ) = −
λW 0 (λ)
2
Green’s function is a Herglotz–Nevanlinna function:
zφ(z, x)ψ(z, x)
z 7→
W (z)
3
Normalization:
φ(0, x) = x,
ψ(0, x) = 1 − x
For each fixed x ∈ (0, 1), the pair (Φ− , Φ+ ) defined by
φ(z, x)
ψ(z, x)
,
Φ+ (z) =
x
1−x
is the solution of a coupling problem
Φ− (z) =
Jonathan Eckhardt
A coupling problem for entire functions
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
−f 00 = z ωf
with a positive Borel measure ω on (0, 1) satisfying (∗)
1 For every eigenvalue λ ∈ σ:
γλ
ψ(λ, · )
φ(λ, · ) = −
λW 0 (λ)
2
Green’s function is a Herglotz–Nevanlinna function:
zφ(z, x)ψ(z, x)
z 7→
W (z)
3
Normalization:
φ(0, x) = x,
4
ψ(0, x) = 1 − x
Read off the measure ω from the solution:
Z xZ
∂
0
x Φ− (0) =
φ(z, x)
=−
∂z
z=0
Jonathan Eckhardt
0
s
r dω(r ) ds
0
A coupling problem for entire functions
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
−f 00 = z ωf
with a positive Borel measure ω on (0, 1) satisfying (∗)
1 For every eigenvalue λ ∈ σ:
γλ
ψ(λ, · )
φ(λ, · ) = −
λW 0 (λ)
2
Green’s function is a Herglotz–Nevanlinna function:
zφ(z, x)ψ(z, x)
z 7→
W (z)
3
Normalization:
φ(0, x) = x,
4
ψ(0, x) = 1 − x
Read off the measure ω from the solution:
Z xZ
∂
0
x Φ− (0) =
φ(z, x)
=−
∂z
z=0
Jonathan Eckhardt
0
s
r dω(r ) ds
0
A coupling problem for entire functions
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
−f 00 = z ωf
with a positive Borel measure ω on (0, 1) satisfying (∗)
Recover the measure ω by solving coupling problems:
For every given x ∈ (0, 1) one has
Z xZ s
r dω(r ) ds = −x Φ0− (0),
0
0
where the pair (Φ− , Φ+ ) is the unique solution of the coupling
problem with (admissible) data η given by
η(λ) = −
γλ 1 − x
,
λW 0 (λ) x
Jonathan Eckhardt
λ∈σ
A coupling problem for entire functions
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
−f 00 = z ωf
with a positive Borel measure ω on (0, 1) satisfying (∗)
Recover the measure ω by solving coupling problems:
For every given x ∈ (0, 1) one has
Z xZ s
r dω(r ) ds = −x Φ0− (0),
0
0
where the pair (Φ− , Φ+ ) is the unique solution of the coupling
problem with (admissible) data η given by
η(λ) = −
γλ 1 − x
,
λW 0 (λ) x
λ∈σ
⇒ The measure ω is uniquely determined by the spectral data
Jonathan Eckhardt
A coupling problem for entire functions
Applications I: Inverse spectral theory
Spectral problem for a vibrating string:
−f 00 = z ωf
with a positive Borel measure ω on (0, 1) satisfying (∗)
Recover the measure ω by solving coupling problems:
For every given x ∈ (0, 1) one has
Z xZ s
r dω(r ) ds = −x Φ0− (0),
0
0
where the pair (Φ− , Φ+ ) is the unique solution of the coupling
problem with (admissible) data η given by
η(λ) = −
γλ 1 − x
,
λW 0 (λ) x
λ∈σ
⇒ The measure ω is uniquely determined by the spectral data
...even when ω is a real-valued measure
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Model for shallow water waves (Camassa–Holm, 1993)
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Model for shallow water waves (Camassa–Holm, 1993)
Finite time blow-up that resembles wave breaking
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Model for shallow water waves (Camassa–Holm, 1993)
Finite time blow-up that resembles wave breaking
u(x, t0 )
x
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Model for shallow water waves (Camassa–Holm, 1993)
Finite time blow-up that resembles wave breaking
Traveling wave solutions; solitons (peakons)
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Model for shallow water waves (Camassa–Holm, 1993)
Finite time blow-up that resembles wave breaking
Traveling wave solutions; solitons (peakons)
f (x)
u(x, t) = ce−|x−ct+ξ|
x
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Model for shallow water waves (Camassa–Holm, 1993)
Finite time blow-up that resembles wave breaking
Traveling wave solutions; solitons (peakons)
Amenable to the Inverse Spectral Method
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Model for shallow water waves (Camassa–Holm, 1993)
Finite time blow-up that resembles wave breaking
Traveling wave solutions; solitons (peakons)
Amenable to the Inverse Spectral Method
1
−f 00 + f = z ω( · , t)f
4
Jonathan Eckhardt
with ω = u − uxx
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Model for shallow water waves (Camassa–Holm, 1993)
Finite time blow-up that resembles wave breaking
Traveling wave solutions; solitons (peakons)
Amenable to the Inverse Spectral Method
1
−f 00 + f = z ω( · , t)f
4
St0
St
6
u( · , t0 )
with ω = u − uxx
6
Camassa–Holm flowJonathan Eckhardt
u( · , t)
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Model for shallow water waves (Camassa–Holm, 1993)
Finite time blow-up that resembles wave breaking
Traveling wave solutions; solitons (peakons)
Amenable to the Inverse Spectral Method
1
−f 00 + f = z ω( · , t)f
4
St0
St
6
u( · , t0 )
with ω = u − uxx
St = (σt , {γλ (t)}λ∈σt )
6
Camassa–Holm flowJonathan Eckhardt
u( · , t)
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Model for shallow water waves (Camassa–Holm, 1993)
Finite time blow-up that resembles wave breaking
Traveling wave solutions; solitons (peakons)
Amenable to the Inverse Spectral Method
1
−f 00 + f = z ω( · , t)f
4
St0
linear flow
-
6
u( · , t0 )
with ω = u − uxx
St
St = (σt , {γλ (t)}λ∈σt )
6
Camassa–Holm flowJonathan Eckhardt
u( · , t)
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Model for shallow water waves (Camassa–Holm, 1993)
Finite time blow-up that resembles wave breaking
Traveling wave solutions; solitons (peakons)
Amenable to the Inverse Spectral Method
1
−f 00 + f = z ω( · , t)f
4
St0
linear flow
-
6
u( · , t0 )
with ω = u − uxx
St
6
Camassa–Holm flowJonathan Eckhardt
St = (σt , {γλ (t)}λ∈σt )
Spectrum invariant
u( · , t)
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Model for shallow water waves (Camassa–Holm, 1993)
Finite time blow-up that resembles wave breaking
Traveling wave solutions; solitons (peakons)
Amenable to the Inverse Spectral Method
1
−f 00 + f = z ω( · , t)f
4
St0
linear flow
-
6
u( · , t0 )
with ω = u − uxx
St
6
Camassa–Holm flowJonathan Eckhardt
St = (σ, {γλ (t)}λ∈σ )
Spectrum invariant
u( · , t)
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Model for shallow water waves (Camassa–Holm, 1993)
Finite time blow-up that resembles wave breaking
Traveling wave solutions; solitons (peakons)
Amenable to the Inverse Spectral Method
1
−f 00 + f = z ω( · , t)f
4
St0
linear flow
-
6
u( · , t0 )
with ω = u − uxx
St
6
Camassa–Holm flowJonathan Eckhardt
u( · , t)
St = (σ, {γλ (t)}λ∈σ )
Spectrum invariant
1
and γ˙λ = 2λ
γλ
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Model for shallow water waves (Camassa–Holm, 1993)
Finite time blow-up that resembles wave breaking
Traveling wave solutions; solitons (peakons)
Amenable to the Inverse Spectral Method
1
−f 00 + f = z ω( · , t)f
4
St0
linear flow
-
with ω = u − uxx
St
6
u( · , t0 )
Camassa–Holm flowJonathan Eckhardt
u( · , t)
St = (σ, {γλ (t)}λ∈σ )
Spectrum invariant
1
and γ˙λ = 2λ
γλ
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Model for shallow water waves (Camassa–Holm, 1993)
Finite time blow-up that resembles wave breaking
Traveling wave solutions; solitons (peakons)
Amenable to the Inverse Spectral Method
1
−f 00 + f = z ω( · , t)f
4
St0
linear flow
-
with ω = u − uxx
St
6
u( · , t0 )
Camassa–Holm flowJonathan Eckhardt
?
u( · , t)
St = (σ, {γλ (t)}λ∈σ )
Spectrum invariant
1
and γ˙λ = 2λ
γλ
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
For any given x and t, we have
u(x, t) =
Φ0− (0) + Φ0+ (0) 1 X 1
+
,
2
2
λ
λ∈σ
where the pair (Φ− , Φ+ ) is the unique solution of the coupling
problem with (admissible) data η given by
η(λ) = −
γλ (0) t −x
e 2λ ,
λW 0 (λ)
Jonathan Eckhardt
λ∈σ
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
For any given x and t, we have
u(x, t) =
Φ0− (0) + Φ0+ (0) 1 X 1
+
,
2
2
λ
λ∈σ
where the pair (Φ− , Φ+ ) is the unique solution of the coupling
problem with (admissible) data η given by
η(λ) = −
γλ (0) t −x
e 2λ ,
λW 0 (λ)
λ∈σ
→ Recover solution u by means of solving coupling problems
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
For any given x and t, we have
u(x, t) =
Φ0− (0) + Φ0+ (0) 1 X 1
+
,
2
2
λ
λ∈σ
where the pair (Φ− , Φ+ ) is the unique solution of the coupling
problem with (admissible) data η given by
η(λ) = −
γλ (0) t −x
e 2λ ,
λW 0 (λ)
λ∈σ
→ Recover solution u by means of solving coupling problems
...up to (possible) blow-up
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
For any given x and t, we have define
u(x, t) =
Φ0− (0) + Φ0+ (0) 1 X 1
+
,
2
2
λ
λ∈σ
where the pair (Φ− , Φ+ ) is the unique solution of the coupling
problem with (admissible) data η given by
η(λ) = −
γλ (0) t −x
e 2λ ,
λW 0 (λ)
Jonathan Eckhardt
λ∈σ
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
For any given x and t, we have define
u(x, t) =
Φ0− (0) + Φ0+ (0) 1 X 1
+
,
2
2
λ
λ∈σ
where the pair (Φ− , Φ+ ) is the unique solution of the coupling
problem with (admissible) data η given by
η(λ) = −
γλ (0) t −x
e 2λ ,
λW 0 (λ)
λ∈σ
Possible due to unique solvability of the coupling problem
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
For any given x and t, we have define
u(x, t) =
Φ0− (0) + Φ0+ (0) 1 X 1
+
,
2
2
λ
λ∈σ
where the pair (Φ− , Φ+ ) is the unique solution of the coupling
problem with (admissible) data η given by
η(λ) = −
γλ (0) t −x
e 2λ ,
λW 0 (λ)
λ∈σ
Theorem (Global weak solutions)
The function u is a global weak solution of the CH equation
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
What to expect after long time?
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
What to expect after long time?
Solution asymptotically becomes a superposition of solitons,
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
What to expect after long time?
Solution asymptotically becomes a superposition of solitons, each
of which corresponding to an eigenvalue of the spectral problem
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
What to expect after long time?
Solution asymptotically becomes a superposition of solitons, each
of which corresponding to an eigenvalue of the spectral problem
f (x)
u(x, t) = c e−|x−ct+ξ|
x
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
What to expect after long time?
Solution asymptotically becomes a superposition of solitons, each
of which corresponding to an eigenvalue of the spectral problem
f (x)
u(x, t) = c e−|x−ct+ξ|
x
In contrast to the KdV equation: Infinitely many eigenvalues
Nonlinear steepest decent method does not apply
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
What to expect after long time?
Solution asymptotically becomes a superposition of solitons, each
of which corresponding to an eigenvalue of the spectral problem
Stability for the coupling problem
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
What to expect after long time?
Solution asymptotically becomes a superposition of solitons, each
of which corresponding to an eigenvalue of the spectral problem
Stability for the coupling problem
Consider the solution u along a ray x = ct:
u(ct, t) =
Φ0− (0) + Φ0+ (0) 1 X 1
+
,
2
2
λ
λ∈σ
where (Φ− , Φ+ ) is the solution of the coupling problem with data
γλ (0) ( 1 −c )t
η(λ) = −
, λ∈σ
e 2λ
λW 0 (λ)
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
What to expect after long time?
Solution asymptotically becomes a superposition of solitons, each
of which corresponding to an eigenvalue of the spectral problem
Stability for the coupling problem gives:
Theorem (Long-time asymptotics)
For our global weak solution u we have
u(x, t) =
X 1
t
e−|x− 2λ +ξλ | + o (1) ,
2λ
t → ∞,
λ∈σ
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
Camassa–Holm equation
ut − uxxt = uuxxx − 3uux + 2ux uxx
What to expect after long time?
Solution asymptotically becomes a superposition of solitons, each
of which corresponding to an eigenvalue of the spectral problem
Stability for the coupling problem gives:
Theorem (Long-time asymptotics)
For our global weak solution u we have
u(x, t) =
X 1
t
e−|x− 2λ +ξλ | + o (1) ,
2λ
t → ∞,
λ∈σ
with phase shifts ξλ given in terms of σ and {γλ (0)}λ∈σ
Jonathan Eckhardt
A coupling problem for entire functions
Applications II: Nonlinear wave equations
λ−2
λ−3
λ−4
t
λ4
λ3
λ2
u(x, t)
λ−1
λ1
u(x, t0 )
x
u(x, t) =
X 1
t
e−|x− 2λ +ξλ | + o(1),
2λ
t→∞
λ∈σ
Jonathan Eckhardt
A coupling problem for entire functions
x
Proofs
−1
For real ω ∈ Hloc
(0, 1) and υ a positive Borel measure on (0, 1):
−f 00 = z ωf + z 2 υf
Jonathan Eckhardt
A coupling problem for entire functions
Proofs
−1
For real ω ∈ Hloc
(0, 1) and υ a positive Borel measure on (0, 1):
−f 00 = z ωf + z 2 υf
• Let σ be a discrete set of nonzero reals with
X 1
<∞
|λ|
λ∈σ
Jonathan Eckhardt
A coupling problem for entire functions
Proofs
−1
For real ω ∈ Hloc
(0, 1) and υ a positive Borel measure on (0, 1):
−f 00 = z ωf + z 2 υf
• Let σ be a discrete set of nonzero reals with
X 1
<∞
|λ|
λ∈σ
• Define isospectral set:
Iso(σ) = {(ω, υ) | corresponding spectrum coincides with σ}
Jonathan Eckhardt
A coupling problem for entire functions
Proofs
−1
For real ω ∈ Hloc
(0, 1) and υ a positive Borel measure on (0, 1):
−f 00 = z ωf + z 2 υf
• Let σ be a discrete set of nonzero reals with
X 1
<∞
|λ|
λ∈σ
• Define isospectral set:
Iso(σ) = {(ω, υ) | corresponding spectrum coincides with σ}
• Solutions: φ(z, x) ∼ x as x → 0 and ψ(z, x) ∼ 1 − x as x → 1
Jonathan Eckhardt
A coupling problem for entire functions
Proofs
−1
For real ω ∈ Hloc
(0, 1) and υ a positive Borel measure on (0, 1):
−f 00 = z ωf + z 2 υf
• Let σ be a discrete set of nonzero reals with
X 1
<∞
|λ|
λ∈σ
• Define isospectral set:
Iso(σ) = {(ω, υ) | corresponding spectrum coincides with σ}
• Solutions: φ(z, x) ∼ x as x → 0 and ψ(z, x) ∼ 1 − x as x → 1
• Wronskian:
Y
z
0
0
ψ(z, x)φ (z, x) − ψ (z, x)φ(z, x) =
1−
= W (z)
λ
λ∈σ
Jonathan Eckhardt
A coupling problem for entire functions
Proofs
−1
For real ω ∈ Hloc
(0, 1) and υ a positive Borel measure on (0, 1):
−f 00 = z ωf + z 2 υf
• Let σ be a discrete set of nonzero reals with
X 1
<∞
|λ|
λ∈σ
• Define isospectral set:
Iso(σ) = {(ω, υ) | corresponding spectrum coincides with σ}
• Solutions: φ(z, x) ∼ x as x → 0 and ψ(z, x) ∼ 1 − x as x → 1
• Wronskian:
Y
z
0
0
ψ(z, x)φ (z, x) − ψ (z, x)φ(z, x) =
1−
= W (z)
λ
λ∈σ
• Define norming constants γλ ∈ R+ for λ ∈ σ:
Z 1
Z 1
0
2
γλ =
|φ (λ, x)| dx +
|λφ(λ, x)|2 dυ(x)
0
0
Jonathan Eckhardt
A coupling problem for entire functions
Proofs
−1
For real ω ∈ Hloc
(0, 1) and υ a positive Borel measure on (0, 1):
−f 00 = z ωf + z 2 υf
• Let σ be a discrete set of nonzero reals with
X 1
<∞
|λ|
λ∈σ
• Define isospectral set:
Iso(σ) = {(ω, υ) | corresponding spectrum coincides with σ}
• Solutions: φ(z, x) ∼ x as x → 0 and ψ(z, x) ∼ 1 − x as x → 1
• Wronskian:
Y
z
0
0
ψ(z, x)φ (z, x) − ψ (z, x)φ(z, x) =
1−
= W (z)
λ
λ∈σ
• Define norming constants γλ ∈ R+ for λ ∈ σ:
Z 1
Z 1
X
0
2
γλ =
|φ (λ, x)| dx +
|λφ(λ, x)|2 dυ(x) → µ =
γλ−1 δλ
0
0
Jonathan Eckhardt
λ∈σ
A coupling problem for entire functions
Proofs
−1
For real ω ∈ Hloc
(0, 1) and υ a positive Borel measure on (0, 1):
−f 00 = z ωf + z 2 υf
Inverse problem (IP)
Given γ ∈ Rσ+ , find a pair (ω, υ) in Iso(σ) such that the
corresponding norming constants are precisely γλ
Jonathan Eckhardt
A coupling problem for entire functions
Proofs
−1
For real ω ∈ Hloc
(0, 1) and υ a positive Borel measure on (0, 1):
−f 00 = z ωf + z 2 υf
Inverse problem (IP)
Given γ ∈ Rσ+ , find a pair (ω, υ) in Iso(σ) such that the
corresponding norming constants are precisely γλ
Theorem (Existence and uniqueness)
The inverse problem (IP) has a unique solution for every γ ∈ Rσ+
Jonathan Eckhardt
A coupling problem for entire functions
Proofs
−1
For real ω ∈ Hloc
(0, 1) and υ a positive Borel measure on (0, 1):
−f 00 = z ωf + z 2 υf
Inverse problem (IP)
Given γ ∈ Rσ+ , find a pair (ω, υ) in Iso(σ) such that the
corresponding norming constants are precisely γλ
Theorem (Existence and uniqueness)
The inverse problem (IP) has a unique solution for every γ ∈ Rσ+
The isospectral set Iso(σ) is homeomorphic to Rσ+ by means of
(ω, υ) 7→ {γλ }λ∈σ
Jonathan Eckhardt
A coupling problem for entire functions
Proofs
−1
For real ω ∈ Hloc
(0, 1) and υ a positive Borel measure on (0, 1):
−f 00 = z ωf + z 2 υf
Inverse problem (IP)
Given γ ∈ Rσ+ , find a pair (ω, υ) in Iso(σ) such that the
corresponding norming constants are precisely γλ
Theorem (Existence and uniqueness)
The inverse problem (IP) has a unique solution for every γ ∈ Rσ+
The isospectral set Iso(σ) is homeomorphic to Rσ+ by means of
(ω, υ) 7→ {γλ }λ∈σ
Uniqueness for the coupling problem ⇒ Uniqueness for (IP)
Jonathan Eckhardt
A coupling problem for entire functions
Proofs
−1
For real ω ∈ Hloc
(0, 1) and υ a positive Borel measure on (0, 1):
−f 00 = z ωf + z 2 υf
Inverse problem (IP)
Given γ ∈ Rσ+ , find a pair (ω, υ) in Iso(σ) such that the
corresponding norming constants are precisely γλ
Theorem (Existence and uniqueness)
The inverse problem (IP) has a unique solution for every γ ∈ Rσ+
The isospectral set Iso(σ) is homeomorphic to Rσ+ by means of
(ω, υ) 7→ {γλ }λ∈σ
Uniqueness for the coupling problem ⇒ Uniqueness for (IP)
Existence for (IP) ⇒ Existence for the coupling problem
Jonathan Eckhardt
A coupling problem for entire functions
Proofs – Uniqueness for the coupling problem
For a solution (Φ− , Φ+ ) of the coupling problem with data η,
there are two de Branges spaces B1 , B2 such that:
Jonathan Eckhardt
A coupling problem for entire functions
Proofs – Uniqueness for the coupling problem
For a solution (Φ− , Φ+ ) of the coupling problem with data η,
there are two de Branges spaces B1 , B2 such that:
1
B1 and B2 are isometrically embedded in L2 (R; µ) with
δ0 X 1 |η(λ)|
µ=
+
δλ
2
2 |λW 0 (λ)|
λ∈σ
Jonathan Eckhardt
A coupling problem for entire functions
Proofs – Uniqueness for the coupling problem
For a solution (Φ− , Φ+ ) of the coupling problem with data η,
there are two de Branges spaces B1 , B2 such that:
1
B1 and B2 are isometrically embedded in L2 (R; µ) with
δ0 X 1 |η(λ)|
µ=
+
δλ
2
2 |λW 0 (λ)|
λ∈σ
2
The corresponding reproducing kernels K1 and K2 satisfy
K2 (0, 0) ≥ 1 ≥ K1 (0, 0)
Jonathan Eckhardt
A coupling problem for entire functions
Proofs – Uniqueness for the coupling problem
For a solution (Φ− , Φ+ ) of the coupling problem with data η,
there are two de Branges spaces B1 , B2 such that:
1
B1 and B2 are isometrically embedded in L2 (R; µ) with
δ0 X 1 |η(λ)|
µ=
+
δλ
2
2 |λW 0 (λ)|
λ∈σ
2
3
The corresponding reproducing kernels K1 and K2 satisfy
K2 (0, 0) ≥ 1 ≥ K1 (0, 0)
B1 has codimension at most one in B2
Jonathan Eckhardt
A coupling problem for entire functions
Proofs – Uniqueness for the coupling problem
For a solution (Φ− , Φ+ ) of the coupling problem with data η,
there are two de Branges spaces B1 , B2 such that:
1
B1 and B2 are isometrically embedded in L2 (R; µ) with
δ0 X 1 |η(λ)|
µ=
+
δλ
2
2 |λW 0 (λ)|
λ∈σ
2
3
The corresponding reproducing kernels K1 and K2 satisfy
K2 (0, 0) ≥ 1 ≥ K1 (0, 0)
B1 has codimension at most one in B2 with
Φ+ (z) = K1 (0, z)
if B1 coincides with B2 and
1 − K1 (0, 0)
Φ+ (z) = K1 (0, z) + Θ(z)
Θ(0)
else, when B2 = B1 ⊕ span{Θ}
Jonathan Eckhardt
A coupling problem for entire functions
Proofs – Uniqueness for the coupling problem
For a solution (Φ− , Φ+ ) of the coupling problem with data η,
there are two de Branges spaces B1 , B2 such that:
1
B1 and B2 are isometrically embedded in L2 (R; µ) with
δ0 X 1 |η(λ)|
µ=
+
δλ
2
2 |λW 0 (λ)|
λ∈σ
2
3
The corresponding reproducing kernels K1 and K2 satisfy
K2 (0, 0) ≥ 1 ≥ K1 (0, 0)
B1 has codimension at most one in B2 with
Φ+ (z) = K1 (0, z)
if B1 coincides with B2 and
1 − K1 (0, 0)
Φ+ (z) = K1 (0, z) + Θ(z)
Θ(0)
else, when B2 = B1 ⊕ span{Θ}
→ de Branges’ subspace ordering theorem
Jonathan Eckhardt
A coupling problem for entire functions
Proofs – Existence for the (IP) in the finite case
For finite σ and positive γλ
Jonathan Eckhardt
A coupling problem for entire functions
Proofs – Existence for the (IP) in the finite case
For finite σ and positive γλ we consider the rational function
1 X γλ
− +
z
λ−z
λ∈σ
Jonathan Eckhardt
A coupling problem for entire functions
Proofs – Existence for the (IP) in the finite case
For finite σ and positive γλ we consider the rational function
1
1 X γλ
=
− +
z
λ−z
1
λ∈σ
−l0 z +
1
ω1 + zυ1 +
1
..
.+
ωN + zυN −
Jonathan Eckhardt
A coupling problem for entire functions
1
lN z
Proofs – Existence for the (IP) in the finite case
For finite σ and positive γλ we consider the rational function
1
1 X γλ
=
− +
z
λ−z
1
λ∈σ
−l0 z +
1
ω1 + zυ1 +
1
..
.+
ωN + zυN −
Define ω =
PN
n=1 ωn δxn
x1
and υ =
x2
PN
n=1 υn δxn
x3
0
with xn =
Pn−1
x4
1
point masses
Jonathan Eckhardt
1
lN z
A coupling problem for entire functions
k=0 lk
x
Proofs – Existence for the (IP) in the finite case
For finite σ and positive γλ we consider the rational function
1
1 X γλ
=
− +
z
λ−z
1
λ∈σ
−l0 z +
1
ω1 + zυ1 +
1
..
.+
ωN + zυN −
Define ω =
PN
n=1 ωn δxn
x1
and υ =
x2
PN
n=1 υn δxn
x3
0
with xn =
Pn−1
x4
1
point masses
−f 00 = z ωf + z 2 υf
Jonathan Eckhardt
1
lN z
A coupling problem for entire functions
k=0 lk
x
Proofs – Existence for the (IP) in the finite case
For finite σ and positive γλ we consider the rational function
1
1 X γλ
=
− +
z
λ−z
1
λ∈σ
−l0 z +
1
ω1 + zυ1 +
1
..
.+
ωN + zυN −
Define ω =
PN
n=1 ωn δxn
x1
and υ =
x2
PN
n=1 υn δxn
with xn =
x3
0
Pn−1
x4
1
point masses
f (xn+1 ) − f (xn ) = f 0 (xn+1 −)ln
f 0 (xn+1 −) − f 0 (xn −) = −(zωn + z 2 υn )f (xn )
Jonathan Eckhardt
1
lN z
A coupling problem for entire functions
k=0 lk
x
Proofs – Existence for the (IP) in the finite case
For finite σ and positive γλ we consider the rational function
1
1 X γλ
=
− +
z
λ−z
1
λ∈σ
−l0 z +
1
ω1 + zυ1 +
1
..
.+
ωN + zυN −
Define ω =
PN
n=1 ωn δxn
and υ =
PN
n=1 υn δxn
with xn =
Pn−1
Weyl–Titchmarsh function:
m(z) =
ψ 0 (z, 0)
zψ(z, 0)
Jonathan Eckhardt
1
lN z
A coupling problem for entire functions
k=0 lk
Proofs – Existence for the (IP) in the finite case
For finite σ and positive γλ we consider the rational function
1
1 X γλ
=
− +
z
λ−z
1
λ∈σ
−l0 z +
1
ω1 + zυ1 +
1
..
.+
ωN + zυN −
Define ω =
PN
n=1 ωn δxn
and υ =
PN
n=1 υn δxn
with xn =
Pn−1
Weyl–Titchmarsh function:
m(z) =
1 X γλ
ψ 0 (z, 0)
=− +
zψ(z, 0)
z
λ−z
λ∈σ
Jonathan Eckhardt
1
lN z
A coupling problem for entire functions
k=0 lk
Proofs – Existence for the (IP) in the finite case
For finite σ and positive γλ we consider the rational function
1
1 X γλ
=
− +
z
λ−z
1
λ∈σ
−l0 z +
1
ω1 + zυ1 +
1
..
.+
ωN + zυN −
Define ω =
PN
n=1 ωn δxn
and υ =
PN
n=1 υn δxn
with xn =
1
lN z
Pn−1
k=0 lk
Weyl–Titchmarsh function:
m(z) =
1 X γλ
ψ 0 (z, 0)
=− +
zψ(z, 0)
z
λ−z
λ∈σ
(ω, υ) ∈ Iso(σ) and γλ are the corresponding norming constants
Jonathan Eckhardt
A coupling problem for entire functions
Proofs – Existence for the coupling problem
...with admissible data η
Jonathan Eckhardt
A coupling problem for entire functions
Proofs – Existence for the coupling problem
...with admissible data η
• Take solutions (Φk,− , Φk,+ ) of the coupling problem with the
cut-off set σk = σ ∩ [−k, k] and truncated coupling constants η|σk
Jonathan Eckhardt
A coupling problem for entire functions
Proofs – Existence for the coupling problem
...with admissible data η
• Take solutions (Φk,− , Φk,+ ) of the coupling problem with the
cut-off set σk = σ ∩ [−k, k] and truncated coupling constants η|σk
• Recall the a priori bounds
Y
Y
|z|
|z|
|Φk,± (z)| ≤
1+
≤
1+
|λ|
|λ|
λ∈σk
Jonathan Eckhardt
λ∈σ
A coupling problem for entire functions
Proofs – Existence for the coupling problem
...with admissible data η
• Take solutions (Φk,− , Φk,+ ) of the coupling problem with the
cut-off set σk = σ ∩ [−k, k] and truncated coupling constants η|σk
• Recall the a priori bounds
Y
Y
|z|
|z|
|Φk,± (z)| ≤
1+
≤
1+
|λ|
|λ|
λ∈σk
λ∈σ
• Choose a convergent subsequence
Jonathan Eckhardt
A coupling problem for entire functions
Proofs – Existence for the coupling problem
...with admissible data η
• Take solutions (Φk,− , Φk,+ ) of the coupling problem with the
cut-off set σk = σ ∩ [−k, k] and truncated coupling constants η|σk
• Recall the a priori bounds
Y
Y
|z|
|z|
|Φk,± (z)| ≤
1+
≤
1+
|λ|
|λ|
λ∈σk
λ∈σ
• Choose a convergent subsequence
• The limit is a solution of the coupling problem with data η
Jonathan Eckhardt
A coupling problem for entire functions
A coupling problem for entire functions
Jonathan Eckhardt
University of Vienna
[email protected]
May 23, 2017
Jonathan Eckhardt
A coupling problem for entire functions