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Sfb 288 Differential Geometry and Quantum Physics |
Abstract for Sfb Preprint No. 269
Spectral Analysis of Volterra Operators and Inverse Problems for Systems of Ordinary Differential Equations M. M. Malamud
This paper consists of two parts. The purpose of the first part is a treatment of integral Volterra operators with matrix kernels. The following questions are considered: description of cyclic, invariant and hyperinvariant subspaces, spectral multiplicity and similarity. But the principle aim of the paper is to investigate inverse problems for some systems of ordinary differential equation (ODE) on a finite interval. The approach provided is a further developement of the one applied in [M4] to the generalization of the well-known Borg result [Bo] for higher order ODE, that is to the unique recovery of an n-th order ODE from n spectra of the boundary value problems. The main idea of the approach just mentioned is to reduce the questions of uniqueness in inverse problems to the investigation of the uniqueness of Goursat type problems for a certain partial differential equation for the kernel of a triangular transformation operator. Now the connection between two topics under investigation becomes clear. In order to sudy the inverse problems we need a trianglular transformation operator for systems of ODE. The existence of such an operator is closely connected with the similarity of integral Volterra operators with kernel $k(x,t)$ being the Green function of the Cauchy problem for ODE with zero initial data and the simple Volterra operator being the tensor product of the integration operator and a matrix.
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