EPSRC Reference: 
EP/H046240/1 
Title: 
Statistics of spectra of quantum graphs 
Principal Investigator: 
Winn, Dr B 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
School of Mathematics 
Organisation: 
Loughborough University 
Scheme: 
Standard Research 
Starts: 
01 July 2010 
Ends: 
30 June 2013 
Value (£): 
231,919

EPSRC Research Topic Classifications: 
Mathematical Analysis 
Mathematical Physics 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 
Panel Date  Panel Name  Outcome 
04 Mar 2010

Mathematics Prioritisation Panel

Announced


Summary on Grant Application Form 
Quantum graphs may be thought of as vibrating networks of connected wires, perhaps connected like a spider's web, or else in more complicated ways. At certain frequencies of vibration, standing waves may be formed. These frequencies are called eigenvalues, and the shapes of the standing waves are called eigenfunctions. What is remarkable is that these eigenvalues and eigenfunctions serve as a model for the energy levels and wavefunctions of complex quantum mechanical systems, if statistical indicators are considered such as the distribution of spacings between levels. This correspondence is largely a numerical observation, but has been supported by heuristic and rigorous evidence.The present proposal aims to understand rigorously some aspects of eigenfunction shape for quantum graphs, with the intention of using the understanding gained to make predictions about other complex quantum systems. One of the central mysteries of semiclassical quantum mechanics is to understand how wavefunctions behave at large energies. It is known that almost all of the wavefunctions for quantum systems which have chaotic classical dynamics become uniformly distributed in the large energy limit. This means that for a quantum mechanical particle prepared in such a state, the probability of finding it in any region is merely proportional to the volume of that region. What is yet far from being fully understood is the behaviour of possible exceptional wavefunctions. One possibility is that they do not even exist; that is, all wavefunctions become uniformly distributed. Another possibility is that the wavefunctions become enhanced around unstable classical periodic orbits, which would be very far from uniform distribution. This is an important question, since physical intuition does not give a clear prediction one way or the other. Moreover there is mathematical evidence pointing to both outcomes, in different systems. My belief is that quantum graphs will be a suitable testing ground for these questions, avoiding the technical difficulties associated with complex quantum systems.Using quantum graphs as models has already resulted in some successes in understanding the distribution of spacings between energy levels. I hope to make similar progress in understanding localisation in quantum wavefunctions. This will build upon the 70plus year history of graphlike models as key tools in understanding spectral features of mathematical chemistry and physics. Recent progresses in semiclassical structures in quantum wavefunctions emphasise the timeliness of the proposed project.

Key Findings 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Potential use in nonacademic contexts 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Impacts 
Description 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk 
Summary 

Date Materialised 


Sectors submitted by the Researcher 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Project URL: 

Further Information: 

Organisation Website: 
http://www.lboro.ac.uk 