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Abstract: We consider a quantum dot in a semi-conductor device P and a train of probes sent to it. A projective measurement is applied to each probe, just after it interacts with P. This produces a discrete density-matrices-valued stochastic process ρ_{n}. We prove that the density matrices, ρ_{n}, purify as n tends to infinity. Our results can be used to mathematically understand situations in the spirit of experiments of Haroche and Wineland (which led to the Nobel Prize in Physics 2012).
Abstract: Starting from the wave equation for a medium with material properties that vary periodically, we study a system of recurrence relations that describe propagation of wave packets that oscillate on the microscale (i.e. on lengths of the order of the period of the medium) and vary slowly on the macroscale (i.e. on lengths that contain a large number of periods). The resulting equations contain a version of the geometric optics and the overall energy transport description for periodic media.
Abstract: Photonic Crystal Fibres are periodically structured electromagnetic media with a direction of continuous translational symmetry. In this talk, we discuss the existence of time-harmonic solutions to the Maxwell system that propagate down the direction of symmetry. This leads to a two-parameter problem and we analyse the "gaps" in this frequency-wavenumber plane for which no propagating solutions exist. The location of these gaps are characterised in terms of the electromagnetic parameters and underlying geometry of the periodic structure. This is joint work with Ilia Kamotski (UCL) and Valery Smyshlyaev (UCL).
Abstract: We will discuss the problem of homogenisation (classical, or low-contrast) in periodic structures which are periodic metric graphs embedded in the d-dimensional Euclidean space. Our analysis is based on the combination of spectral theory of second order differential operators on graphs (so called quantum graphs) and the classical bounded triples theory. "Magnetic" effects which may arise in this problem will be considered. We will also discuss the possibility of extending our method to the so-called high-frequency homogenisation setting.
Abstract: Toeplitz operators over weighted Bergmann spaces on the unit disk in the complex plane and the open ball in C^{n} provide a simple and natural example of quantization and have rich connections with both functional analysis and differential geometry. In this talk, we will discuss some results of the author and his collaborators which take advantage of a previously under-utilized connection with the discrete series of irreducible unitary representations of Hermitian semisimple Lie groups to construct new C^{*}-algebras generated by commuting Toeplitz operators and in some cases compute the spectrum.
Abstract: The homogenisation procedure in critical contrast setting will be revisited from a non-traditional perspective. Namely, we will argue that the homogenisation limit has a lot to do with an explicitly constructible generalised resolvent. This resolvent, in turn, gives rise to the corresponding dilation via the well-known functional model approach. Although this approach, owing to its abstract operator-theoretical background, seems to be rather universal, a concrete example that will be discussed is essentially one-dimensional and thus the presentation will hopefully be very explicit. The talk is based on joint research with Kirill Cherednichenko (Bath, UK).
Abstract: Hormander-type operators are a general class of second-order hypoelliptic partial differential equations. These operators appear naturally in many research fields, both theoretical and applied, including, CR geometry, geometric theory of several complex variables, mathematical models in finance and human vision. The goal of this talk is to study the Dirichlet and Neumann problem for Hormander-type operators. The study of these problems in this more general setting differs substantially from the ordinary Laplacian due to the presence of the so-called characteristic points on the boundary.
Abstract: In this talk I will review a recent trend in the stability analysis of spatially periodic waves based on Evans function techniques, which provide rigorous results on cases previously studied by the (formal) physical modulation theory of Whitham. Stability is considered from the point of view of spectral analysis of the linearized problem (spectral stability), from the point of view of wave modulation theory (the strongly nonlinear theory due to Whitham as well as the weakly nonlinear theory of wave packets), and in the orbital (nonlinear) framework. The connection between these two different approaches is made through a "modulational instability index". As an example I will present a detailed analysis of stability properties of periodic traveling wave solutions of nonlinear Klein-Gordon equations with periodic potential. We analyse waves of both subluminal and superluminal propagation velocities, as well as waves of both librational and rotational types. We prove that only subluminal rotational waves are spectrally stable and establish exponential instability in the other three cases. The proof corrects a frequently cited result given by Scott (1965). In addition, I will show that the spectral information is crucial in the establishment of the orbital (nonlinear) stability for subluminal rotational wavetrains. These results are joint work with C.K.R.T. Jones (North Carolina), P.D. Miller (Michigan), J. Angulo (Sao Paulo) and R. Marangell (Sydney).
Abstract: In the talk I will present a Hilbert space perspective to existence, uniqueness and continuous dependence on the data of solutions for a certain class of linear equations involving time. For this, we define an appropriate exponentially weighted space-time Hilbert space and prove that the operator in question is continuously invertible in this space-time Hilbert space. Further, we will provide conditions on the coefficients for the solution operator thus defined being causal.
Abstract: The matrix Schrödinger equation with a selfadjoint matrix potential is considered on the half line with the general selfadjoint boundary condition at the origin. When the matrix potential is integrable, the high-energy asymptotics are established for the Jost matrix, its inverse, and the scattering matrix. Under the additional assumption that the matrix potential has a first moment, it is shown that the scattering matrix is continuous at zero energy, and a explicit formula is provided for its value at zero energy. The small-energy asymptotics are established also for the Jost matrix, its inverse, and various other quantities relevant to the corresponding direct and inverse scattering problems. When the potential has a second moment more detailed results are obtained. Furthermore, Levinson's theorem is proven, the generalized Fourier maps are constructed, the stationary formulas for the wave operators are stablished and the existence and completeness of the wave operators are proven. Also, Krein's spectral shift function is studied and trace formulas of the Buslaev-Faddeev type are obtained. Finally a Marchenko theory is developed that gives a characterization of the scattering data.
Work in collaboration with Tuncay Aktosun (University of Texas Arlington) and Martin Klaus (Virginia Polytechnic Institute).