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 of Stieltjes type , IEOT 87 (2017) Inverse spectral problem for regular indefinite Krein–Stieltjes strings − f ′′ = z ω f on [0 , L ] with ω = � ∞ n =0 ω n δ x n 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 ω f + z 2 υ f on [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 ω f + z 2 υ f on [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 ω f on [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 ω f + z 2 υ f on [0 , L ) H. Langer, Spektralfunktionen einer Klasse von Differential- operatoren 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 ω f + z 2 υ f on [0 , L ) H. Langer, Spektralfunktionen einer Klasse von Differential- operatoren 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 Π κ . 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 ω f + z 2 υ f on [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 ω f + z 2 υ f on [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 ω f + z 2 υ f on [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 ω f + z 2 υ f on [0 , L ) • Define a Weyl–Titchmarsh function m on C \ R by m ( z ) = ψ ′ ( z , 0 − ) z ψ ( 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 ω f + z 2 υ f on [0 , L ) • Define a Weyl–Titchmarsh function m on C \ R by m ( z ) = ψ ′ ( z , 0 − ) z ψ ( 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 ω f + z 2 υ f on [0 , L ) • Define a Weyl–Titchmarsh function m on C \ R by m ( z ) = ψ ′ ( z , 0 − ) z ψ ( z , 0) → is a Herglotz–Nevanlinna function • Measure µ in the integral representation is a spectral measure � 1 λ � 1 m ( z ) = az + b + λ − z − 1 + λ 2 d µ ( λ ) , 1 + λ 2 d µ ( λ ) < ∞ R R 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 ω f + z 2 υ f on [0 , L ) • Define a Weyl–Titchmarsh function m on C \ R by m ( z ) = ψ ′ ( z , 0 − ) z ψ ( 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 ω f + z 2 υ f on [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 ω f + z 2 υ f on [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 ω f + z 2 υ f on [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 ω n δ x n , 0 = x 0 < x 1 < · · · < x N < x N +1 = L , ω n ≥ 0 Jonathan Eckhardt Generalized indefinite strings

Generalized indefinite strings Krein string ( L , ω ) with ω = � N n =0 ω n δ x n , 0 = x 0 < x 1 < · · · < x N < x N +1 = L , ω n ≥ 0 Differential equation − f ′′ = z ω f reduces to a difference equation f ( x ) ω 2 ω 1 ω 3 L x 1 x 2 x 3 f ( x 1 − ) = f ( x 1 +) − f ′′ = 0 f ′ ( x 1 − ) = f ′ ( x 1 +) + z ω 1 f ( x 1 ) Jonathan Eckhardt Generalized indefinite strings