Quasiconformal mappings and periodic spectral problems in dimension two

We study spectral properties of second-order elliptic operators with periodic coefficients in dimension two. These operators act in periodic simply-connected waveguides, with either Dirichlet, or Neumann, or the third boundary condition. The main result is the absolute continuity of the spectra of such operators. The cornerstone of the proof is an isothermal change of variables, reducing the metric to a flat one and the waveguide to a straight strip. The main technical tool is the quasiconformal variant of the Riemann mapping theorem. This work is supported by The Royal Society.