Evgeny Korotyaev – Inverse Problems for vector-valued Sturm-Liouville Operators on the Unit Interval

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Date(s) - 23 juin 2010

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Abstract. We consider the inverse problem for the Schr odinger operator $Hy = ?y?? + V y$ on the unit interval $[0, 1]$ with the Dirichlet boundary condition $ y(1) = y(1)=0$. Here $y=(y_j)_1^N, N >3 $ is a vector-valued function and $V =V(x)$ is a $N \times N$ matrix valued self-adjoint potential from $L^2 (0,1)$. The collection of the eigenvalues of $H$ and some subspaces (associated with eigenvalues) is called the spectral data of the operator. The complete characterization of spectral data corresponding to $N \times N$ self-adjoint square-integrable matrix-valued potentials is given. Moreover, for any $V$ we describe the so-called iso-spectral set, i.e., the set of all potentials with the same spectrum of the corresponding the Schrödinger operator. The talk is a joint with Dmitri Chelkak and is based on the papers 1) Chelkak, Dmitri; Korotyaev, Evgeny Weyl-Titchmarsh functions of vector-valued Sturm-Liouville operators on the unit interval, Jour. Func. Anal., 257 (2009), 1546- 1588. 2) Chelkak, D.; Korotyaev, E. Parametrization of the isospectral set for the vector- valued Sturm-Liouville problem, J. Funct. Anal., 241(2006), 359-373.