### University College London members and associates

- S. Kurylev
- L. Parnovski
- V. Smyshlyaev
- A. Sobolev

### About the UCL node

Our plan is to study spectra of finite-difference operators on graphs, and initially on lattices and possibly trees. The main focus will be on operators with discrete spectrum, and hence the main issue will be to describe the distribution of eigenvalues for large value of the spectral parameter. The principal difficulty lies in the fact that the lattices are multi-dimensional, which is expected to lead to new interesting effects as well as new technical obstacles.

V. Smyshlyaev plans to study spectral properties of regular, in particular periodic, lattices whose components may have highly contrasting properties. Analogous continuum problems have recently been intensively studied using tools of a 'non-classical' (high-contrast) homogenisation, and are known to display band gap and other non-standard effects due to a'`micro-resonant' nature of some components coupled to macroscopic properties of the others. This prompts in turn development and application of analytic tools of 'two-scale' operator and spectral convergence and of related (two-scale) compactness. We aim at addressing similar discrete problems, anticipating that some relevant simplifications in the model will allow making further progress both on the analytical side and on revealing new effects. This is hoped to be of interest in its own right, but also to shed new light on the `limitâ€™ continuous problems.