Corona-type theorems and division in some function algebras on planar domains

Let $A$ be an algebra of bounded smooth functions on the interior of a compact set in the plane. We study the following problem: if $f,f_1,\dots,f_n\in A$ satisfy $|f|\leq \sum_{j=1}^n |f_j|$, does there exist $g_j\in A$ and a constant $N\in\N$ such that $f^N=\sum_{j=1}^n g_j f_j$? A prominent role in our proofs is played by a new space, C_{\dbar, 1}(K), which we call the algebra of \dbar-smooth functions. In the case $n=1$, a complete solution is given for the algebras $A^m(K)$ of functions holomorphic in $K^\circ$ and whose first $m$-derivatives extend continuously to \ov{K^\circ}. This necessitates the introduction of a special class of compacta, the so-called locally L-connected sets. We also present another constructive proof of the Nullstellensatz for $A(K)$, that is only based on elementary \dbar-calculus and Wolff's method.Comment: 23 pages, 6 figure

Siciak-Zahariuta extremal functions, analytic discs and polynomial hulls

We prove two disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space. We use these formulas to characterize the polynomial hull of an arbitrary compact subset of complex affine space in terms of analytic discs. Similar results in previous work of ours required the subsets to be connected

Interface modes of two-dimensional composite structures

The surface modes of a composite consisting of aligned metallic wires with square cross sections are investigated, on the basis of photonic band structure calculations. The effective long-wavelength dielectric response function is computed, as a function of the filling fraction. The dependence of the optical absorption on the shape of the wires and the polarization of light is discussed, and the effect of sharp corners analyzed. The effect of the interaction between the wires on the localization of surface plasmons is also addressed.Comment: 12 pages, 4 figures, to appear in Surf. Sc

Reconstruction of Bandlimited Functions from Unsigned Samples

We consider the recovery of real-valued bandlimited functions from the absolute values of their samples, possibly spaced nonuniformly. We show that such a reconstruction is always possible if the function is sampled at more than twice its Nyquist rate, and may not necessarily be possible if the samples are taken at less than twice the Nyquist rate. In the case of uniform samples, we also describe an FFT-based algorithm to perform the reconstruction. We prove that it converges exponentially rapidly in the number of samples used and examine its numerical behavior on some test cases

A Weighted Estimate for the Square Function on the Unit Ball in \C^n

We show that the Lusin area integral or the square function on the unit ball of \C^n, regarded as an operator in weighted space $L^2(w)$ has a linear bound in terms of the invariant $A_2$ characteristic of the weight. We show a dimension-free estimate for the ``area-integral'' associated to the weighted $L^2(w)$ norm of the square function. We prove the equivalence of the classical and the invariant $A_2$ classes.Comment: 11 pages, to appear in Arkiv for Matemati

Electron energy loss and induced photon emission in photonic crystals

The interaction of a fast electron with a photonic crystal is investigated by solving the Maxwell equations exactly for the external field provided by the electron in the presence of the crystal. The energy loss is obtained from the retarding force exerted on the electron by the induced electric field. The features of the energy loss spectra are shown to be related to the photonic band structure of the crystal. Two different regimes are discussed: for small lattice constants $a$ relative to the wavelength of the associated electron excitations $\lambda$, an effective medium theory can be used to describe the material; however, for $a\sim\lambda$ the photonic band structure plays an important role. Special attention is paid to the frequency gap regions in the latter case.Comment: 12 pages, 7 figure

Causality, particle localization and positivity of the energy

Positivity of the Hamiltonian alone is used to show that particles, if initially localized in a finite region, immediately develop infinite tails.Comment: To appear in: Irreversibility and Causality in Quantum Theory -- Semigroups and Rigged Hilbert Spaces, edited by A. Bohm, H.-D. Doebner and P. Kielanowski, Springer Lecture Notes in Physics, Vol. 504 (1998

Boundedness, compactness and Schatten-class membership of weighted composition operators

The boundedness and compactness of weighted composition operators on the Hardy space ${\mathcal H}^2$ of the unit disc is analysed. Particular reference is made to the case when the self-map of the disc is an inner function. Schatten-class membership is also considered; as a result, stronger forms of the two main results of a recent paper of Gunatillake are derived. Finally, weighted composition operators on weighted Bergman spaces $\mathcal{A}^2 \alpha(\mathbb{D})$ are considered, and the results of Harper and Smith, linking their properties to those of Carleson embeddings, are extended to this situation.Comment: 12 page

Recurrence for discrete time unitary evolutions

We consider quantum dynamical systems specified by a unitary operator U and an initial state vector \phi. In each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state. We call the system recurrent if this eventually happens with probability one. We show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of U with respect to \phi. We also show that in the recurrent case the expected first return time is an integer or infinite, for which we give a topological interpretation. A key role in our theory is played by the first arrival amplitudes, which turn out to be the (complex conjugated) Taylor coefficients of the Schur function of the spectral measure. On the one hand, this provides a direct dynamical interpretation of these coefficients; on the other hand it links our definition of first return times to a large body of mathematical literature.Comment: 27 pages, 5 figures, typos correcte

Zeros of analytic functions, with or without multiplicities

The classical Mason-Stothers theorem deals with nontrivial polynomial solutions to the equation $a+b=c$. It provides a lower bound on the number of distinct zeros of the polynomial $abc$ in terms of the degrees of $a$, $b$ and $c$. We extend this to general analytic functions living on a reasonable bounded domain $\Omega\subset\mathbb C$, rather than on the whole of $\mathbb C$. The estimates obtained are sharp, for any $\Omega$, and a generalization of the original result on polynomials can be recovered from them by a limiting argument.Comment: This is a retitled and slightly revised version of my paper arXiv:1004.359