20 research outputs found

    Local spectral asymptotics and heat kernel bounds for Dirac and Laplace operators

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    In this dissertation we study non-negative self-adjoint Laplace type operators acting on smooth sections of a vector bundle. First, we assume base manifolds are compact, boundaryless, and Riemannian. We start from the Fourier integral operator representation of half-wave operators, continue with spectral zeta functions, heat and resolvent trace asymptotic expansions, and end with the quantitative Wodzicki residue method. In particular, all of the asymptotic coefficients of the microlocalized spectral counting function can be explicitly given and clearly interpreted. With the auxiliary pseudo-differential operators ranging all smooth endomorphisms of the given bundle, we obtain certain asymptotic estimates about the integral kernel of heat operators. As applications, we study spectral asymptotics of Dirac type operators such as characterizing those for which the second coefficient vanishes. Next, we assume vector bundles are trivial and base manifolds are Euclidean domains, and study non-negative self-adjoint extensions of the Laplace operator which acts component-wise on compactly supported smooth functions. Using finite propagation speed estimates for wave equations and explicit Fourier Tauberian theorems obtained by Yuri Safarov, we establish the principle of not feeling the boundary estimates for the heat kernel of these operators. In particular, the implied constants are independent of self-adjoint extensions. As a by-product, we affirmatively answer a question about upper estimate for the Neumann heat kernel. Finally, we study some specific values of the spectral zeta function of two-dimensional Dirichlet Laplacians such as spectral determinant and Casimir energy. For numerical purposes we substantially improve the short-time Dirichlet heat trace asymptotics for polygons. This could be used to measure the spectral determinant and Casimir energy of polygons whenever the first several hundred or one thousand Dirichlet eigenvalues are known with high precision by other means

    Light-Matter Interaction Models in Relativistic Quantum Information

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    In this thesis, we study the interaction of a first-quantized atomic system with the quantum electromagnetic field within the context of relativistic quantum information (RQI). To that end, we examine common classes of Hamiltonians -- in particular, the dipolar coupling model as well as scalar field analogues (Unruh-DeWitt models) -- for their applicability in RQI setups. Firstly, we investigate how quantum randomness generation based on unbiased measurements on an atom can get compromised by an adversary that has access to the electromagnetic field. We show that preparing the atom in the ground state in the presence of no field excitations is, in general, not the best choice to generate randomness. Secondly, at the study of light-matter interactions inside optical cavities, we show that frequently employed approximations, such as the single-mode approximation and dimensional reduction, fail for relativistic regimes but can be already ill-behaved for non-relativistic scenarios. In particular, we show how approximating a very long and thin cavity by a one-dimensional system can be understood by recasting the D+1 dimensional quantum field inside the cavity as an infinite sum of massive 1+1 dimensional fields. The dimensional reduction approximation can subsequently be identified with ignoring all but one of these subfields or, equivalently, with a change of the atomic localization. 


 Up to this point, we have treated the atomic center of mass classically -- a feature that is shared by usual Hamiltonians in RQI and quantum optics. We therefore revisit the interaction of atoms and light by considering all atomic degrees of freedom to be quantum. Further, we discuss subtleties with respect to the gauge nature of light and the effect that multipole approximations have. This allows us to connect the multipolar Hamiltonian with the common effective models of quantum optics and relativistic quantum information. In particular, we discuss the influence of atomic center-of-mass delocalization and the presence of the so-called Roentgen term. Significantly, Unruh-DeWitt models fail to account for the entangling interaction between all atomic and field degrees of freedom, and we present then a scalar analogue of the Roentgen term. Finally, we demonstrate how the usual dipole model preserves covariance when considering atoms on relativistic trajectories and how this model can be used as a qualitative means to study RQI scenarios

    Revivals in time evolution problems

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    Subject to periodic boundary conditions, it is known that the solution to a certain family of linear dispersive partial differential equations, such as the free linear Schr¨odinger and Airy evolution, exhibits a dichotomy at rational and irrational times. At rational times, the solution is decomposed into a finite number of translated copies of the initial condition. Consequently, when the initial function has a jump discontinuity, then the solution also exhibits finitely many jump discontinuities. On the other hand, at irrational times the solution becomes a continuous, but nowhere differentiable function. These two effects form the revival and fractalisation phenomenon at rational and irrational times, respectively. The main aim of the thesis is to further investigate the phenomenon of revivals in time evolution problems posed under appropriate boundary conditions on a finite interval. We consider both first-order and second-order in time problems. For the former, we examine the influence of non-periodic boundary conditions on the revival effect. For the latter, we study the revivals under periodic and non-periodic boundary conditions. In terms of first-order in time evolution problems, we show that the revival phenomenon persists in the free linear Schr¨odinger equation under pseudo-periodic and Robin-type boundary conditions. Moreover, we prove that under quasi-periodic boundary conditions, the Airy equation does not in general exhibit revivals. With respect to second-order in time equations, we first formulate an abstract setting for the revival phenomenon, which we then apply to establish that the periodic, even-order poly-harmonic wave equation exhibits revivals. Finally, following the lack of revivals in Airy’s quasi-periodic problem, we characterise quasi-periodic and periodic problems, either of first-order or second-order in time, for which the revival effect breaks. In general, our approach relies on identifying the canonical periodic components of the generalised Fourier series representations of solutions, in order to utilise the classical periodic theory of revivals

    High-Order Numerical Methods in Lake Modelling

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    The physical processes in lakes remain only partially understood despite successful data collection from a variety of sources spanning several decades. Although numerical models are already frequently employed to simulate the physics of lakes, especially in the context of water quality management, improved methods are necessary to better capture the wide array of dynamically important physical processes, spanning length scales from ~ 10 km (basin-scale oscillations) - 1 m (short internal waves). In this thesis, high-order numerical methods are explored for specialized model equations of lakes, so that their use can be taken into consideration in the next generation of more sophisticated models that will better capture important small scale features than their present day counterparts. The full three-dimensional incompressible density-stratified Navier-Stokes equations remain too computationally expensive to be solved for situations that involve both complicated geometries and require resolution of features at length-scales spanning four orders of magnitude. The main source of computational expense lay with the requirement of having to solve a three-dimensional Poisson equation for pressure at every time-step. Simplified model equations are thus the only way that numerical lake modelling can be carried out at present time, and progress can be made by seeking intelligent parameterizations as a means of capturing more physics within the framework of such simplified equation sets. In this thesis, we employ the long-accepted practice of sub-dividing the lake into vertical layers of different constant densities as an approximation to continuous vertical stratification. We build on this approach by including weakly non-hydrostatic dispersive correction terms in the model equations in order to parameterize the effects of small vertical accelerations that are often disregarded by operational models. Favouring the inclusion of weakly non-hydrostatic effects over the more popular hydrostatic approximation allows these models to capture the emergence of small-scale internal wave phenomena, such as internal solitary waves and undular bores, that are missed by purely hydrostatic models. The Fourier and Chebyshev pseudospectral methods are employed for these weakly non-hydrostatic layered models in simple idealized lake geometries, e.g., doubly periodic domains, periodic channels, and annular domains, for a set of test problems relevant to lake dynamics since they offer excellent resolution characteristics at minimal memory costs. This feature makes them an excellent benchmark to compare other methods against. The Discontinuous Galerkin Finite Element Method (DG-FEM) is then explored as a mid- to high-order method that can be used in arbitrary lake geometries. The DG-FEM can be interpreted as a domain-decomposition extension of a polynomial pseudospectral method and shares many of the same attractive features, such as fast convergence rates and the ability to resolve small-scale features with a relatively low number of grid points when compared to a low-order method. The DG-FEM is further complemented by certain desirable attributes it shares with the finite volume method, such as the freedom to specify upwind-biased numerical flux functions for advection-dominated flows, the flexibility to deal with complicated geometries, and the notion that each element (or cell) can be regarded as a control volume for conserved fluid quantities. Practical implementation details of the numerical methods used in this thesis are discussed, and the various modelling and methodology choices that have been made in the course of this work are justified as the difficulties that these choices address are revealed to the reader. Theoretical calculations are intermittently carried out throughout the thesis to help improve intuition in situations where numerical methods alone fall short of giving complete explanations of the physical processes under consideration. The utility of the DG-FEM method beyond purely hyperbolic systems is also a recurring theme in this thesis. The DG-FEM method is applied to dispersive shallow water type systems as well as incompressible flow situations. Furthermore, it is employed for eigenvalue problems where orthogonal bases must be constructed from the eigenspaces of elliptic operators. The technique is applied to the problem calculating the free modes of oscillation in rotating basins with irregular geometries where the corresponding linear operator is not self-adjoint

    Eigenstructure of the equilateral triangle. Part III. The Robin problem

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    Lamé's formulas for the eigenvalues and eigenfunctions of the Laplacian on an equilateral triangle under Dirichlet and Neumann boundary conditions are herein extended to the Robin boundary condition. They are shown to form a complete orthonormal system. Various properties of the spectrum and modal functions are explored

    Hybrid routing in delay tolerant networks

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    This work addresses the integration of today\\u27s infrastructure-based networks with infrastructure-less networks. The resulting Hybrid Routing System allows for communication over both network types and can help to overcome cost, communication, and overload problems. Mobility aspect resulting from infrastructure-less networks are analyzed and analytical models developed. For development and deployment of the Hybrid Routing System an overlay-based framework is presented

    Hybrid Routing in Delay Tolerant Networks

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    This work addresses the integration of today\u27s infrastructure-based networks with infrastructure-less networks. The resulting Hybrid Routing System allows for communication over both network types and can help to overcome cost, communication, and overload problems. Mobility aspect resulting from infrastructure-less networks are analyzed and analytical models developed. For development and deployment of the Hybrid Routing System an overlay-based framework is presented
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