Almost periodic and related multi-dimensional spectral problems
Lead Research Organisation:
University College London
Department Name: Mathematics
Abstract
We plan to study operators with almost-periodic coefficients acting in multidimensional spaces.
Such operators are very interesting from physical point of view. They are also very interesting
and challenging mathematically, since at present only very little is known about their spectra.
We plan to develop a new method for working with such operators, called the milti-scale variational
approach. We also plan to study a number of other problems, which we consider to be of lower risk
than the main problem.
Such operators are very interesting from physical point of view. They are also very interesting
and challenging mathematically, since at present only very little is known about their spectra.
We plan to develop a new method for working with such operators, called the milti-scale variational
approach. We also plan to study a number of other problems, which we consider to be of lower risk
than the main problem.
Planned Impact
We expect to produce high-quality mathematical results important to world-wide analytic community.
A post-doctoral research associate will be trained.
We plan to bring six overseas researches to the UK; during their visits they will give talks at UK universities.
A five-days workshop will be organised in London, bringing to the UK world leading experts in spectral theory of ergodic operators.
A post-doctoral research associate will be trained.
We plan to bring six overseas researches to the UK; during their visits they will give talks at UK universities.
A five-days workshop will be organised in London, bringing to the UK world leading experts in spectral theory of ergodic operators.
Organisations
Publications
Artemev A
(2013)
Inverse electrostatic and elasticity problems for checkered distributions
in Inverse Problems
Boscain U
(2015)
Spectral analysis and the Aharonov-Bohm effect on certain almost-Riemannian manifolds
in Communications in Partial Differential Equations
Exner P
(2014)
Asymptotic eigenvalue estimates for a Robin problem with a large parameter
in Portugaliae Mathematica
Filonov N
(2015)
On the spectrum of an "even" periodic Schrödinger operator with a rational magnetic flux
in Journal of Spectral Theory
Girouard A
(2013)
The Steklov spectrum of surfaces: asymptotics and invariants
GIROUARD A
(2014)
The Steklov spectrum of surfaces: asymptotics and invariants
in Mathematical Proceedings of the Cambridge Philosophical Society
Kachkovskiy I
(2022)
Convergence of perturbation series for unbounded monotone quasiperiodic operators
in Advances in Mathematics
Kachkovskiy I
(2020)
Convergence of perturbation series for unbounded monotone quasiperiodic operators
| Description | We have solved several important problems mentioned in the proposal. The most important result is the complete asymptotic expansion of the kernel of the spectral projection of multi-dimensional almost-periodic Schroedinger operators. We also have proved the absolute continuity of the spectrum and the lack of spectral gaps for large energies for the large class of two-dimensional quasi-periodic Schroedinger operator; we have also made a significant step towards proving these results in higher dimensions. To put these results in context, the multi-dimensional quasi-periodic operators is the area where until recently, no results existed whatsoever. We have also proved several important and surprising results about periodic and quasi-periodic operators. We also have proved that generically the dispersion relation of two-dimensional periodic operators near the spectral edge is non-degenerate. Finally, we have established some perturbation theoretical results for one-dimensional almost-periodic operators. |
| Exploitation Route | The analytic (especially spectral theoretical) community will benefit from results obtained within this project. As a part of the project we have established a variant of the perturbation theory of almost-periodic problems, both for abstract operators and for differential and pseudo-differential operators. This general theory is likely to be applicable to many other problems outside the scope of our proposal. |
| Sectors | Other |
| Description | The Research Associate who had been employed on this grant has found a permanent job in academia. He is a lecturer at Groningen. Scientific connections established as a result of research related to this grant allowed us to organise a six months programme at the Newton Institute during January-June 2015 (this programme was not funded by the grant). Additionally, we organised a three days workshop in 2016 on periodic, almost-periodic, and random operators. We have published a number of research papers and made numerous presentations at conferences and research seminars. |
| Sector | Other |