Spectral theory for non-self-adjoint differential operators

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A central topic in modern theoretical physics is the inconsistency between the standard model and general relativity. The quest for a Grand Unified Theory led to the development of new mathematical models, for instance in Quantum Mechanical non-self-adjoint operators were considered instead of self-adjoint.A prominent class of non-self-adjoint operators are self-adjoint operators in Krein spaces and PT-symmetric operators. Unlike self-adjoint operators in Hilbert spaces, the spectral properties of PT-symmetric operators are fundamentally different, for example, they may have non-real spectrum with accumulation points.In the present project we investigate the spectral properties of non-self-adjoint differential operators. The focus of this research program is on the spectral properties of indefinite singular Sturm-Liouville operators and indefinite elliptic differential operators. One aim is to localize the position of the non-real point spectrum. Furthermore, we want to study the accumulation of the point spectrum against the essential spectrum. In addition to the so-called WKB approximation for solutions of second order differential equations, we use oscillation techniques for Sturm-Liouville operators and perturbation results for operators in Krein spaces. Moreover, for PT--symmetric quantum mechanics we develop by means of the WKB approximation criteria for the limit point and the limit circle case for Sturm-Liouville operators with complex-valued coefficients.The present proposal is a continuation of the current Project (sign removed). Initially, the project (sign removed) was designed for 3 years, but finally it was approved for 18 month. Here we apply for another 18 month in order to complete all the intended research topics of the project (sign removed). We added two new research topics (classification of the limit point and limit circle case for Sturm-Liouville operators with complex coefficient, and indefinite elliptic differential operators), which seems to us very natural and closelyconnected to the results we already obtained in the first (18 month) period of the project (sign removed).