Continued fraction expansions and generalized indefinite strings - - PowerPoint PPT Presentation

Continued fraction expansions and generalized indefinite strings

Jonathan Eckhardt

Loughborough University

Jonathan Eckhardt Generalized indefinite strings

Operator Theory and Indefinite Inner Product Spaces

TU Wien – December, 2016

• A. Fleige & H. Winkler, An indefinite inverse spectral problem
• f Stieltjes type, IEOT 87 (2017)

Inverse spectral problem for regular indefinite Krein–Stieltjes strings −f ′′ = z ωf

• n [0, L]

with ω = ∞

n=0 ωnδxn

ω:

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Generalized indefinite string (L, ω, υ)

• L is a positive number or infinity
• ω is a real distribution in H−1

loc[0, L)

• υ is a non-negative Borel measure on [0, L)

Spectral problem: −f ′′ = z ωf + z2υf

• n [0, L)

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Generalized indefinite string (L, ω, υ)

• L is a positive number or infinity
• ω is a real distribution in H−1

loc[0, L)

• υ is a non-negative Borel measure on [0, L)

Spectral problem: −f ′′ = z ωf + z2υf

• n [0, L)

Krein string (L, ω) Mark Krein 1950s

• L is a positive number or infinity
• ω is a non-negative Borel measure on [0, L)

Spectral problem: −f ′′ = z ωf

• n [0, L)

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Generalized indefinite string (L, ω, υ)

• L is a positive number or infinity
• ω is a real distribution in H−1

loc[0, L)

• υ is a non-negative Borel measure on [0, L)

Spectral problem: −f ′′ = z ωf + z2υf

• n [0, L)
• H. Langer, Spektralfunktionen einer Klasse von Differential-
• peratoren zweiter Ordnung ..., Ann. Acad. Sci. Fenn. (1976)

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Generalized indefinite string (L, ω, υ)

• L is a positive number or infinity
• ω is a real distribution in H−1

loc[0, L)

• υ is a non-negative Borel measure on [0, L)

Spectral problem: −f ′′ = z ωf + z2υf

• n [0, L)
• H. Langer, Spektralfunktionen einer Klasse von Differential-
• peratoren zweiter Ordnung ..., Ann. Acad. Sci. Fenn. (1976)
• M. Krein & H. Langer, On some extension problems which are

closely connected with the theory of Hermitian operators in a space Πκ. III. Indefinite analogues of the Hamburger and Stieltjes moment problems, Beitr¨ age Anal. (1979/80) ...for indefinite analogues of moment problems

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Generalized indefinite string (L, ω, υ)

• L is a positive number or infinity
• ω is a real distribution in H−1

loc[0, L)

• υ is a non-negative Borel measure on [0, L)

Spectral problem: −f ′′ = z ωf + z2υf

• n [0, L)

Relevant for completely integrable nonlinear PDEs: Camassa–Holm equation, Hunter–Saxton equation, Dym equation Eckhardt & Kostenko, An isospectral problem for global conservative multi-peakon ..., Comm. Math. Phys. (2014) Eckhardt, The inverse spectral transform for the conservative Camassa–Holm flow with ..., Arch. Ration. Mech. Anal. (2017)

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Generalized indefinite string (L, ω, υ)

• L is a positive number or infinity
• ω is a real distribution in H−1

loc[0, L)

• υ is a non-negative Borel measure on [0, L)

Spectral problem: −f ′′ = z ωf + z2υf

• n [0, L)

Relevant for completely integrable nonlinear PDEs: Camassa–Holm equation, Hunter–Saxton equation, Dym equation Eckhardt & Kostenko, An isospectral problem for global conservative multi-peakon ..., Comm. Math. Phys. (2014) Eckhardt, The inverse spectral transform for the conservative Camassa–Holm flow with ..., Arch. Ration. Mech. Anal. (2017) ...where υ appears because of blow-up of solutions

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Generalized indefinite string (L, ω, υ)

• L is a positive number or infinity
• ω is a real distribution in H−1

loc[0, L)

• υ is a non-negative Borel measure on [0, L)

Spectral problem: −f ′′ = z ωf + z2υf

• n [0, L)

Relevant for completely integrable nonlinear PDEs: Camassa–Holm equation, Hunter–Saxton equation, Dym equation Eckhardt & Kostenko, An isospectral problem for global conservative multi-peakon ..., Comm. Math. Phys. (2014) Eckhardt, The inverse spectral transform for the conservative Camassa–Holm flow with ..., Arch. Ration. Mech. Anal. (2017) ...where υ appears because of blow-up of solutions Inverse spectral theory is important

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Generalized indefinite string (L, ω, υ)

• L is a positive number or infinity
• ω is a real distribution in H−1

loc[0, L)

• υ is a non-negative Borel measure on [0, L)

Spectral problem: −f ′′ = z ωf + z2υf

• n [0, L)
• Define a Weyl–Titchmarsh function m on C\R by

m(z) = ψ′(z, 0−) zψ(z, 0)

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Generalized indefinite string (L, ω, υ)

• L is a positive number or infinity
• ω is a real distribution in H−1

loc[0, L)

• υ is a non-negative Borel measure on [0, L)

Spectral problem: −f ′′ = z ωf + z2υf

• n [0, L)
• Define a Weyl–Titchmarsh function m on C\R by

m(z) = ψ′(z, 0−) zψ(z, 0) → is a Herglotz–Nevanlinna function

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Generalized indefinite string (L, ω, υ)

• L is a positive number or infinity
• ω is a real distribution in H−1

loc[0, L)

• υ is a non-negative Borel measure on [0, L)

Spectral problem: −f ′′ = z ωf + z2υf

• n [0, L)
• Define a Weyl–Titchmarsh function m on C\R by

m(z) = ψ′(z, 0−) zψ(z, 0) → is a Herglotz–Nevanlinna function

• Measure µ in the integral representation is a spectral measure

m(z) = az+b+

• R

1 λ − z − λ 1 + λ2 dµ(λ),

• R

1 1 + λ2 dµ(λ) < ∞

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Generalized indefinite string (L, ω, υ)

• L is a positive number or infinity
• ω is a real distribution in H−1

loc[0, L)

• υ is a non-negative Borel measure on [0, L)

Spectral problem: −f ′′ = z ωf + z2υf

• n [0, L)
• Define a Weyl–Titchmarsh function m on C\R by

m(z) = ψ′(z, 0−) zψ(z, 0) → is a Herglotz–Nevanlinna function

• Measure µ in the integral representation is a spectral measure
• For a Krein string, m is moreover a Stieltjes function, so that

z → z m(z) is a Herglotz–Nevanlinna function as well

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Generalized indefinite string (L, ω, υ)

• L is a positive number or infinity
• ω is a real distribution in H−1

loc[0, L)

• υ is a non-negative Borel measure on [0, L)

Spectral problem: −f ′′ = z ωf + z2υf

• n [0, L)

Theorem (Eckhardt–Kostenko 2016) The mapping (L, ω, υ) → m is a homeomorphism between generalized indefinite strings and Herglotz–Nevanlinna functions

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Generalized indefinite string (L, ω, υ)

• L is a positive number or infinity
• ω is a real distribution in H−1

loc[0, L)

• υ is a non-negative Borel measure on [0, L)

Spectral problem: −f ′′ = z ωf + z2υf

• n [0, L)

Theorem (Eckhardt–Kostenko 2016) The mapping (L, ω, υ) → m is a homeomorphism between generalized indefinite strings and Herglotz–Nevanlinna functions Theorem (Krein 1950s, de Branges 1960s) The mapping (L, ω) → m is a homeomorphism between Krein strings and Stieltjes functions

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Generalized indefinite string (L, ω, υ)

• L is a positive number or infinity
• ω is a real distribution in H−1

loc[0, L)

• υ is a non-negative Borel measure on [0, L)

Spectral problem: −f ′′ = z ωf + z2υf

• n [0, L)

Theorem (Eckhardt–Kostenko 2016) The mapping (L, ω, υ) → m is a homeomorphism between generalized indefinite strings and Herglotz–Nevanlinna functions Theorem (Krein 1950s, de Branges 1960s) The mapping (L, ω) → m is a homeomorphism between Krein strings and Stieltjes functions Which m correspond to coefficients with discrete support?

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Krein string (L, ω) with ω = N

n=0 ωnδxn,

0 = x0 < x1 < · · · < xN < xN+1 = L, ωn ≥ 0

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Krein string (L, ω) with ω = N

n=0 ωnδxn,

0 = x0 < x1 < · · · < xN < xN+1 = L, ωn ≥ 0 Differential equation −f ′′ = z ωf reduces to a difference equation L f (x) ω2 x2 ω3 x3 ω1 x1 −f ′′ = 0 f (x1−) = f (x1+) f ′(x1−) = f ′(x1+) + z ω1f (x1)

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Krein string (L, ω) with ω = N

n=0 ωnδxn,

0 = x0 < x1 < · · · < xN < xN+1 = L, ωn ≥ 0 Differential equation −f ′′ = z ωf reduces to a difference equation L f (x) ω2 x2 ω3 x3 ω1 x1 −f ′′ = 0 f (x1−) = f (x1+) f ′(x1−) = f ′(x1+) + z ω1f (x1) → Recurrence relations for solutions

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Krein string (L, ω) with ω = N

n=0 ωnδxn,

0 = x0 < x1 < · · · < xN < xN+1 = L, ωn ≥ 0 m(z) = ψ′(z, 0−) zψ(z, 0)

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Krein string (L, ω) with ω = N

n=0 ωnδxn,

0 = x0 < x1 < · · · < xN < xN+1 = L, ωn ≥ 0 m(z) = ψ′(z, 0−) zψ(z, 0) ψ′(z, xn−) = ψ′(z, xn+1−) + z ωnψ(z, xn) ψ(z, xn−1) = ψ(z, xn) − lnψ′(z, xn−) ln = xn − xn−1

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Krein string (L, ω) with ω = N

n=0 ωnδxn,

0 = x0 < x1 < · · · < xN < xN+1 = L, ωn ≥ 0 m(z) = ω0 + 1 −l1z + 1 ω1 + 1 ... + 1 −lNz + 1 ωN − 1 lN+1z ln = xn − xn−1

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Krein string (L, ω) with ω = N

n=0 ωnδxn,

0 = x0 < x1 < · · · < xN < xN+1 = L, ωn ≥ 0 m(z) = ω0 + 1 −l1z + 1 ω1 + 1 ... + 1 −lNz + 1 ωN − 1 lN+1z ln = xn − xn−1 → m is a rational Stieltjes function

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Converse: Given a rational Stieltjes function m, does it have a continued fraction expansion?

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Converse: Given a rational Stieltjes function m, does it have a continued fraction expansion? m(z) = ω0 + 1 −l1z + 1 ω1 + 1 ... + 1 −lNz + 1 ωN − 1 lN+1z

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Converse: Given a rational Stieltjes function m, does it have a continued fraction expansion? m(z) = ω0 + 1 −l1z + 1 ω1 + 1 ... + 1 −lNz + 1 ωN − 1 lN+1z Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (1894/95)

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Converse: Given a rational Stieltjes function m, does it have a continued fraction expansion? m(z) = ω0 + 1 −l1z + 1 ω1 + 1 ... + 1 −lNz + 1 ωN − 1 lN+1z Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (1894/95) ...explicit formulas in terms of moments of µ

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Converse: Given a rational Stieltjes function m, does it have a continued fraction expansion? m(z) = ω0 + 1 −l1z + 1 ω1 + 1 ... + 1 −lNz + 1 ωN − 1 lN+1z Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (1894/95) ...explicit formulas in terms of moments of µ m(z) = ω0 +

• [0,∞)

1 λ − z dµ(λ)

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Converse: Given a rational Stieltjes function m, does it have a continued fraction expansion? m(z) = ω0 + 1 −l1z + 1 ω1 + 1 ... + 1 −lNz + 1 ωN − 1 lN+1z Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (1894/95) ...explicit formulas in terms of moments of µ m(z) = ω0 +

• [0,∞)

1 λ − z dµ(λ) = ω0 −

∞

• k=0

1 zk+1

• [0,∞)

λkdµ(λ)

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Converse: Given a rational Stieltjes function m, does it have a continued fraction expansion? m(z) = ω0 + 1 −l1z + 1 ω1 + 1 ... + 1 −lNz + 1 ωN − 1 lN+1z Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (1894/95) ...explicit formulas in terms of moments of µ Define Krein string (L, ω) with finitely many point masses

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Converse: Given a rational Stieltjes function m, does it have a continued fraction expansion? m(z) = ω0 + 1 −l1z + 1 ω1 + 1 ... + 1 −lNz + 1 ωN − 1 lN+1z Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (1894/95) ...explicit formulas in terms of moments of µ Define Krein string (L, ω) with finitely many point masses ...whose Weyl–Titchmarsh function coincides with m

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Converse: Given a rational Stieltjes function m, does it have a continued fraction expansion? m(z) = ω0 + 1 −l1z + 1 ω1 + 1 ... + 1 −lNz + 1 ωN − 1 lN+1z Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (1894/95) ...explicit formulas in terms of moments of µ Define Krein string (L, ω) with finitely many point masses ...whose Weyl–Titchmarsh function coincides with m Compactness & continuity arg. → lift to all Stieltjes functions

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Stieltjes moment problem Given {sk}k∈N0, find a Borel measure µ on [0, ∞) such that sk =

• [0,∞)

λkdµ(λ), k ∈ N0 Krein–Stieltjes strings are Krein strings (L, ω) such that ω is supported on a discrete set in [0, L), so that ω =

∞

• n=0

ωnδxn

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Stieltjes moment problem Given {sk}k∈N0, find a Borel measure µ on [0, ∞) such that sk =

• [0,∞)

λkdµ(λ), k ∈ N0 Krein–Stieltjes strings are Krein strings (L, ω) such that ω is supported on a discrete set in [0, L), so that ω =

∞

• n=0

ωnδxn {sk}k∈N0

formulas of Stieltjes Stieltjes moment sequence

(L, ω)

Krein–Stieltjes string

µ

spectral measure

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Stieltjes moment problem Given {sk}k∈N0, find a Borel measure µ on [0, ∞) such that sk =

• [0,∞)

λkdµ(λ), k ∈ N0 Krein–Stieltjes strings are Krein strings (L, ω) such that ω is supported on a discrete set in [0, L), so that ω =

∞

• n=0

ωnδxn {sk}k∈N0

formulas of Stieltjes Stieltjes moment sequence

(L, ω)

Krein–Stieltjes string

µ

spectral measure

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Stieltjes moment problem Given {sk}k∈N0, find a Borel measure µ on [0, ∞) such that sk =

• [0,∞)

λkdµ(λ), k ∈ N0 Krein–Stieltjes strings are Krein strings (L, ω) such that ω is supported on a discrete set in [0, L), so that ω =

∞

• n=0

ωnδxn {sk}k∈N0

formulas of Stieltjes Stieltjes moment sequence

(L, ω)

Krein–Stieltjes string

µ

spectral measure

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Stieltjes moment problem Given {sk}k∈N0, find a Borel measure µ on [0, ∞) such that sk =

• [0,∞)

λkdµ(λ), k ∈ N0 Krein–Stieltjes strings are Krein strings (L, ω) such that ω is supported on a discrete set in [0, L), so that ω =

∞

• n=0

ωnδxn {sk}k∈N0

formulas of Stieltjes Stieltjes moment sequence

(L, ω)

Krein–Stieltjes string

µ

spectral measure

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Stieltjes moment problem Given {sk}k∈N0, find a Borel measure µ on [0, ∞) such that sk =

• [0,∞)

λkdµ(λ), k ∈ N0 Krein–Stieltjes strings are Krein strings (L, ω) such that ω is supported on a discrete set in [0, L), so that ω =

∞

• n=0

ωnδxn {sk}k∈N0

formulas of Stieltjes Stieltjes moment sequence

(L, ω)

Krein–Stieltjes string

µ

spectral measure

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Stieltjes moment problem Given {sk}k∈N0, find a Borel measure µ on [0, ∞) such that sk =

• [0,∞)

λkdµ(λ), k ∈ N0 Krein–Stieltjes strings are Krein strings (L, ω) such that ω is supported on a discrete set in [0, L), so that ω = ∞

n=0 ωnδxn

Theorem (Krein 1950s) Stieltjes formulas establish a bijection between Stieltjes moment sequences and Krein–Stieltjes strings (with ω({0}) = 0)

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Stieltjes moment problem Given {sk}k∈N0, find a Borel measure µ on [0, ∞) such that sk =

• [0,∞)

λkdµ(λ), k ∈ N0 Krein–Stieltjes strings are Krein strings (L, ω) such that ω is supported on a discrete set in [0, L), so that ω = ∞

n=0 ωnδxn

Theorem (Krein 1950s) Stieltjes formulas establish a bijection between Stieltjes moment sequences and Krein–Stieltjes strings (with ω({0}) = 0)

1 The Stieltjes moment problem is indeterminate

⇔ the corresponding string (L, ω) is regular

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Stieltjes moment problem Given {sk}k∈N0, find a Borel measure µ on [0, ∞) such that sk =

• [0,∞)

λkdµ(λ), k ∈ N0 Krein–Stieltjes strings are Krein strings (L, ω) such that ω is supported on a discrete set in [0, L), so that ω = ∞

n=0 ωnδxn

Theorem (Krein 1950s) Stieltjes formulas establish a bijection between Stieltjes moment sequences and Krein–Stieltjes strings (with ω({0}) = 0)

1 The Stieltjes moment problem is indeterminate

⇔ the corresponding string (L, ω) is regular

2 In this case, the solutions of the Stieltjes moment problem

correspond to (Krein string) extensions of (L, ω)

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

After Stieltjes (1894/95): attempts to generalize to measures on R Hamburger moment problem Given {sk}k∈N0, find a Borel measure µ on R such that sk =

• R

λkdµ(λ), k ∈ N0 ω0 + 1 −l1z + 1 ω1 + 1 ... + 1 −lNz + 1 ωN − 1 lN+1z With ln > 0, ωn ∈ R Indefinite Krein–Stieltjes string

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

After Stieltjes (1894/95): attempts to generalize to measures on R Hamburger moment problem Given {sk}k∈N0, find a Borel measure µ on R such that sk =

• R

λkdµ(λ), k ∈ N0 ω0 + 1 −l1z + 1 ω1 + 1 ... + 1 −lNz + 1 ωN − 1 lN+1z With ln > 0, ωn ∈ R Indefinite Krein–Stieltjes string Grommer (1914), Hamburger (1919/20) → Jacobi matrices

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Modified continued fraction expansion: υ0z + ω0 + 1 −l1z + 1 υ1z + ω1 + 1 ... + 1 −lNz + 1 υNz + ωN − 1 lN+1z

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Modified continued fraction expansion: υ0z + ω0 + 1 −l1z + 1 υ1z + ω1 + 1 ... + 1 −lNz + 1 υNz + ωN − 1 lN+1z Theorem (Krein–Langer 1979/80) Every rational H–N function admits an expansion of this form ...explicit formulas in terms of moments of µ

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Modified continued fraction expansion: υ0z + ω0 + 1 −l1z + 1 υ1z + ω1 + 1 ... + 1 −lNz + 1 υNz + ωN − 1 lN+1z Theorem (Krein–Langer 1979/80) Every rational H–N function admits an expansion of this form ...explicit formulas in terms of moments of µ In terms of strings: ω = N

n=0(υnz + ωn)δxn

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Modified continued fraction expansion: υ0z + ω0 + 1 −l1z + 1 υ1z + ω1 + 1 ... + 1 −lNz + 1 υNz + ωN − 1 lN+1z Theorem (Krein–Langer 1979/80) Every rational H–N function admits an expansion of this form ...explicit formulas in terms of moments of µ In terms of strings: ω = N

n=0(υnz + ωn)δxn

→ leads to −f ′′ = z ωf + z2υf

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Modified continued fraction expansion: υ0z + ω0 + 1 −l1z + 1 υ1z + ω1 + 1 ... + 1 −lNz + 1 υNz + ωN − 1 lN+1z Theorem (Krein–Langer 1979/80) Every rational H–N function admits an expansion of this form ...explicit formulas in terms of moments of µ Given a rational Herglotz–Nevanlinna function m, there is a GIS... ω =

N

• n=0

ωnδxn, υ =

N

• n=0

υnδxn

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

Modified continued fraction expansion: υ0z + ω0 + 1 −l1z + 1 υ1z + ω1 + 1 ... + 1 −lNz + 1 υNz + ωN − 1 lN+1z Theorem (Krein–Langer 1979/80) Every rational H–N function admits an expansion of this form ...explicit formulas in terms of moments of µ Given a rational Herglotz–Nevanlinna function m, there is a GIS... → solution of the inverse spectral problem for GIS

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn {sk}k∈N0

formulas of Krein–Langer Hamburger moment sequence

(L, ω)

Krein–Langer string

µ

spectral measure

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn {sk}k∈N0

formulas of Krein–Langer Hamburger moment sequence

(L, ω, υ)

Krein–Langer string

µ

spectral measure

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn {sk}k∈N0

formulas of Krein–Langer Hamburger moment sequence

(L, ω, υ)

Krein–Langer string

µ

spectral measure

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn {sk}k∈N0

formulas of Krein–Langer Hamburger moment sequence

(L, ω, υ)

Krein–Langer string

µ

spectral measure

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn {sk}k∈N0

formulas of Krein–Langer Hamburger moment sequence

(L, ω, υ)

Krein–Langer string

µ

spectral measure

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn Theorem (Eckhardt–Kostenko 2018) These formulas establish a bijection between Hamburger moment sequences and Krein–Langer strings (with ω({0}) = υ({0}) = 0) {sk}k∈N0

formulas of Krein–Langer Hamburger moment sequence

(L, ω, υ)

Krein–Langer string

µ

spectral measure

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn Theorem (Eckhardt–Kostenko 2018) These formulas establish a bijection between Hamburger moment sequences and Krein–Langer strings (with ω({0}) = υ({0}) = 0)

1 The Hamburger moment problem is indeterminate

⇔ the corresponding string (L, ω, υ) is regular

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn Theorem (Eckhardt–Kostenko 2018) These formulas establish a bijection between Hamburger moment sequences and Krein–Langer strings (with ω({0}) = υ({0}) = 0)

1 The Hamburger moment problem is indeterminate

⇔ the corresponding string (L, ω, υ) is regular

2 In this case, the solutions of the Hamburger moment problem

correspond to (gen. ind. string) extensions of (L, ω, υ)

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn Theorem (Eckhardt–Kostenko 2018) These formulas establish a bijection between Hamburger moment sequences and Krein–Langer strings (with ω({0}) = υ({0}) = 0)

1 The Hamburger moment problem is indeterminate

⇔ the corresponding string (L, ω, υ) is regular

2 In this case, the solutions of the Hamburger moment problem

correspond to (gen. ind. string) extensions of (L, ω, υ) ω: υ:

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn Theorem (Eckhardt–Kostenko 2018) These formulas establish a bijection between Hamburger moment sequences and Krein–Langer strings (with ω({0}) = υ({0}) = 0)

1 The Hamburger moment problem is indeterminate

⇔ the corresponding string (L, ω, υ) is regular

2 In this case, the solutions of the Hamburger moment problem

correspond to (gen. ind. string) extensions of (L, ω, υ) ω: υ:

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn Theorem A generalized indefinite string (L, ω, υ) is a Krein–Langer string if and only if all moments of the spectral measure µ exist and µ({0}) = lim

k→∞

∆0,k ∆2,k−1 ∆0,k =

• s0

s1 · · · sk−1 s1 s2 · · · sk . . . . . . ... . . . sk−1 sk · · · s2k−2

• ,

∆2,k =

• s2

s3 · · · sk+1 s3 s4 · · · sk+2 . . . . . . ... . . . sk+1 sk+2 · · · s2k

• Jonathan Eckhardt

Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn Theorem A generalized indefinite string (L, ω, υ) is a Krein–Langer string if and only if all moments of the spectral measure µ exist and µ({0}) = lim

k→∞

∆0,k ∆2,k−1 All moments of µ exist if and only if ω: υ: R L

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn Theorem A generalized indefinite string (L, ω, υ) is a Krein–Langer string if and only if all moments of the spectral measure µ exist and µ({0}) = lim

k→∞

∆0,k ∆2,k−1 The moments of µ exist up to order 2K if and only if ω: υ:

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn Theorem A generalized indefinite string (L, ω, υ) is a Krein–Langer string if and only if all moments of the spectral measure µ exist and µ({0}) = lim

k→∞

∆0,k ∆2,k−1

1 Proof: a H–N can be expanded in a continued fraction

⇔ moments of the measure µ exist

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn Theorem A generalized indefinite string (L, ω, υ) is a Krein–Langer string if and only if all moments of the spectral measure µ exist and µ({0}) = lim

k→∞

∆0,k ∆2,k−1

1 Proof: a H–N can be expanded in a continued fraction

⇔ moments of the measure µ exist

2 Solves the inverse spectral problem for Krein–Langer strings Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn Theorem A generalized indefinite string (L, ω, υ) is a Krein–Langer string if and only if all moments of the spectral measure µ exist and µ({0}) = lim

k→∞

∆0,k ∆2,k−1

1 Proof: a H–N can be expanded in a continued fraction

⇔ moments of the measure µ exist

2 Solves the inverse spectral problem for Krein–Langer strings 3 Explicit formulas for the solution in terms of moments Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn Theorem A generalized indefinite string (L, ω, υ) is a Krein–Langer string if and only if all moments of the spectral measure µ exist and µ({0}) = lim

k→∞

∆0,k ∆2,k−1

1 Proof: a H–N can be expanded in a continued fraction

⇔ moments of the measure µ exist

2 Solves the inverse spectral problem for Krein–Langer strings 3 Explicit formulas for the solution in terms of moments

→ This covers the result of Fleige–Winkler (where υ = 0)

Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings

More generally, Krein–Langer strings are generalized indefinite strings such that ω and υ are supported on discrete sets in [0, L) ω =

∞

• n=0

ωnδxn, υ =

∞

• n=0

υnδxn Theorem A generalized indefinite string (L, ω, υ) is a Krein–Langer string if and only if all moments of the spectral measure µ exist and µ({0}) = lim

k→∞

∆0,k ∆2,k−1

1 Proof: a H–N can be expanded in a continued fraction

⇔ moments of the measure µ exist

2 Solves the inverse spectral problem for Krein–Langer strings 3 Explicit formulas for the solution in terms of moments

→ This covers the result of Fleige–Winkler (where υ = 0) → Explicit infinite-soliton solutions for corresponding PDEs

Jonathan Eckhardt Generalized indefinite strings

Continued fraction expansions and generalized indefinite strings

Jonathan Eckhardt

Loughborough University

Jonathan Eckhardt Generalized indefinite strings