. Instutute of Mathematics, Romanian Academy of Sciences
Spectral Gaps for Periodic Hamiltonians in Slowly Varying Magnetic Fields.
I report on some work done in collaboration with H. Cornean and B. Helffer. We
consider a periodic Schrödinger operator in two dimensions, perturbed by a
weak magnetic field whose intensity slowly varies around a positive mean. We
show in great generality that the bottom of the spectrum of the corresponding
magnetic Schrödinger operator develops spectral islands separated by gaps,
reminding of a Landau-level structure.
First, we construct an effective magnetic matrix which accurately describes
the low lying spectrum of the full operator. The construction of this
effective magnetic matrix does not require a gap in the spectrum of the
non-magnetic operator, only that the first and the second Bloch eigenvalues
Second, we perform a detailed spectral analysis of the effective matrix using
a gauge-covariant magnetic pseudo-differential calculus adapted for slowly
varying magnetic fields.