Asymptotics, Operators, and Functionals

Speaking in 2023:

23 January Christoph Fischbacher (Baylor) Title: Complete non-selfadjointness for Schrödinger operators on the semi-axis. Abstract: We investigate complete non-selfadjointness for all maximally dissipative extensions of a Schrödinger operator on a half-line with dissipative bounded potential and dissipative boundary condition. We show that all maximally dissipative extensions that preserve the differential expression are completely non-selfadjoint. However, it is possible for maximally dissipative extensions to have a one-dimensional reducing subspace on which the operator is selfadjoint. We give a characterization of these extensions and the corresponding subspaces and present a specific example.

Away: [30 January Analysis Seminar, University of Birmingham]

13 February Kirill Cherednichenko (Bath) Title: Frequency dispersion in stratified elastic media.

20 February Basant lal Sharma (IIT Kanpur) Title: Some illustrative problems in the discrete scattering theory. Abstract: The talk concerns my research over the last decade on the topic of scattering of (scalar) waves in lattices, in the presence of an idealised line defect (rigid constraint, crack, surface step, or interface). Exact solutions are possible, under certain simplified conditions, typically in the form of a contour integral (thereby reducing of the problem to quadrature), which can be used to obtain far-field approximations as well as few other physically relevant objects. There exists a close connection of these problems with the theory of Toeplitz operators. Potential applications of the analysis include wave scattering by sharp-edged defects in meta-materials, high-frequency elastic wave scattering in elastic media by sharp corners, acoustic wave propagation in structured media, phononic and electronic transport problems in nanoribbon-nanotube junctions, etc. The talk will also highlight a few open questions concerning mathematical aspects of such problems.

6 March Martin Sieber (Bristol) Title: Flux conservation in ray methods with diffractive contributions. Abstract: Ray methods are very useful for propagating properties of wave fields in the high frequency domain. They obtain the wave field at a receiver location by summing over rays from transmitter to receiver. For an accurate description, effects of diffraction have to be taken into account. That can be done by including diffracted rays whose theory is based on the geometrical theory of diffraction or the uniform theory of diffraction. One question that arises is how these methods satisfy flux conservation. For small scatterers this follows from the optical theorem. However, for diffraction at a corner or a magnetic flux line the corresponding quantities diverge. I will discuss what replaces them in an asymptotic analysis of uniform approximations.

20 March Tristan Lawrie (Nottingham) Title: A quantum graph approach to metamaterial design. Abstract: Since the turn of the century, metamaterials have gained a large amount of attention due to their potential for possessing highly nontrivial and exotic properties - such as cloaking or perfect lensing. There has been a great push to create reliable mathematical models that accurately describe the required material composition. Here, we consider a quantum graph approach to metamaterial design. An infinite square periodic quantum graph, constructed from vertices and edges, acts as a paradigm for a 2D metamaterial. Wave transport occurs along the edges with vertices acting as scatterers modelling sub-wavelength resonant elements. These resonant elements are constructed with the help of finite quantum graphs attached to each vertex of the lattice with customisable properties controlled by a unitary scattering matrix. The metamaterial properties are understood and engineered by manipulating the band diagram of the periodic structure. The engineered properties are then demonstrated in terms of the reflection and transmission behaviour of Gaussian beam solutions at an interface between two different metamaterials. We extend this treatment to N layered metamaterials using the Transfer Matrix Method. We demonstrate both positive and negative refraction and beam steering. Our proposed quantum graph modelling technique is very flexible and can be easily adjusted making it an ideal design tool for creating metamaterials with exotic band diagram properties or testing promising multi-layer set ups and wave steering effects.

3-6 April British Mathematical Colloquium, University of Bath

Away: [10 April Applied Analysis Seminar, Louisiana State University]

Away: [14 April Mathematical Physics and Harmonic Analysis Seminar, Texas A&M University]

Away: [15 May Department of Computer Science, University of Verona]

22 May Benjamin Fehrman (Oxford) Title: Non-equilibrium fluctuations and parabolic-hyperbolic PDE with irregular drift. Abstract Non-equilibrium behavior in physical systems is widespread. A statistical description of these events is provided by macroscopic fluctuation theory, a framework for non-equilibrium statistical mechanics that postulates a formula for the probability of a space-time fluctuation based on the constitutive equations of the system. This formula is formally obtained via a zero noise large deviations principle for the associated fluctuating hydrodynamics, which postulates a conservative, singular stochastic PDE to describe the system far-from-equilibrium. In this talk, we will focus particularly on the fluctuations of the zero range process about its hydrodynamic limit. We will show how the associated MFT and fluctuating hydrodynamics lead to a class of conservative SPDEs with irregular coefficients, and how the study of large deviations principles for the particles processes and SPDEs leads to the analysis of parabolic-hyperbolic PDEs in energy critical spaces. The analysis makes rigorous the connection between MFT and fluctuating hydrodynamics in this setting, and provides a positive answer to a long-standing open problem for the large deviations of the zero range process.

29 May - 2 June CUWB-I Conference ("CUWB"="CIMAT+UNAM+Warwick+Bath") Department of Mathematics, Faculty of Science, University of Split

12 June Ian Wood (Kent) Title: Spectrum of the Maxwell equations for the flat interface between homogeneous dispersive media. Abstract: We determine and classify the spectrum of a non-selfadjoint operator pencil generated by the time-harmonic Maxwell problem with a nonlinear dependence on the frequency. More specifically, we consider one- and two-dimensional reductions for the case of two homogeneous materials joined at a planar interface. The dependence on the spectral parameter, i.e. the frequency, is in the dielectric function and we make no assumptions on its form. In order to allow also for non-conservative media, the dielectric function is allowed to be complex, yielding a non-selfadjoint problem. This is joint work with Malcolm Brown (Cardiff), Tomas Dohnal (Halle) and Michael Plum (Karlsruhe).

19 June Matteo Capoferri (Heriot-Watt) Title: Invariant subspaces of elliptic systems. Abstract

4 July Gregory Berkolaiko (TAMU) Title: Duistermaat index and eigenvalue interlacing for self-adjoint extensions of a symmetric operator Abstract: Eigenvalue interlacing is a tremendously useful tool in linear algebra and spectral analysis. In its simplest form, the interlacing inequality states that a rank-one positive perturbation shifts the eigenvalue up, but not further than the next unperturbed eigenvalue. For different types of perturbations, this idea is known as the "Weyl interlacing" (additive perturbations), "Cauchy interlacing" (for principal submatrices of Hermitian matrices), "Dirichlet-Neumann bracketing" and so on. We discuss the extension of this idea to general "perturbations in boundary conditions", encoded as interlacing between eigenvalues of two self-adjoint extensions of a fixed symmetric operator with finite (and equal) defect numbers. In this context, even the terms such as "signature of the perturbation" are not immediately clear, since one cannot take the difference of two operators with different domains. However, it turns out that definitive answers can be obtained, and they are expressed most concisely in terms of the Duistermaat index, an integer-valued topological invariant describing the relative position of three Lagrangian planes in a symplectic space. Two of the Lagrangian planes describe the two self-adjoint extensions being compared, while the third one corresponds to the distinguished Friedrichs extension. We will illustrate our general results with simple examples, avoiding technicalities as much as possible and giving intuitive explanations of the Duistermaat index, the rank and signature of the perturbation in the self-adjoint extension, and the curious role of the third extension (Friedrichs) appearing in the answers. Based on a work in progress with Graham Cox, Yuri Latushkin and Selim Sukhtaiev.

2 October Alexander Kiselev (Bath) Title: A generalisation of Cayley identity to the case of (unbounded) operators in Hilbert spaces. Abstract

9 October Yi Sheng Lim (Bath) Title: Wave dispersion in periodic composites with annular resonators. Abstract: We will discuss the problem of high-contrast homogenization of a periodic composite. The composite we consider will consist of a "soft" and a "stiff" material, arranged in a "stiff-soft-stiff" setup (soft annulus in a stiff material). We will look at an effective description of # this composite, in the sense of operator norms of the resolvents. This description is expressed in terms of various objects associated to the Ryzhov boundary triple (2009), and we will discuss ways to extract information about effective wave propagation behaviour of the composite.

16 October Andrew Morris (Birmingham) Title: The minimal regularity Dirichlet problem for degenerate elliptic PDEs beyond symmetric coefficients. Abstract

23 October Jakob Reiffenstein (Vienna) Title: An interlacing criterion for matrix-valued meromorphic Herglotz functions. Abstract

30 October Aaron Pim (Bath) Title: The application of optimal control to treatment planning in proton radiation therapy. Abstract