EPSRC Reference: 
EP/F029721/1 
Title: 
Periodic Spectral Problems 
Principal Investigator: 
Parnovski, Professor L 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
UCL 
Scheme: 
Standard Research 
Starts: 
01 June 2008 
Ends: 
30 September 2011 
Value (£): 
425,556

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 
Panel Date  Panel Name  Outcome 
06 Sep 2007

Mathematics Prioritisation Panel (Science)

Announced


Summary on Grant Application Form 
Periodic differential operators arise in many areas of physics and mathematics, and studying their spectral properties is very important. Spectra of periodic operators have a bandgap structure, that is, they consist of a collection of closed intervals possibly separated by gaps. There is a famous hypothesis, called the BetheSommerfeld 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 in a much more general setting. The solution is expected to require the use of the pseudodifferential calculus, geometry of lattices and geometrical combinatorics. An important quantitative characteristic of differential operators acting on a noncompact manifold in the socalled integrated density of states. This function is a natural analogue of the spectral counting function. We plan to study the behaviour of this function for large values of energy and, in particular, to prove that the density of states has a complete asymptotic expansion in the (negative as well as positive) powers of energy.We also plan to study more basic properties of the nature of the spectrum, namely whether the spectrum is absolutely continuous. We plan to give establish a wide range of sufficient conditions which guarantee the absolute continuity of the spectrum. Finally, we plan to study limitperiodic problems. These problems are natural generalisation of the periodic ones. While the class of limitperiodic operators is not as wide as the class of quasiperiodic or almostperiodic operators, some of the methods of the periodic theory are applicable to the limitperiodic case. We intend to prove the BetheSommerfeld conjecture in the limitperiodic setting.

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