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Quadratic operator pencils associated with the conservative Camassa-Holm flow. (Pinceaux quadratiques d’opérateurs associés au flot Camassa-Holm conservateur.) (English. French summary) Zbl 1484.34090

Summary: We discuss direct and inverse spectral theory for a Sturm-Liouville type problem with a quadratic dependence on the eigenvalue parameter, \[ -f^{\prime \prime}+\frac{1}{4}f=z\omega f+z^2vf \text{,} \] which arises as the isospectral problem for the conservative Camassa-Holm flow. In order to be able to treat rather irregular coefficients (that is, when \(\omega\) is a real-valued Borel measure on \(\mathbb{R}\) and \(v\) is a non-negative Borel measure on \(\mathbb{R}\)), we employ a novel approach to study this spectral problem. In particular, we provide basic self-adjointness results for realizations in suitable Hilbert spaces, develop (singular) Weyl-Titchmarsh theory and prove several basic inverse uniqueness theorems for this spectral problem.

MSC:

34B24 Sturm-Liouville theory
34L05 General spectral theory of ordinary differential operators
34B20 Weyl theory and its generalizations for ordinary differential equations
34A55 Inverse problems involving ordinary differential equations
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