My scientific interests center around a rigorous analysis of physical problems, in particular adiabatic, topological, magnetic and semiclassical problems motivated by condensed matter physics and electromagnetism.
Photonic crystals & photonic topological insulators
Recently, I became very interested in light dynamics in matter. One of the motivations to study such effects is the analogy to quantum mechanics: under certain conditions, light behaves akin to a quantum particle. Light propagating in a photonic crystal (PhC), for instance, behaves in many respects akin to a quantum particle (an electron or a hole) in a periodic potential. PhCs are made up of A PhC is made up of a dielectric material that is arranged in a periodic fashion, e. g. by alternating two materials. These materials can be tailor-made to have certain properties for light of specific wavelengths (e. g. microwaves and optical frequencies). One of the early successes was the construction of a material with a photonic band gap. The similarity to quantum systems can be used to anticipate physical effects in PhCs. Edge currents, for instance, have been predicted and experimentally observed in PhCs.
One of my main research interests is to find out if and when this analogy can be made rigorous. The most natural starting point is to reformulate the Maxwell equations as a Schrödinger-type equation. Such a reformulation not only allows one to explore whether the behavior of the electromagnetic waves mimics that of a quantum particle, but also to gives access to the rich toolbox of techniques initially developed for quantum systems. One such technique is space-adiabatic perturbation theory which Giuseppe De Nittis and I have used to derived simpler, effective models for light propagating in adiabatically perturbed PhCs. Turns out that there are a few differences. To name one, the symmetry realized by complex conjugation is a particle-hole type symmetry (in the language of classification of topological insulators), meaning that Bloch waves are complex and come in conjugate pairs. Hence, to obtain real solutions, one needs to consider Bloch functions from two distinct bands. This also complicates a derivation of effective ray optics equations: there are no states supported by a single, isolated band so that one has to control interference terms between the isolated band and its conjugate twin band. By virtue of the spectral gap, band transitions are suppressed to leading order at least, but all the interesting topological effects are sub-leading effects.
The main technique in the derivation of effective dynamics is space-adiabatic perturbation theory, a very versatile technique which exploits the unique structure of adiabatic systems. The characterizing properties of adiabatic systems are (i) the existence of two inherent scales, a microscopic and a macroscopic scale; (ii) the adiabatic parameter is then the ratio of these scales and quantifies the strength of the perturbation; and (iii) a family of relevant states of the unperturbed system. Besides time-adiabatic problems, the most well-known example, many other systems share this structure: the non- and semi-relativistic limit of the Dirac equation and the piezoelectric effect can be viewed in this fashion. Most applications make use of pseudodifferential theory so that one naturally obtains a semiclassical limit in the sense of an Egorov theorem.
Apart from my interest in photonic cyrstals, I have also worked on magnetic pseudodifferential operators and am still interested in classical and quantum systems with magnetic fields.
Study of simpler model operators
Usually, even the effective models derived from full models are too complicated to analyze explicitly. So one usually studies simpler operators which retain certain essential features, e. g. a two-band Dirac-type operator stands in for an effective two-band operator with a conical intersection. And Maxwell-Harper operators can be used to study two topologically non-trivial bands in such a way that the operator still has a particle-hole-type symmetry.