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