Trace formulas and inverse spectral theory for generalized indefinite strings

Abstract

Generalized indefinite strings provide a canonical model for self-adjoint operators with simple spectrum (other classical models are Jacobi matrices, Krein strings and 2x2 canonical systems). We prove a number of Szegő-type theorems for generalized indefinite strings and related spectral problems (including Krein strings, canonical systems and Dirac operators). More specifically, for several classes of coefficients (that can be regarded as Hilbert-Schmidt perturbations of model problems), we provide a complete characterization of the corresponding set of spectral measures. In particular, our results also apply to the isospectral Lax operator for the conservative Camassa-Holm flow and allow us to establish existence of global weak solutions with various step-like initial conditions of low regularity via the inverse spectral transform.

arXiv comment96 pages. Main additions: Corollaries 7.1, 7.2, 10.1, 10.2 and Section 13.3

References in corpus (28)

• An integrable shallow water equation with peaked solitons
• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• On an integrable two-component Camassa-Holm shallow water system
• Inverse Scattering Transform for the Camassa-Holm equation
• A 2-Component Generalization of the Camassa-Holm Equation and Its Solutions
• Long-Time Asymptotics for the Camassa-Holm Equation
• Global solutions for the two-component Camassa-Holm system
• An isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation
• Generalised Fourier transform for the Camassa-Holm hierarchy
• The inverse spectral problem for indefinite strings
• On the isospectral problem of the dispersionless Camassa-Holm equation
• Sturm-Liouville operators with measure-valued coefficients
• The inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data
• Global conservative solutions of the Camassa-Holm equation for initial data nonvanishing asymptotics
• Direct and inverse spectral theory of singular left-definite Sturm-Liouville operators
• Spectral asymptotics for canonical systems
• Quadratic operator pencils associated with the conservative Camassa-Holm flow
• Jacobi Polynomials, Bernstein-type Inequalities and Dispersion Estimates for the Discrete Laguerre Operator
• Trace formulas and continuous dependence of spectra for the periodic conservative Camassa-Holm flow
• The Classical Moment Problem and Generalized Indefinite Strings
• Discrete Nonlocal Waves
• The Decay of Unstable Noncommutative Solitons
• On the absolutely continuous spectrum of generalized indefinite strings
• De Branges canonical systems with finite logarithmic integral
• Dynamics of Noncommutative Solitons I: Spectral Theory and Dispersive Estimates
• Heat kernels of the discrete Laguerre operators
• Continued fraction expansions of Herglotz-Nevanlinna functions and generalized indefinite strings of Stieltjes type
• Generalized indefinite strings with purely discrete spectrum

Cited by in corpus (1)

• The conservative Camassa-Holm flow with step-like irregular initial data