A unified mode decomposition method for physical fields in homogeneous cosmology

The methods of mode decomposition and Fourier analysis of classical and quantum fields on curved spacetimes previously available mainly for the scalar field on Friedman- Robertson-Walker (FRW) spacetimes are extended to arbitrary vector bundle fields on general spatially homogeneous spacetimes. This is done by developing a rigorous unified framework which incorporates mode decomposition, harmonic analysis and Fourier anal- ysis. The limits of applicability and uniqueness of mode decomposition by separation of the time variable in the field equation are found. It is shown how mode decomposition can be naturally extended to weak solutions of the field equation under some analytical assumptions. It is further shown that these assumptions can always be fulfilled if the vector bundle under consideration is analytic. The propagator of the field equation is explicitly mode decomposed. A short survey on the geometry of the models considered in mathematical cosmology is given and it is concluded that practically all of them can be represented by a semidirect homogeneous vector bundle. Abstract harmonic analytical Fourier transform is introduced in semidirect homogeneous spaces and it is explained how it can be related to the spectral Fourier transform. The general form of invariant bi-distributions on semidirect homogeneous spaces is found in the Fourier space which generalizes earlier results for the homogeneous states of the scalar field on FRW spacetimes

Tunneling between corners for Robin Laplacians

We study the Robin Laplacian in a domain with two corners of the same opening, and we calculate the asymptotics of the two lowest eigenvalues as the distance between the corners increases to infinity.Comment: 27 pages, 5 figure

Lower bounds on the lowest spectral gap of singular potential Hamiltonians

We analyze Schr\"odinger operators whose potential is given by a singular interaction supported on a sub-manifold of the ambient space. Under the assumption that the operator has at least two eigenvalues below its essential spectrum we derive estimates on the lowest spectral gap. In the case where the sub-manifold is a finite curve in two dimensional Euclidean space the size of the gap depends only on the following parameters: the length, diameter and maximal curvature of the curve, a certain parameter measuring the injectivity of the curve embedding, and a compact sub-interval of the open, negative energy half-axis which contains the two lowest eigenvalues.Comment: 24 pages. To appear in slightly different form in Annales Henri Poincar

Patterson-Sullivan distributions and quantum ergodicity

We relate two types of phase space distributions associated to eigenfunctions $\phi_{ir_j}$ of the Laplacian on a compact hyperbolic surface $X_{\Gamma}$: (1) Wigner distributions \int_{S^*\X} a dW_{ir_j}=< Op(a)\phi_{ir_j}, \phi_{ir_j}>_{L^2(\X)}, which arise in quantum chaos. They are invariant under the wave group. (2) Patterson-Sullivan distributions $PS_{ir_j}$, which are the residues of the dynamical zeta-functions \lcal(s; a): = \sum_\gamma \frac{e^{-sL_\gamma}}{1-e^{-L_\gamma}} \int_{\gamma_0} a (where the sum runs over closed geodesics) at the poles $s = {1/2} + ir_j$. They are invariant under the geodesic flow. We prove that these distributions (when suitably normalized) are asymptotically equal as $r_j \to \infty$. We also give exact relations between them. This correspondence gives a new relation between classical and quantum dynamics on a hyperbolic surface, and consequently a formulation of quantum ergodicity in terms of classical ergodic theory.Comment: 54 pages, no figures. Added some reference

Distinguishability revisited: depth dependent bounds on reconstruction quality in electrical impedance tomography

The reconstruction problem in electrical impedance tomography is highly ill-posed, and it is often observed numerically that reconstructions have poor resolution far away from the measurement boundary but better resolution near the measurement boundary. The observation can be quantified by the concept of distinguishability of inclusions. This paper provides mathematically rigorous results supporting the intuition. Indeed, for a model problem lower and upper bounds on the distinguishability of an inclusion are derived in terms of the boundary data. These bounds depend explicitly on the distance of the inclusion to the boundary, i.e. the depth of the inclusion. The results are obtained for disk inclusions in a homogeneous background in the unit disk. The theoretical bounds are verified numerically using a novel, exact characterization of the forward map as a tridiagonal matrix.Comment: 25 pages, 6 figure