Schrödinger operators with complex potentials

The Schrödinger equation describes the motion of quantum mechanical particles. The spatial part of the equation is governed by a so-called Schrödinger operator, a partial differential operator that can be viewed as a quantization of the classical Hamiltonian. The eigenvalues (or more generally, the spectrum) of this operator are the possible energies of the system. Conservation of energy requires that the Schrödinger operator is self-adjoint ("symmetric/hermitian"); in particular, the potential must be real-valued in this case. However, many interesting physical phenomena (resonances, dissipation of energy etc.) are modelled by Schrödinger operators with complex-valued potentials. Mathematically, complex potentials pose a significant challenge, and the theory is much less developed than its classical counterpart dealing only with real-valued potentials. Even though rapid progress has been made in recent years, there is a need for more (counter)-examples.

The aim of this project is to construct explicit (counter)-examples of complex potentials leading to spectral behaviour that is "unexpected" from the point of view of the classical theory. The candidate will have the opportunity to participate in workshops of our LMS Joint Research Group "Challenges in Non-Self-Adjoint Spectral Theory".