| EPSRC Reference: |
EP/D00022X/2 |
| Title: |
Periodic operators and related spectral distribution problems |
| Principal Investigator: |
Sobolev, Professor A |
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| Department: |
Mathematics |
| Organisation: |
UCL |
| Scheme: |
Standard Research (Pre-FEC) |
| Starts: |
01 September 2007 |
Ends: |
30 September 2009 |
Value (£): |
93,046
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| No relevance to Underpinning Sectors |
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Summary on Grant Application Form |
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Spectra of periodic operators have a band-gap structure, that is, they consist of a collection of closed intervals possibly separated by gaps. There is a famous hypothesis, called the Bethe-Sommerfeld conjecture, which claims that the number of gaps is finite. It has been justified for the Schroedinger operator with an electric field. One aim of the present project is to prove the conjecture for the magnetic fields. This is a much more challenging problem, since the magnetic field induces a stronger perturbation than the electric one. The solution is expected to require the use of the pseudo-differential calculus and the geometry of lattices. The second objective is to investigate the distribution of the eigenvalues for some differential operators on manifolds of ``simple'' structure, for example, on the torus. Here there is a number of hypothesis and partial results describing subtle properties of the eigenvalue distribution function for the Laplace operator. The aim of the project is to find out how far these results can be extended to the case of the perturbed Laplace operator, for instance, to the Schroedinger operator with electric and magnetic fields.
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Key Findings |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
| Summary |
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| Date Materialised |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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