Jonathan Eckhardt

Sturm-Liouville operators with measure-valued coefficients

We give a comprehensive treatment of Sturm-Liouville operators whose coefficients are measures, including a full discussion of self-adjoint extensions and boundary conditions, resolvents, and Weyl-Titchmarsh-Kodaira theory. We avoid previous technical restrictions and, at the same time, extend all results to a larger class of operators. Our operators include classical Sturm-Liouville operators, Sturm-Liouville operators with (local and non-local) δ and δ′ interactions or transmission conditions as well as eigenparameter dependent boundary conditions, Krein string operators, Lax operators arising in the treatment of the Camassa-Holm equation, Jacobi operators, and Sturm-Liouville operators on time scales as special cases.

Supersymmetry and Schrodinger-type operators with distributional matrix-valued potentials

Building on work on Miuras transformation by Kappeler, Perry, Shubin, and Topalov, we develop a detailed spectral theoretic treatment of Schrodinger operators with matrix-valued potentials, with special emphasis on distributional potential coecients. Our principal method relies on a supersymmetric (factorization) formalism underlying Miuras transformation, which intimately connects the triple of operators (D;H1;H2) of the form D = 0 A A 0 in L 2 (R) 2m and H1 = A A; H2 = AA in L 2 (R) m : Here A = Im(d=dx) + in L2(R)m, with a matrix-valued coecient = 2 L1 (R) m m, m2 N, thus explicitly permitting distributional potential coecients Vj in Hj, j = 1; 2, where Hj = Im d 2 dx2 +Vj(x); Vj(x) = (x) 2 + ( 1) j 0 (x); j = 1; 2: Upon developing Weyl{Titchmarsh theory for these generalized Schrodinger operators Hj, with (possibly, distributional) matrix-valued potentials Vj, we provide some spectral theoretic applications, including a derivation of the cor- responding spectral representations for Hj, j = 1; 2. Finally, we derive a local Borg{Marchenko uniqueness theorem for Hj, j = 1; 2, by employing the underlying supersymmetric structure and reducing it to the known local Borg{Marchenko uniqueness theorem for D.

Uniqueness results for one-dimensional Schrödinger operators with purely discrete spectra

We provide an abstract framework for singular one-dimensional Schrödinger operators with purely discrete spectra to show when the spectrum plus norming constants determine such an operator completely. As an example we apply our findings to prove new uniqueness results for perturbed quantum mechanical harmonic oscillators. In addition, we also show how to establish a Hochstadt–Lieberman type result for these operators. Our approach is based on the singular Weyl–Titchmarsh–Kodaira theory which is extended to cover the present situation.We provide an abstract framework for singular one-dimensional Schroedinger operators with purely discrete spectra to show when the spectrum plus norming constants determine such an operator completely. As an example we apply our findings to prove a new uniqueness results for perturbed quantum mechanical harmonic oscillators. In addition, we also show how to establish a Hochstadt-Liebermann type result for these operators. Our approach is based on the singular Weyl-Titchmarsh theory which is extended to cover the present situation.

Direct and inverse spectral theory of singular left-definite Sturm–Liouville operators

Abstract We discuss direct and inverse spectral theory of self-adjoint Sturm–Liouville relations with separate boundary conditions in the left-definite setting. In particular, we develop singular Weyl–Titchmarsh theory for these relations. Consequently, we apply de Brangesʼ subspace ordering theorem to obtain inverse uniqueness results for the associated spectral measure. The results can be applied to solve the inverse spectral problem associated with the Camassa–Holm equation.

An Isospectral Problem for Global Conservative Multi-Peakon Solutions of the Camassa–Holm Equation

We introduce a generalized isospectral problem for global conservative multi-peakon solutions of the Camassa–Holm equation. Utilizing the solution of the indefinite moment problem given by M. G. Krein and H. Langer, we show that the conservative Camassa–Holm equation is integrable by the inverse spectral transform in the multi-peakon case.

Inverse Uniqueness Results for Schrödinger Operators Using de Branges Theory

We utilize the theory of de Branges spaces to show when certain Schrödinger operators with strongly singular potentials are uniquely determined by their associated spectral measure. The results are applied to obtain an inverse uniqueness theorem for perturbed spherical Schrödinger operators.

On the isospectral problem of the dispersionless Camassa-Holm equation

Abstract We discuss direct and inverse spectral theory for the isospectral problem of the dispersionless Camassa–Holm equation, where the weight is allowed to be a finite signed measure. In particular, we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral problem for the class of measures which are sign definite. The results are applied to deduce several facts for the dispersionless Camassa–Holm equation. In particular, we show that initial conditions with integrable momentum asymptotically split into a sum of peakons as conjectured by McKean.

The Inverse Spectral Transform for the Conservative Camassa–Holm Flow with Decaying Initial Data

We establish the inverse spectral transform for the conservative Camassa–Holm flow with decaying initial data. In particular, it is employed to prove the existence of weak solutions for the corresponding Cauchy problem.

Singular Weyl–Titchmarsh–Kodaira theory for one-dimensional Dirac operators

We develop singular Weyl–Titchmarsh–Kodaira theory for one-dimensional Dirac operators. In particular, we establish existence of a spectral transformation as well as local Borg–Marchenko and Hochstadt–Lieberman type uniqueness results. Finally, we give some applications to the case of radial Dirac operators.

Quadratic operator pencils associated with the conservative Camassa-Holm flow

We discuss direct and inverse spectral theory for a Sturm-Liouville type problem with a quadratic dependence on the eigenvalue parameter, which arises as the isospectral problem for the conservative Camassa-Holm flow.