Nonselfadjoint Inverse Problems

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Direct and inverse spectral and scattering problems of selfadjoint Sturm-Liouville operators are among the most studied subjects in mathematics. The status of nonselfadjoint Sturm-Liouville operators is by far not as complete even though they are currently under intensive investigation by a number of people. The present research project aspires to make a contribution in this area, in particular with regard to inverse problems. The main tool which sets the treatment of selfadjoint problems apart from others is the spectral theorem. The dire consequences of its absence can sometimes be overcome when it is assumed that the di.erential equation has (or its solutions have) certain structural properties. For example, Floquet theory guarantees a certain structure for the solutions of periodic equations, which in turn allows to draw conclusions for the spectrum which are very similar to the selfadjoint case (intervals become analytic arcs). Another class of such potentials are the so called algebro-geometric potentials which have been intensively investigated in the past few decades by many people including the PI. It is planned to apply the expertise gathered to obtain results for this kind of potentials and certain perturbations of them. In particular, recovery of the potential of a Schrodinger equation from the location of eigenvalues and resonances will be investigated.Physical laws are encoded by differential equations. The problem of obtaining solutions (or at least some of their properties) knowing the coefficients of the differential equation is usually called a direct problem. The inverse problem, on the other hand, is the problem of obtaining the coefficients from a certain knowledge about the solutions (often knowledge about spectral properties). The goal of the project is to investigate certain aspects of such problems. The differential equations investigated have widespread applications in physics and engineering, e.g. recovering material properties inside an object from measurements on the outside of the object. The solution of inverse problems is at the heart of medical and industrial imaging, mineral exploration, and earth quake studies to name a few.