## Analysis Seminars 2014 - 2015

### 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**

##### 29 September 2014

**Speaker:** Alexander Kiselev (Department of Mathematical Physics, St.

Petersburg State University)

**Title:** An inverse spectral problem on quantum graphs: reconstruction

of matching conditions at graph vertices.

**Abstract:** We will discuss one of the possible inverse spectral problems for quantum graphs. A quantum graph we study is a compact finite metric graph with an associated second-order differential operator defined on it. The matching conditions at graph vertices which reflect the graph connectivity are used to specify the domain of the corresponding operator. The class of matching conditions we allow is the following: at each graph vertex the coupling is assumed to be of either delta or delta-prime type. It has to be noted that the standard, or Kirchhoff, matching conditions are a particular case of delta-type coupling when all coupling constants zero out.

The inverse spectral problem we have in mind is this: does the spectrum of the operator on a graph (be it a Laplace or Schrodinger operator) uniquely determine matching at graph vertices? This type of inverse spectral problem is not as well-studied as, say, the inverse spectral problem of reconstructing the graph connectivity and metric properties based on the spectrum of a Laplace of Schrodinger operator on it. It turns out however that the mathematical apparatus we develop in order to study the former inverse problem can in fact be used in the study of the latter one.

In the simplest case of graph Laplacians, we derive a series of trace formulae which link together matching conditions of two operators under the assumption that their spectra coincide counting multiplicities. Thus necessary conditions of isospectrality of two graph Laplacians are obtained. Under the additional restriction that graph edge lengths are rationally independent, we are able to obtain necessary and sufficient conditions of the named isospectrality. It turns out that it can only occur in simplest graphs (e.g., chains or pure cycles).

The results in the case of Schrodinger operators appear less complete. We will argue however that in the case of infinitely smooth edge potentials one can advance virtually as far as in the case of graph Laplacians using more or less the same mathematical toolbox.

##### 6 October 2014

**Speaker:** Michael Ruzhansky (Imperial College).

**Title:** Quantization on Lie groups.

**Abstract:** TBC.

##### 13 October 2014

**Speaker:** Oleg Karpenkov** (**Liverpool).

**Title: **Toric singularities of surfaces in terms of lattice trigonometry.

**Abstract:** Continued fractions plays an important role in lattice trigonometry. From one hand this subject is a natural and therefore interesting to be considered by itself. From the other hand lattice trigonometry helps to describe singularities of toric varieties (which gives first results toward the solution of so-called "IKEA problem"). In this talk I will give a general introduction to the subject with various examples. I will try to avoid complicated technical details explaining main ideas behind them.

##### 20 October 2014

**Speaker:** Charles Batty (Oxford).

**Title:** Tauberian theorems, operator semigroups, and rates of decay.

**Abstract:** A Tauberian theorem, due to Ingham and Karamata in 1935, says that if $f$ is a bounded function on $[0,\infty)$ and its Laplace transform extends holomorphically across the imaginary axis then the improper integral of $f$ exists. This result can be applied in the abstract theory of operator semigroups to establish decay of solutions of certain linear evolution equations of parabolic or hyperbolic type. Recently there has been interest in the rate of convergence in these results. I shall describe the abstract results and some applications to damped wave equations and dynamical systems.

##### 27 October 2014

**Speaker: **Aleksander Pushnitski** (**King's College).

**Title:** Spectral asymptotics for compact Hankel operators.

**Abstract:** : I will give a short introduction into spectral analysis of Hankel operators. After this, I will describe a class of Hankel operators with a power asymptotics of eigenvalues. I will discuss the similarity with the Weyl law for differential operators. The talk is based on my joint work with Dmitri Yafaev (University of Rennes 1).

##### 3 November 2014

**Speaker: **Christian Kühn (Graz University**)**.

**Title: **Schrödinger operators with delta-potentials on manifolds.

**Abstract: **We will present an approach for the definition and investigation of Schrödinger operators with delta-potentials on manifolds. In particular we will consider the case when the manifold is a closed curve in R^3.

##### 10 November 2014

**Speaker:** Daniel Grieser (University of Oldenburg).

**Title: **Eigenvalues of the Laplacian on triangles.

**Abstract: **We study the spectrum of the Laplace operator with Dirichlet boundary conditions on Euclidean triangles. I will discuss two results: The first result, joint with S. Maronna, is a new proof of the fact that a triangle is (among the set of all triangles) uniquely determined by the spectrum. The only previously known proof of this uses wave invariants. The study of these is technically difficult. Our new proof uses heat invariants and is technically simpler, and also involves a curious and interesting – and apparently new – geometric fact about triangles. The second result, joint with R. Melrose, that I will discuss is a description of the full asymptotic behavior of the eigenvalues when the triangle degenerates into a line. This may happen in various ways. More precisely, there are two parameters describing the degeneration, and we give a complete asymptotic expansion in terms of both parameters. This involves a rather intricate and unexpected blow-up of the parameter space, which will be explained in the talk.

##### 17 November 2014

**Speaker:** Christoph Fischbacher (Kent).

**Title: **On the spectrum of the XXZ spin chain.

**Abstract: **We consider the Heisenberg XXZ spin chain in the Ising phase, which means that the anisotropy parameter $1/\Delta$ is strictly less than $1$. After having discussed some of its properties in the finite case, we extend our considerations to the infinite case. Using its conservation of total magnetization, we restrict the operator to subspaces of fixed total magnetization. After having shown that these restrictions are equivalent to fermionic many-particle Schrödinger operators with attractive interaction, we compute the lowest energy band, which is called droplet band. An HVZ type theorem allows us to determine higher band contributions to the spectrum. After a brief discussion of the structure of these higher band contributions, we show the existence of a gap above the droplet band uniformly in the particle number under the assumption that $1/\Delta < 1/2$. This work was done with Prof. G. Stolz, UAB.

##### 24 November 2014

**Speaker:** Lauri Oksanen (UCL).

**Title: **Local reconstruction of a first order perturbation from a restricted

hyperbolic Dirichlet-to-Neumann map.

**Abstract: **We consider a wave equation on a smooth compact Riemannian manifold
with boundary and show that acoustic measurements with sources and
receivers on disjoints sets on the boundary determine the lower order

terms in the wave equation near the set of receivers assuming that the
wave equation is exactly controllable from the set of sources and that
the set of receivers is strictly convex.

##### 1 December 2014

**Speaker: **Ian Wood** (**Kent).

**Title: **Some spectral results for waveguides.

**Abstract: **We study a spectral problem for the Laplacian in a weighted space which is related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic medium. The defect is infinitely extended and aligned with one of the coordinate axes. The perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. We prove that guided mode spectrum can be created by arbitrarily small perturbations. After performing a Floquet decomposition in the axial direction of the waveguide, we study the spectrum created by the perturbation for any fixed value of the quasi-momentum. We will also briefly discuss extending the results to a similar problem for divergence form elliptic operators.

##### 8 December 2014

**Speaker: **Beatrice Pelloni (Reading).

**Title: **TBC.

**Abstract: **TBC.