## Analysis Seminars 2008 - 2009

### Programme

#### 3 September 2008

**Speaker:** Aaron Hiscox (Cardiff)

#### Analysis of Regge-poles for the Schrodinger equation

##### 24 September 2008

Alexandre Girouard (Cardiff)

**Shape optimisation for lower eigenvalues of the Laplace operator**.

The Polya conjecture (from 1954) states that the k-th Neumann eigenvalue of a planar domain is bounded above by 4k Pi.

In this talk I will present a sharp isoperimetric inequality for the second nonzero eigenvalue. This implies Polya conjecture for k=2. I will also discuss similar results for the Steklov spectrum and for the spectrum of the Laplace-Beltrami operator on a closed surface.

This is joint work with Nikolai Nadirashvili and Iosif Polterovich

##### 1 October 2008

Svetlana Jitomirskaia (University of California, Irvine; visiting Isaac Newton Institute, Cambridge)

#### Quantitative Aubry Duality and sharp results on absolutely continuous spectra of 1D quasiperiodic operators.

Abstract: Aubry duality (a Fourier-type transform acting on families of quasiperiodic operators) has previously been understood only on the qualitative level. We develop a quantitative version that, along with a localization-type statement, allows to prove several sharp results on the regime of absolutely continuous spectrum of 1D Schrodinger operators with analytic quasiperiodic potentials with Diophantine frequencies: full non-perturbative version of Eliasson theory, exact modulus of continuity of the integrated density of states and individual spectral measures, dry Ten martini problem. The talk is based on joint work with Artur Avila.

##### 22 October 2008

Evgeny Korotyaev (Cardiff)

#### Inverse problem for 1D Schroedinger operators with periodic distributions

Consider the 1D Schroedinger operator on the real line, where the periodic

potential is a distribution. The spectrum of this operator is purely absolutely continuous and
consists of intervals separated by gaps. We solve the inverse
problem (including characterization) both in terms of vertical slits on the quasimomentum domain and in terms of gap lengths. Furthermore, we obtain a priori two-sided estimates for these maps.

##### 29 October 2008

Timo Betcke (Manchester)

#### Accurate wave computations

The accurate and fast computation of wave phenomena is an important problem in a variety of applications in engineering, mathematics and hysics. The key to this is to approximate solutions from basis functions that already reflect the oscillatory behaviour of the solution. In this talk we will analyse different types of basis functions and their convergence properties. We will then show how they can be used in a finite element like approach to obtain accurate solutions of Laplace eigenvalue problems, interior Helmholtz, exterior Helmholtz and transmission problems.

##### 5 November 2008

Vassili Zhikov (Vladimir)

#### An approach to solvability of generalized Navier-Stokes equations

##### 12 November 2008

Yaroslav Kurylev (University College London)

#### Invisibility and Inverse Problems

In recent years there appeared much research in physical literature, theoretical and also experimental, on the phenomenon of "invisibility", namely the impossibility for an external observer to realise, judging from scattering of waves, that there is an inclusion inside a domain of interest. In this talk, joint with A.Greenleaf, M.Lassas and G.Uhlmann, we discuss some mathematical aspects of this (and similar) phenomena.

##### 20 November 2008 4:15 p.m.* (Note non-standard date / time)*

Jim Wright (Edinburgh)

#### Isoperimetric (type) inequalities in harmonic analysis

##### 26 November 2008

Valery Smyshlyaev (Bath)

#### Homogenisation of PDEs and spectral problems with "partial" degeneracies and applications

We analyse the Floquet-Bloch spectrum for elliptic operators with periodic coefficients, for a generic class of (scalar or vector) operators with small periodicity and a "partial" degeneracy in (some components of) the tensorial coefficients. The employed tools are those of "non-classical" (high contrast) homogenisation, specifically building on some ideas of recent joint work with K.Cherednichenko and V.Zhikov. This leads to interesting effects physically (e.g. allowing "directional localisation", with no wave propagation in certain directions), and mathematically allows treating form a unified perspective "classical", high-contrast homogenizations and intermediate cases. We discuss some related analytic issues.

##### 21 January 2009

South-West Analysis Regional Meeting at the University of Bath, organizer Kirill Cherednichenko.

##### 4 February 2009

Nadia Sidorova (University College London)

#### A two cities theorem for the parabolic Anderson model

The parabolic Anderson problem is the Cauchy problem for the heat equation on the d-dimentional lattice with random potential. We consider independent and identically distributed potentials such that their distribution function converges polynomially at infinity. If the solution is initially localised in the origin, we show that, as time goes to infinity, the solution is completely localised in two points almost surely and in one point with high probability. We also identify the asymptotic behaviour of the concentration sites in terms of a weak limit theorem.

##### 11 February 2009

##### Fifth WIMCS Analysis Workshop

##### "Problems in infinite domains with regular ends."

Contact: Michael Levitin (Cardiff), e-mail Levitin@cardiff.ac.uk

Timetable:

12:00-13:00 (Room M/2.06) Lunch

13:00-13:50 (Room E/0.15) Pavel Exner (Nuclear Physics Institute, Prague)

14:00-14:50 (Room E/0.15) Marco Marletta (Cardiff)

15:00-15:30 (Room M/2.06) Tea

15:30-16:20 (Room E/0.15)Leonid Parnovski (University College London)

16:30-17:20 (Room E/0.15)Pierre Duclos (Toulon and Marseilles)

17:30-18:00 (Room E/0.15) Pavel Exner (Nuclear Physics Institute, Prague)

Full Details including abstracts can be found here.

##### 18 February 2009

Alexander Pushnitski (King's College London)

#### The Birman-Krein formula and related identities

The Birman-Krein formula in mathematical scattering theory relates the spectral shift function to the scattering matrix. This formula requires some trace class assumptions. I will discuss a certain integer valued analogue of the spectral shift function which can be defined outside the trace class scheme. An identity resembling the Birman-Krein formula is valid in this situation.

##### 4 March 2009 3:45 p.m. *(Note non-standard date / time)*

Natalia Babych (Bath)

#### Elastic problems with sharp irregularities.

The talk will be focused on a discussion how sharp irregularities influence the spectral properties and vibrations of systems modelling elastic media in bounded domains.

##### 17 March 2009 1:00 p.m. *(Note non-standard date / time)*

Thomas Kriecherbauer (Bochum)

#### On the universal laws of random matrices.

Eigenvalues of random matrices display universal behavior in two ways. On the one hand, local eigenvalue statistics depend for large matrix dimensions only on the symmetries of the matrices but not on the details of the chosen probability measure. On the other hand, these distributions appear in a number of seemingly unrelated combinatorial models and even in number theory! In this talk mainly the first aspect of universality will be discussed.

##### 25 March 2009 (room M/0.34)

Yiannis Petridis (University College London)

#### Embedded eigenvalues for hyperboic surfaces and resonances

##### 20 April 2009

Joachim Puig (Universitat Politècnica de Catalunya, Barcelona, visiting Cardiff)

#### One-Dimensional Quasi-Periodic Schrödinger Operators. Spectral Theory and Dynamics

One-dimensional quasi-periodic Schrödinger operators arise naturally in several models of mathematical physics and through the linearization around quasi-periodic orbits in dynamical systems. In this talk we will see how many spectral properties of these operators can be derived through an analysis of the dynamics of the corresponding eigenvalue equations and the skew-products they define (and vice-versa). There are several questions in spectral theory which can be studied by through fruitful interaction of these two points of view. We will consider, in particular, the issue of Cantor spectrum, which has been described for the Almost Mathieu and other models, together with some of its implications for the solutions of the eigenvalue equation.

##### 22 April 2009

Tom ter Elst (Auckland)

#### Does diffusion determine the manifold?

The famous question of Kac is whether one can hear the shape of a drum. Or more precisely, whether all eigen frequencies of a drum determine the drum. In general the answer to the latter question is negative. The eigen frequencies are equal if and only if there exists a unitary operator which maps the Laplacian on the first drum onto the Laplacian on the second drum. In this talk we discuss what happens if the unitary operator is replaced by an order isomorphism, i.e., if it maps positive functions to positive functions. Or equivalently, if the diffusion processes on the two drums are equal.

This is joint work with M. Biegert and W. Arendt.

##### 22 April 2009

##### 15 May 2009 *(Note non-standard date)*

Grigori Rozenblum (Chalmers University, Gothenborg)

Eigenvalue estimates for the discrete Schrodinger operator