## Analysis Seminars 2010 - 2011

### Programme

##### 1 October 2010

**Speaker:** Christian Jaekel (Cardiff)

**Location:** Room M/2.30 (Please note this unusual location)

**Title:** Quantum Field Theory on de Sitter Space – part 1

**Abstract:** Taking advantage of previous work by Figari, Hoegh-Krohn and Nappi, I will provide a complete, non-perturbative description of interacting quantum fields describing scalar bosons of mass m>0.

In the first talk I will present the relation of our work to differential geometry, harmonic analysis, complex analysis in several variables, the Fourier-Helgason transformation and the representation theory of semi-simple Lie groups. I will also discuss (quantum) dynamical systems on the de Sitter space, both in their canonical and covariant formulation.

##### 8 October 2010

**Speaker:** Christian Jaekel (Cardiff)

**Title: **Quantum Field Theory on de Sitter Space – part 2

**Abstract:** In the second talk I we will discuss the relation of our work to axiomatic quantum field theory, local quantum field theory and constructive quantum field theory.

This second part will also discuss aspects of spectral theory and Markov processes, and if there is time, some new results in micro-local analysis.

##### 15 October 2010

**Speaker: **Lyonell Boulton (Heriot-Watt, Edinburgh)

**Title: **On the limit of second order spectra and the eigenvalues of self-adjoint operators.

**Abstract:** Let A be a self-adjoint operator acting on a Hilbert space. The notion of second order spectrum of A relative to a given finite-dimensional subspace of its domain has been studied recently in connection with the phenomenon of spectral pollution in the Galerkin method. We will examine in this talk a general framework allowing us to determine how the second order spectrum encodes precise information about the multiplicity of the isolated
eigenvalues of A. The theoretical findings presented will be supported by
various numerical experiments on the computation of inclusions for
eigenvalues in benchmark models via finite element bases.

##### 22 October 2010

Day Session

##### 29 October 2010

**Speaker:** Hillel Raz (Cardiff)

**Title:** Minimal Partitions of Quantum Graphs

**Abstract:**The eigenfunction of the $n$th eigenvalue of the Laplacian ($-d^2/dx^2$) on a bounded regular domain partitions that domain into $k$ subdomains (known as nodal domains). Courant's nodal domain theorem states that $k\le n$. We study the relationship between the $n$th eigenfunction of the Laplacian on a quantum graph (a graph where each edge has a positive length with specific boundary conditions at the vertices) and partitions of the graph into $n$ parts. We describe a procedure of attaining eigenfunctions, and hence the spectrum, by investigating these partitions, in particular minimal ones. The minimal partitions are found by assigning a score to each partition which is the maximum of the first eigenvalue of the Laplacian over each part of the partition. No prior knowledge of quantum graphs is necessary. This is joint work with Ram Band, Gregory Berkolaiko and Uzy Smilansky and is based partially on results by Bernard Helffer, Thomas Hoffmann-Ostenhoff, Susanna Terracini et al

##### 5 November 2010

**Speaker:** Sergey Morozov (University College London)

**Title:** High-energy properties of the density of states of (almost-)periodic Schrödinger operators

**Abstract:** In the last years, a number of results on the asymptotic behaviour of the density of states of multidimensional periodic and almost periodic Schrödinger operators have been obtained. We will review these results, and briefly discuss some of the modern techniques used in their proofs.

##### 12 November 2010

**Speaker:** Par Kurlberg (KTH Stockholm)

**Title:** Number theory related to quantum chaos

**Abstract:** Quantum chaos is concerned with properties of eigenvalues and

eigenfunctions of "quantized Hamiltonians". For instance, can classical

chaos be detected by looking at the spacings between eigenvalues?

Another problem is if classical ergodicity forces eigenfunctions to be

equidistributed in a certain sense. We will give a short introduction

to quantized Hamiltonians, and then show that the study of the above

mentioned questions for some simple dynamical systems gives rise to

interesting problems in number theory.

##### 19 November 2010

**Speaker:** Eugen Varvaruca (Reading)

**Title:** Existence of Steady Free-Surface Waves with Corners of 120 Degrees at Their Crests in the Presence of Vorticity

**Abstract:** We present some recent results on singular solutions of the problem of traveling gravity surface water waves on flows with vorticity. It has been known since the work of Constantin and Strauss (2004) that there exist spatially periodic waves of large amplitude for any vorticity distribution. We show that, for any nonpositive vorticity distribution, a sequence of large-amplitude regular waves converges in a weak sense to an extreme wave with stagnation points at its crests. The proof is based on new a priori estimates, obtained by means of the maximum principle, for the fluid velocity and the wave height along the family of regular waves whose existence was proved by Constantin and Strauss. We also show, by new geometric methods, that this extreme wave has corners of 120 degrees at its crests, as conjectured by Stokes in 1880. The results were obtained in collaboration with Ovidiu Savin (Columbia University) and Georg Weiss (University of Tokyo).

##### 26 November 2010

**Speaker:** Isaac Chenchiah (Bristol)

**Title:** Evolution of brittle damage in solids

**Abstract:** We present a variational formulation for the quasi-static evolution of brutal brittle damage for isotropic materials. Along the way we will survey contemporary variational approaches to describing the existence and quasi-static evolution of microstructures in solids. This is joint work with Christopher Larsen, Worcester Polytechnic Institute.

##### 3 December 2010

**Speaker:** Lillian Pierce (Oxford)

**Title:** Discrete Analogues in Harmonic Analysis

**Abstract:** Discrete problems have a habit of being beautiful but difficult. This can be true even of discrete problems whose continuous analogues are easy. For example: computing the surface area of a sphere of radius N^{1/2} in k-dimensional Euclidean space (easy). Counting the number of representations of an integer N as a sum of k squares (historically hard). In this talk we'll survey a menagerie of discrete analogues of operators arising in harmonic analysis, including singular integral operators (such as the Hilbert transform), maximal functions, and fractional integral operators. In certain cases we can learn everything we want to know about the analogous discrete operator immediately, from its continuous analogue. In other cases (such as Radon transforms), the discrete operator requires a completely new approach. We'll see what makes a discrete operator easy/hard to treat, and outline some of the methods that are breaking new ground, key aspects of which come from number theory. In particular, we will highlight the roles played by theta functions, exponential sums, Waring's problem, and the circle method of Hardy and Littlewood. No specialist knowledge of singular integral operators or the circle method will be assumed.

##### 10 December 2010

**Speaker:** Igor Wigman (Cardiff)

**Title:** Fluctuations of the nodal length of random spherical harmonics

**Abstract:** Using the multiplicities of the Laplace eigenspace on the sphere (the space of spherical harmonics) we endow the space with Gaussian probability measure. This induces a notion of random Gaussian spherical harmonics of degree $n$ having Laplace eigenvalue $E=n(n+1)$. We study the length distribution of the nodal lines of random spherical harmonics. It is known that the expected length is of order $n$. It is natural to conjecture that the variance should be of order $n$, due to the natural scaling. Our principal result is that, due to an unexpected cancelation, the variance of the nodal length of random spherical harmonics is of order $\log{n}$. This behaviour is consistent with the one predicted by Berry for nodal lines on chaotic billiards (Random Wave Model). In addition we find that a similar result is applicable for "generic" linear statistics of the nodal lines

##### 10 December 2010

**Speaker:** Anatoly Zhigljavsky (Cardiff)

**Title: **Uniformity of sequences in [0,1] and in the circle

**Abstract: **I will discuss several uniformity characteristics of point sequences in [0,1] and in the circle and the problems of construction of the sequences which are optimal with respect to these characteristics. It will be shown that the sequences that are optimal with respect to one natural criterion may be awfully bad with respect to another one. Some of the problems to be discussed are classical but some other are much less known. The discussion will be of survey character with vague mathematical formulation of the results. The main emphasis will be on ideas. Close interaction with the audience is expected.

##### 4 February 2011

**Speaker: **Ram Band (Bristol)

**Title: **Scattering from isospectral graphs

**Abstract: **In 1966 Marc Kac asked 'Can one hear the shape of a drum?'. The answer was given only in 1992, when Gordon et al. found a pair of drums with the same spectrum. The study of isospectrality and inverse problems is obviously not limited to drums and treats various objects such as molecules, quantum dots and graphs.

In 2005 Okada et al. conjectured that isospectral drums can be distinguished by their scattering poles (resonances). We prove that this is not the case for isospectral quantum graphs, i.e., isospectral quantum graphs share the same resonance distribution.

This is a joint work with Adam Sawicki and Uzy Smilansky.

##### 11 February 2010

**Speaker:** Alexander Pushnitsky (King’s College London)

**Title: **Asymptotics of Eigenvalue Clusters of the Perturbed Landau Hamiltonians

**Abstract: **I will discuss the spectrum of the Landau Hamiltonian (i.e. the two-dimensional Schrodinger operator with a constant homogeneous magnetic field) perturbed by an electric potential which decays at infinity. The spectrum of this Hamiltonian consists of clusters of eigenvalues around the Landau levels. It turns out that the the density of eigenvalues in the N'th cluster for large N can be described by means of a simple semiclassical formula. The formula has been inspired by the classical result of A.Weinstein about the eigenvalue clusters for a Laplacian plus a potential on a sphere. The talk is based on a recent joint work with Georgi Raikov and Carlos Villegas-Blas.

##### 18 February 2011

**Speaker: **Yves Tourigny (Bristol)

**Title: **Krein's theory of strings and continued fractions

**Abstract:**In a series of papers published in the Soviet Union during the
1950's, M. G. Krein made a detailed study of the existence of spectral
expansions associated with the vibrating string equation. This work was to
a large extent inspired by Stieltjes' solution of the moment problem that
now bears his name. We shall review these connections, and show how they
lead to the solution of some inverse spectral problems by continued
fractions.

##### 23 February 2010 - 16:10 - 17:10

**Speaker: **Roman Schubert (Bristol)

**Title: **How do coherent states spread? Time evolution on Ehrenfest time scales.

**Abstract:** We will discuss the time dependent Schroedinger equation for a

particular class of initial states, the so called coherent states. The time evolution of coherent states can approximately described using a simple semiclassical approximation and for this reason they are widely used in many applications in chemistry and physics. In recent years the accuracy of the semiclassical approximation for the time evolution of coherent states has been studied in great detail in the mathematical literature, in particular by Combescure and Robert, and by Hagedorn and Joye. The main result of these studies is that the semiclassical approximation is accuarte up to a time called the Ehrenfest time which depends on the semiclassical parameter h and the dynamical properties of the classical system. We will discuss what happens to the state at the Ehrenfest time. We find that the state changes its nature and transforms from a localised coherent state into an extended Lagrangian state, and we develop a uniform semiclassical approximation describing this transition.

This is joint work with Fabricio Toscano and Raul Vallejos

##### 4 March 2011

**Speaker: **Yan Fyodorov (Nottingham)

**Title: **Freezing Transition in Decaying Burgers Turbulence and Random Matrix Dualities

**Abstract:**Freezing transition with decreasing temperature is a generic property of equilibrium statistical mechanics models whose random energy landscapes are logarithmically correlated in space. The extreme value statistics plays an important role in elucidating the nature of such a transition. In the present work we reveal a similar transition to take place in one-dimensional decaying Burgers turbulence with a power-law correlated random profile of Gaussian-distributed initial velocities < v(x; 0)v(x'; 0) > ~ |x -x'|^{-2}, with the role of temperature played by viscosity. The low-viscosity phase exhibits non-Gaussian one-point probability density of velocities, continuously dependent on viscosity, reflecting a spontaneous one step replica symmetry breaking (RSB) in the associated statistical mechanics problem. We obtain the low orders cumulants analytically. Our results, which are checked numerically, are based on combining insights in the mechanism of the freezing transition in random logarithmic potentials with an extension of duality relations discovered recently in Random Matrix Theory.

The presentation is based on the work Y.V. Fyodorov, P. Le Doussal and A. Rosso Europh. Lett. v.90 (2010) 60004.

##### 16 March 2011 - 15:15 in M/2.06

**Speaker: **Stefan Neukamm (MPI Leipzig)

**Title:** On the commutability of linearization and homogenization in finite elasticity.

**Abstract:**We study the question whether homogenization and linearization commute in finite elasticity. In particular, we prove that homogenization and linearization commute at identity for a large class of energy densities in geometrically nonlinear, single well elasticity. This commutativity result holds under very general assumptions on the spatial heterogeneity; e.g. random materials can be treated. In the talk we mainly focus on the case of periodic composites. (joint work with S. Müller & A. Gloria)

##### 18 March 2011

**Speaker: **Alexandre Girouard

**Title:** Geometric control of the Steklov spectrum of a domain.

**Abstract:**The Dirichlet-to-Neumann map is a first order elliptic pseudodifferential operator acting on the boundary of a compact manifold. Its spectrum which is called the Steklov spectrum. with In this talk I will discuss how the geometry of an ambient manifold can be used to control the Steklov spectrum of a bounded domain in terms of its isoperimetric ratio. On surfaces, it is possible to control the spectrum in terms of the genus. The proofs use classical complex analysis and some results in abstract metric geometry.

##### 25 March 2011

**Speaker: **Martin Huxley (Cardiff)

**Title:** The Tiled Circle Problem: an application of uniform distribution.

**Abstract:** TBC

##### 1 April 2011

**Speaker: **Anna Kirpichnikova (Glasgow)

**Title:** Focusing waves in the unknown media

**Abstract:** TBC

##### 8 April 2011 - N/2.25

**Speaker**: Kirill Cherednichenko (Cardiff)

**Title:** Nonlinear elastic plates with finite bending energy'

**Abstract:** I will continue the discussion of the behaviour of deformations with finite bending energy in
the limit as the thickness of the plate goes to zero. This will be based on the theory presented earlier at our seminar by Mikhail. The main result of the present lecture is a variational model from the hierarchy of "nonlinear plate theories'', which was proposed heuristically by Kirchhoff and derived rigorously only a decade ago.

##### 13 May 2011

**Speaker: **Mark Peletier

**Title:** TBC

**Abstract:** TBC

##### 13 May 2011 - 15:40 in M/2.06

**Speaker**: Dmitry Yafaev

**Title:** Exponential decay of eigenfunctions of differential equations

**Abstract:** The first exponential estimate on eigenfunctions $\psi$ of the discrete spectrum for second order self-adjoint elliptic operators $H$ is due to Shnol' (1957) who proved that an eigenfunction corresponding to an eigenvalue $\lambda$ satisfies the estimate

\[\int_{{\Bbb R}^{\rm d}} | \psi(x) |^2 e^{2\delta |x|} dx < \ii.\eqno(*)\] Here $\delta$ depends only on the distance $d(\lambda)=\dist\{\lambda,\sigma_{ess}(H)\}$ between the corresponding eigenvalue and the essential spectrum

$ \sigma_{ess}(H)$ of the operator $H$. Later Agmon (1982) has shown that estimate (*) is true with an arbitrary $\delta \sqrt{d(\lambda)}$, but only for eigenvalues lying below $ \sigma_{ess}(H)$. A natural question to ask is whether such a stronger estimate is true for eigenvalues lying in gaps of $\sigma_{ess}(H)$. We give a negative answer to this question considering a one-dimensional Schr\"odinger operator whose potential is a sum of a periodic function and of a function with compact support.

Another goal of our work is to study exponential decay of eigenfunctions for first order matrix differential operators

\[ H =-i\sum_{j=1}^{\rm d} A_j \frac{\partial}{\partial x_{j}} +V(x) \]
acting in the space ${\cal H}=L_2({\Bbb R}^{\rm d} ; {\Bbb C}^n)$.

Here $A_j=A_j^*$, $j=1,\ldots, {\rm d}$, are constant matrices and $V(x)$ is a symmetric matrix-valued functiion.

Set \[\gamma =\max_{|\xi|=1} {\pmb|} \sum_{j=1}^{\rm d} A_j \xi_{j}{\pmb|}, \q \xi=(\xi_{1},\ldots, \xi_{{\rm d}}) , \q{\pmb|} \cdot{\pmb|}={\pmb|} \cdot{\pmb|}_{{\Bbb C}^n}.
\]

For example, $\gamma=1$ for the Dirac operator. Our main result is the estimate (*) with an arbitrary $ \delta< \gamma^{-1} d(\lambda)$ for all eigenvalues (including those lying in gaps of $\sigma_{ess}(H)$).

##### 3 June 2011 - 15:15 in Room M/2.06

**Speaker: **Alex Iosevich.

**Title: **The paraboloid: a ubiquitous counter-example.

**Abstract: **We shall survey the Falconer distance problem, fractal variants of the regular value theorem in differential geometry and distribution of lattice points near families of surfaces. Connections between these problems shall be explored and the paraboloid, not the sphere, will pop up each time to provide a simple and convenient sharpness example.

##### 10 June 2011 - 14:00 in Room M/2.06

**Speaker: **Serife Faydaoglu (Dokuz Eylül University, Izmir, Turkey).

**Title: **An expansion formula for a Sturm-Liouville eigenvalue problem with impulse

**Abstract: **The presentation is concerned with an eigenvalue problem for second order differential equations with impulse. Such a problem arises when the method of separation of variables applies to the heat conduction equation for two-layered composite. The existence of a countably infinite set of eigenvalues and eigenfunctions is proved and a uniformly convergent expansion formula in the eigenfunctions is established.

##### 14 June 2011 - 11:00 to 15:45 in Room M/0.40

**One Day Workshop: Metamaterials and high-contrast homogenisation: analysis, numerics and application **

##### 15 August 2011 - 14:10 in Room M/2.06

**Speaker: **Dmitry Jakobson (McGill)

**Title: **Curvature of random metrics.

**Abstract: **We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class; our questions are motivated by comparison geometry. Next, analogous questions are considered for the scalar curvature in dimension n>2, and for the Q-curvature of random Riemannian metrics. This is joint work with I. Wigman and Y. Canzani.