Spectral Theory Minisymposium

This is a part of the programme for The Fifth European Congress of Mathematics, to be held in Amsterdam in July 2008. All of the talks in the Spectral Theory Minisymposium are on Friday 18 July.

The speakers in the MiniSymposium will be as follows. Their abstracts are provided below.

Morning Chair: E B Davies

• A Sobolev, University College London, 10:30 - 11:15
  Some aspects of perturbation theory for the periodic Schr\"odinger operators.

• Y Last, Hebrew University of Jerusalem, 11:15 - 12:00
  On the Structure of Hofstadter's Butterfly.

Afternoon Chair: A Sobolev

• M J Esteban, University of Paris IX - Dauphine, 13:30 - 14:00
  Self-adjoint extensions via Hardy-like inequalities.

• G M Graf, ETH Zurich, 14:00 - 14:30
  Quantization of charge transport: equivalence of scattering and Chern number approaches.

• R L Frank, Princeton University, 14:30 - 15:00
  Lieb-Thirring and Hardy-Sobolev inequalities.

The abstracts for the above lectures are as follows.

• A Sobolev,
  The main obstacle in the perturbation theory for periodic operators is the presence of the so-called unstable eigenvalues. In particular, these eigenvalues play a prominent role in the study of the Integrated Density of States (IDS) which is a central objects in spectral analysis of periodic operators. The aim of the talk is to present a "sharp" asymptotic formula for IDS as the spectral parameter $\lambda$ tends to infinity. A two-term formula was known with a remainder estimate which was still far from the predicted next term. We present the three-term asymptotic formula for IDS of the periodic Schr\"odinger operator in dimension $d=2$. Some intermediate results are proved for arbitrary $d\ge 2$, but the extension of the three-term formula to $d >2$ will require further insight into the structure of the unstable eigenvalues.

• Y Last,
  We review some aspects of the spectral theory of the critically coupled Almost Mathieu Operator connected with the structure of the famous associated "Hofstadter's Butterfly." We present a new result (joint with Mira Shamis) establishing that for a topologically generic set of irrational frequencies, the Hausdorff dimension of the spectrum of the critical Almost Mathieu Operator is zero. This result is based a new approach which combines certain inductive WKB-type estimates with Green function techniques and provides more detailed information than what has been previously achieved using more elaborate semiclassical approaches.

• M J Esteban,
  Distinguished selfadjoint extensions of operators which are not semibounded can be deduced from the positivity of the Schur Complement (as a quadratic form). In practical applications this amounts to proving a Hardy-like inequality. Particular cases are Dirac-Coulomb operators where distinguished selfadjoint extensions are obtained for the optimal range of coupling constant

• G M Graf,
  This talk is about mesoscopic devices, which are driven slowly and periodically in time, and about the resulting charge transport across them. Two descriptions are available: one by B\"uttiker et al. in terms of scattering matrices, the other by Thouless in terms of a Chern number. In the first approach the system is viewed as consisting of a finite device connected to infinite leads. It allows for scattering states at Fermi energy and is hence gapless. In the second one the device is idealized as being of infinite extent. It is supposed to have a gap containing the Fermi energy at all times, on which the Chern number crucially depends. We show how to relate the two seemingly disjoint approaches. We prove that they then yield the same transported charge.

• R L Frank,
  Lieb-Thirring inequalities estimate moments of negative eigenvalues of Schr\"odinger operators $-\Delta-V$ in terms of integral norms of the potential $V$. We prove that such inequalities for general operators of the form $T-V$ are, under certain conditions on the `kinetic energy' $T$, equivalent to Sobolev-type inequalities. In particular, any improvement of Sobolev inequalities will lead to an improvement of Lieb-Thirring inequalities. As an application, we show that the classical Lieb-Thirring inequalities remain valid, with possibly different constants, when the critical Hardy-weight $C|x|^{-2}$ is subtracted from the Laplace operator. Similar results are true for fractional powers of the Laplacian, and also in the presence of a magnetic field.

E Brian Davies
King's College London

updated 24 June 2008