Spektraltheorie nichtselbstadjungierter Differentialoperatoren
Mathematik und Naturwissenschaften
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 Mechanics 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 SturmLiouville 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.