## Analysis Seminars 2009 - 2010

### Programme

All seminars are held at 2:10pm in Room M/2.06, Senghennydd Road, Cardiff unless stated otherwise.

Programme Organiser and Contact: **Dr Mikhail Cherdantsev**

##### 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

##### 7 October 2009 - 2:45pm

Mark Kelbert (Swansea)

#### A probabilistic proof of the entropy-power inequality

Consider two independent random variables (RVs) $X_1$ and $X_2$ taking values in $\R^d$, with probability density functions $f_{X_1}(x)$ and $f_{X_2}(x)$, respectively, where $x\in\R^d$. Let $h(X_i)$, $i=1,2$ stand for the differential entropies $$h(X_i)=-\int_{\R^d}f_{X_i}(x)\ln\;f_{X_i}(x){\rm d}x:= -\E\ln\;f_{X_i}(X_i),$$ and assume that $-\infty <+\infty$. The entropy-power inequality states that $$e^{\frac{2}{d}h(X_1+X_2)}\geq e^{\frac{2}{d}h(X_1)}+ e^{\frac{2}{d}h(X_2)}, $$ or, equivalently, $$ h(X_1+X_2)\geq h(Y_1+Y_2) $$ where $Y_1$ and $Y_2$ are {\it any} independent normal RVs with $h(Y_1)=h(X_1)$ and $h(X_2)=h(Y_2)$.

##### 14 October 2009

Thomas Hoffmann-Ostenhof (Edwin Schroedinger Institute, Vienna)

#### Spectral minimal partitions

I give an overview of recent results concerning minimal partitions obtained in collaborations Bernard Helffer and partly with Susanna Terracini and Virginie Bonnaillie Noel.

##### 21 October 2009 - 3pm (note 'non-standard time')

Mette Iversen (Bristol)

#### Minimization of Dirichlet eigenvalues with geometric constraints

We consider the problem of minimizing the k th Dirichlet eigenvalue over all open sets in Euclidean space with either a volume constraint or a Hausdorff measure constraint on the boundary. In particular we obtain upper bounds for the number of components of the various minimizers.

#### 28 October 2009 - 2:45pm (note 'non-standard time')

Jens Wirth (Imperial College London)

#### Decay estimates in anisotropic thermo-elasticity

The equations of thermo-elasticity couple a hyperbolic system (the system of crystal acoustics) to a heat equation. Aim of the talk is to present (sharp and frequency localised) a priori estimates for solutions in terms of geometric properties of the symbol of the elastic operator. The approach is based on an asymptotic decoupling of the system for small and large frequencies. Several examples for the two-dimensional situation will be given and the particular situation of cubic media in three space dimensions discussed in some detail. The two-dimensional treatment is based on joint work with Michael Reissig (Freiberg, Germany).

#### 28 October 2009 - 4:00pm (note 'non-standard time')

Speaker: Lyonell Boulton (Heriot-Watt)

#### On the eigenfunctions of a coupled system of harmonic oscillators

Let $H:=B(-\partial_x^2)+Ax^2$ acting on $L^2({\bf R};{\bf C}^2)$ where $A$ and $B$ are two constant positive definite matrices. Finding the eigenvalues and eigenfunctions of $H$ turns out to be non-trivial when $A$ and $B$ do not commute. In this talk we will examine this problem and characterise a variety of non-commuting pairs $(A,B)$ for which the eigenfunctions can be explicitly determined.

##### 4 November 2009

Nicholas Michalowski (Edinburgh)

#### Weighted Norm Inequalities for Pseudodifferential and Pseudo-Pseudodifferential Operators

I will discuss weighted boundedness results for both pseudodiffential operators with symbols in the H\"ormander classes $S^m_{\rho, \delta}$ and for "pseudo-pseudodifferential operators" introduced by Kenig-Staubach whose symbols lack smoothness in the spacial variables. The weights in question will come from the Muckenhaupt $A_p$ classes. I will discuss how to derive boundedness for the commutators between these operators and functions in BMO. This work is joint with D. Rule and W. Staubach.

##### 25 November 2009 - 3.15pm

Leonid Parnovski (University College London)

#### Periodic problems and the distribution of lattice points

##### 2 December 2009

Mikhail Cherdantsev (Cardiff).

#### Two-Scale Gamma Convergence and Its Applications to Homogenisation of Non-Linear High-Contrast Problems

It is a resent results of Bouchitte, Felbacq, Zhikov and others that passing to the limit in high-contrast elliptic PDEs may lead to non-classical effects, which are due to the two-scale nature of the limit problem. These have so far been studied in the linear setting, or under the assumption of convexity of the stored energy function. It seems of practical interest however to investigate the effect of high-contrast in the general non-linear case, such as of finite elasticity. With this aim in mind, we develop a new tool to study non-linear high-contrast problems, which may be thought of as a hybrid of the classical $\Gamma$-convergence (De Giorgi, Dal Maso, Braides) and two-scale convergence (Allaire, Briane, Zhikov). We demonstrate the need for such a tool by showing that in the high-contrast case the minimizing sequences may be non-compact in $L^p$ space and the corresponding minima may not converge to the minimum of the usual $\Gamma$-limit. We prove a compactness principle for high-contrast functionals with respect to the two-scale $\Gamma$-convergence, which in particular implies convergence of their minima.

##### 17 December 2009

Sergey Naboko (St. Petersburg).

#### Unbounded Jacobi matrices with a few gaps in the absolutely continuous spectrum: constructive examples

We consider a class of constructive examples of unbounded Jacobi matrices with absolutely continuous spectra covering a few intervals. We also analize the asymptotic behaviour of the eigenvalues at intinity for such examples.

##### 27 January 2010

Speaker: Bryan Rynne (Heriot-Watt)

##### 3 February 2010

Speaker: Karl Michael Schmidt (Cardiff)

#### Spectral Properties of Rotationally Symmetric Massless Dirac Operators

##### 17 February 2010

Speaker: Karsten Matthies (Bath)

#### Hard sphere dynamics and Boltzmann equations

#### 24 February 2010

LMS/WIMCS Analytical Computational Mathematics Day: Room M/0.34

##### 3 March 2010 - non standard time - 4:15pm

Speaker: Thomas Sorensen (Imperial)

#### The relativistic Scott correction for molecules

We prove the Scott correction for the ground state energy of molecules when the kinetic energy T(p) of the electrons is treated relativistically (T(p)=sqrt{ (pc)^2+(mc^2)^2 } - mc^2). The proof uses the coherent state calculus introduced by Solovej and Spitzer to give a simpler proof of the non-relativistic Scott correction. This is joint work with J.P. Solovej and W.L. Spitzer.

##### 10 March 2010

Speaker: Mark Dennis (Bristol)

#### Fractals, tangles and knots in laser light

Optical fields propagating in three-dimensional free space are complex scalar fields, and typically contain nodal lines (optical vortices) which may be thought of as interference fringes. Random wave fields, representing speckle patterns randomly scattered from rough surfaces, have a tangled skeleton of nodal lines, some of which are closed loops, and others are infinite, open lines. We conjecture, based on computer simulations of random superpositions of plane waves, that these lines have the fractal properties of brownian random walks with characteristic scaling of the probability that pairs of loops are linked. Holographically-controlled laser beams provide the opportunity to control the form of optical fields and the nodal line within them. Using the theory of fibred knots, we design superpositions of laser modes (effectively solutions of the 2+1 Schršdinger equation) which contain isolated knots and links. I will conclude by explaining how these mathematical fields have been experimentally realized.

##### 17 March 2010

Speaker: Valeriy Slastikov (Bristol)

#### Geometrically constrained walls in two dimensions

We address the effect of extreme geometry on a non-convex variational problem motivated by recent investigations of magnetic domain walls trapped by sharp thin necks. We prove the existence of local minimizers representing geometrically constrained walls under suitable symmetry assumptions on the domains and provide an asymptotic characterization of the wall profile. The asymptotic behavior, which depends critically on the scaling of length and width of the neck, turns out to be qualitatively different from the higher-dimensional case and a richer variety of regimes is shown to exist.

##### 17 March 2010 - 4:30pm

Speaker: Stefan Neukamm (TU Munich)

#### Rigorous derivation of a homogenized, bending-torsion theory for elastic rods from 3d elasticity

##### 24 March 2010

Speaker: Serge Richard (Lyon 1 and Cambridge)

#### A topological version of Levinson's theorem

During this seminar, we shall first recall the definitions of the main objects of scattering theory. We shall then introduce a common version of Levinson's theorem that appears in the literature and discuss its meaning. This theorem establishes a relation between the number of bound states of a quantum system and an expression in terms of the scattering operator. Its precise form, however, depends on various conditions, such as the dimension of space or the existence of resonances at thresholds, and also on a regularisation procedure.

We shall then propose a different approach of this result that takes care of the corrections and of the regularisation automatically. In particular, we shall show how K-theory for C*-algebras leads to a topological version of Levinson's theorem. Our approach means, above all, a change of perspective which makes clear that Levinson's theorem is in fact an index theorem. Finally, various examples will be presented.

##### 31 March 2010

MOPNET Meeting in Nottingham

##### 5 May 2010

Sven Gnutzman (Nottingham)

#### Nonlinear Schroedinger Equation on Graphs

##### 14 May 2010

Chris Smyth (Edinburgh)