Spectrally isometric elementary operators

We present criteria for unital elementary operators (of small length) on unital semisimple Banach algebras to be spectral isometries. The surjective ones among them turn out to be algebra automorphisms.

Elementary operators-still not elementary?

Properties of elementary operators, that is, finite sums of two-sided multiplications on a Banach algebra, have been studied under a vast variety of aspects by numerous authors. In this paper we review recent advances in a new direction that seems not to have been explored before: the question when an elementary operator is spectrally bounded or spectrally isometric. As with other investigations, a number of subtleties occur which show that elementary operators are still not elementary to handle.

Generalized quasi mode theory of macroscopic canonical quantization in cavity quantum electrodynamics and quantum optics II. Application to reflection and refraction at a dielectric interface

The generalization of the quasi mode theory of macroscopic quantization in quantum optics and cavity QED presented in the previous paper, is applied to provide a fully quantum theoretic derivation of the laws of reflection and refraction at a boundary. The quasi mode picture of this process involves the annihilation of a photon travelling in the incident region quasi mode, and the subsequent creation of a photon in either the incident region or transmitted region quasi modes. The derivation of the laws of reflection and refraction is achieved through the dual application of the quasi mode theory and a quantum scattering theory based on the Heisenberg picture. Formal expressions from scattering theory are given for the reflection and transmission coefficients. The behaviour of the intensity for a localized one photon wave packet coming in at time minus infinity from the incident direction is examined and it is shown that at time plus infinity, the light intensity is only significant where the classical laws of reflection and refraction predict. The occurrence of both refraction and reflection is dependent upon the quasi mode theory coupling constants between incident and transmitted region quasi modes being nonzero, and it is seen that the contributions to such coupling constants come from the overlap of the mode functions in the boundary layer region, as might be expected from a microscopic theory.

Theory of pseudomodes in quantum optical processes

This paper deals with non-Markovian behavior in atomic systems coupled to a structured reservoir of quantum electromagnetic field modes, with particular relevance to atoms interacting with the field in high-Q cavities or photonic band-gap materials. In cases such as the former, we show that the pseudomode theory for single-quantum reservoir excitations can be obtained by applying the Fano diagonalization method to a system in which the atomic transitions are coupled to a discrete set of (cavity) quasimodes, which in turn are coupled to a continuum set of (external) quasimodes with slowly varying coupling constants and continuum mode density. Each pseudomode can be identified with a discrete quasimode, which gives structure to the actual reservoir of true modes via the expressions for the equivalent atom-true mode coupling constants. The quasimode theory enables cases of multiple excitation of the reservoir to now be treated via Markovian master equations for the atom-discrete quasimode system. Applications of the theory to one, two, and many discrete quasimodes are made. For a simple photonic band-gap model, where the reservoir structure is associated with the true mode density rather than the coupling constants, the single quantum excitation case appears to be equivalent to a case with two discrete quasimodes.

Critical quantum fluctuations in the degenerate parametric oscillator

We develop a systematic theory of critical quantum fluctuations in the driven parametric oscillator. Our analytic results agree well with stochastic numerical simulations. We also compare the results obtained in the positive-P representation, as a fully quantum-mechanical calculation, with the truncated Wigner phase-space equation, also known as the semiclassical theory. We show when these results agree and differ in calculations taken beyond the linearized approximation. We find that the optimal broadband noise reduction occurs just above threshold. In this region where there are large quantum fluctuations in the conjugate variance and macroscopic quantum superposition states might be expected, we find that the quantum predictions correspond very closely to the semiclassical theory.

On a size-structured two-phase population model with infinite states-at-birth

In this work we introduce and analyze a linear size-structured population model with infinite states-at-birth. We model the dynamics of a population in which individuals have two distinct life-stages: an “active” phase when individuals grow, reproduce and die and a second “resting” phase when individuals only grow. Transition between these two phases depends on individuals’ size. First we show that the problem is governed by a positive quasicontractive semigroup on the biologically relevant state space. Then we investigate, in the framework of the spectral theory of linear operators, the asymptotic behavior of solutions of the model. We prove that the associated semigroup has, under biologically plausible assumptions, the property of asynchronous exponential growth.

Large time behaviour for functional differential equations with dominant eigenvalues of arbitrary order

The spectral theory for linear autonomous neutral functional differential equations (FDE) yields explicit formulas for the large time behaviour of solutions. Our results are based on resolvent computations and Dunford calculus, applied to establish explicit formulas for the large time behaviour of solutions of FDE. We investigate in detail a class of two-dimensional systems of FDE. (C) 2009 Elsevier Inc. All rights reserved.

ON DISCRETE SYSMMETRIES OF THE MULTI-BOSON KP HIERARCHIES

We show that the multi-boson KP hierarchies possess a class of discrete symmetries linking them to discrete Toda systems. These discrete symmetries are generated by the similarity transformation of the corresponding Lax operator. This establishes a canonical nature of the discrete transformations. The spectral equation, which defines both the lattice system and the corresponding Lax operator, plays a key role in determining pertinent symmetry structure. We also introduce the concept of the square root lattice leading to a family of new pseudo-differential operators with covariance under additional Backlund transformations.

On discrete symmetries of the multi-boson KP hierarchies

We show that the multi-boson KP hierarchies possess a class of discrete symmetries linking them to discrete Toda systems. These discrete symmetries are generated by the similarity transformation of the corresponding Lax operator. This establishes a canonical nature of the discrete transformations. The spectral equation, which defines both the lattice system and the corresponding Lax operator, plays a key role in determining pertinent symmetry structure. We also introduce the concept of the square root lattice leading to a family of new pseudo-differential operators with covariance under additional Bäcklund transformations.

Topics in geometry, analysis and inverse problems

The thesis consists of three independent parts. Part I: Polynomial amoebas We study the amoeba of a polynomial, as de ned by Gelfand, Kapranov and Zelevinsky. A central role in the treatment is played by a certain convex function which is linear in each complement component of the amoeba, which we call the Ronkin function. This function is used in two di erent ways. First, we use it to construct a polyhedral complex, which we call a spine, approximating the amoeba. Second, the Monge-Ampere measure of the Ronkin function has interesting properties which we explore. This measure can be used to derive an upper bound on the area of an amoeba in two dimensions. We also obtain results on the number of complement components of an amoeba, and consider possible extensions of the theory to varieties of codimension higher than 1. Part II: Differential equations in the complex plane We consider polynomials in one complex variable arising as eigenfunctions of certain differential operators, and obtain results on the distribution of their zeros. We show that in the limit when the degree of the polynomial approaches innity, its zeros are distributed according to a certain probability measure. This measure has its support on the union of nitely many curve segments, and can be characterized by a simple condition on its Cauchy transform. Part III: Radon transforms and tomography This part is concerned with different weighted Radon transforms in two dimensions, in particular the problem of inverting such transforms. We obtain stability results of this inverse problem for rather general classes of weights, including weights of attenuation type with data acquisition limited to a 180 degrees range of angles. We also derive an inversion formula for the exponential Radon transform, with the same restriction on the angle.

Spectral and variational characterizations of solutions to semilinear eigenvalue problems

Die vorliegende Arbeit befaßt sich mit einer Klasse von nichtlinearen Eigenwertproblemen mit Variationsstrukturin einem reellen Hilbertraum. Die betrachteteEigenwertgleichung ergibt sich demnach als Euler-Lagrange-Gleichung eines stetig differenzierbarenFunktionals, zusätzlich sei der nichtlineare Anteil desProblems als ungerade und definit vorausgesetzt.Die wichtigsten Ergebnisse in diesem abstrakten Rahmen sindKriterien für die Existenz spektral charakterisierterLösungen, d.h. von Lösungen, deren Eigenwert gerade miteinem vorgegeben variationellen Eigenwert eines zugehörigen linearen Problems übereinstimmt. Die Herleitung dieserKriterien basiert auf einer Untersuchung kontinuierlicher Familien selbstadjungierterEigenwertprobleme und erfordert Verallgemeinerungenspektraltheoretischer Konzepte.Neben reinen Existenzsätzen werden auch Beziehungen zwischenspektralen Charakterisierungen und denLjusternik-Schnirelman-Niveaus des Funktionals erörtert.Wir betrachten Anwendungen auf semilineareDifferentialgleichungen (sowieIntegro-Differentialgleichungen) zweiter Ordnung. Diesliefert neue Informationen über die zugehörigenLösungsmengen im Hinblick auf Knoteneigenschaften. Diehergeleiteten Methoden eignen sich besonders für eindimensionale und radialsymmetrische Probleme, während einTeil der Resultate auch ohne Symmetrieforderungen gültigist.

Quadraturformeln für das Qualokationsverfahren

Im Mittelpunkt dieser Arbeit steht Beweis der Existenz- und Eindeutigkeit von Quadraturformeln, die für das Qualokationsverfahren geeignet sind. Letzteres ist ein von Sloan, Wendland und Chandler entwickeltes Verfahren zur numerischen Behandlung von Randintegralgleichungen auf glatten Kurven (allgemeiner: periodische Pseudodifferentialgleichungen). Es erreicht die gleichen Konvergenzordnungen wie das Petrov-Galerkin-Verfahren, wenn man durch den Operator bestimmte Quadraturformeln verwendet. Zunächst werden die hier behandelten Pseudodifferentialoperatoren und das Qualokationsverfahren vorgestellt. Anschließend wird eine Theorie zur Existenz und Eindeutigkeit von Quadraturformeln entwickelt. Ein wesentliches Hilfsmittel hierzu ist die hier bewiesene Verallgemeinerung eines Satzes von Nürnberger über die Existenz und Eindeutigkeit von Quadraturformeln mit positiven Gewichten, die exakt für Tschebyscheff-Räume sind. Es wird schließlich gezeigt, dass es stets eindeutig bestimmte Quadraturformeln gibt, welche die in den Arbeiten von Sloan und Wendland formulierten Bedingungen erfüllen. Desweiteren werden 2-Punkt-Quadraturformeln für so genannte einfache Operatoren bestimmt, mit welchen das Qualokationsverfahren mit einem Testraum von stückweise konstanten Funktionen eine höhere Konvergenzordnung hat. Außerdem wird gezeigt, dass es für nicht-einfache Operatoren im Allgemeinen keine Quadraturformel gibt, mit der die Konvergenzordnung höher als beim Petrov-Galerkin-Verfahren ist. Das letzte Kapitel beinhaltet schließlich numerische Tests mit Operatoren mit konstanten und variablen Koeffizienten, welche die theoretischen Ergebnisse der vorangehenden Kapitel bestätigen.

The Feynman-Kac formula and applications to PDEs

The thesis presents a probabilistic approach to the theory of semigroups of operators, with particular attention to the Markov and Feller semigroups. The first goal of this work is the proof of the fundamental Feynman-Kac formula, which gives the solution of certain parabolic Cauchy problems, in terms of the expected value of the initial condition computed at the associated stochastic diffusion processes. The second target is the characterization of the principal eigenvalue of the generator of a semigroup with Markov transition probability function and of second order elliptic operators with real coefficients not necessarily self-adjoint. The thesis is divided into three chapters. In the first chapter we study the Brownian motion and some of its main properties, the stochastic processes, the stochastic integral and the Itô formula in order to finally arrive, in the last section, at the proof of the Feynman-Kac formula. The second chapter is devoted to the probabilistic approach to the semigroups theory and it is here that we introduce Markov and Feller semigroups. Special emphasis is given to the Feller semigroup associated with the Brownian motion. The third and last chapter is divided into two sections. In the first one we present the abstract characterization of the principal eigenvalue of the infinitesimal generator of a semigroup of operators acting on continuous functions over a compact metric space. In the second section this approach is used to study the principal eigenvalue of elliptic partial differential operators with real coefficients. At the end, in the appendix, we gather some of the technical results used in the thesis in more details. Appendix A is devoted to the Sion minimax theorem, while in appendix B we prove the Chernoff product formula for not necessarily self-adjoint operators.

Applications of elliptic functions to solve differential equations

In questa tesi si mostrano alcune applicazioni degli integrali ellittici nella meccanica Hamiltoniana, allo scopo di risolvere i sistemi integrabili. Vengono descritte le funzioni ellittiche, in particolare la funzione ellittica di Weierstrass, ed elenchiamo i tipi di integrali ellittici costruendoli dalle funzioni di Weierstrass. Dopo aver considerato le basi della meccanica Hamiltoniana ed il teorema di Arnold Liouville, studiamo un esempio preso dal libro di Moser-Integrable Hamiltonian Systems and Spectral Theory, dove si prendono in considerazione i sistemi integrabili lungo la geodetica di un'ellissoide, e il sistema di Von Neumann. In particolare vediamo che nel caso n=2 abbiamo un integrale ellittico.

Microscopic theory for quantum mirages in quantum corrals

Scanning tunneling microscopy permits us to image the Kondo resonance of a single magnetic atom adsorbed on a metallic surface. When the magnetic impurity is placed at the focus of an elliptical quantum corral, a Kondo resonance has been recently observed both on top of the impurity and on top of the focus where no magnetic impurity is present. This projection of the Kondo resonance to a remote point on the surface is referred to as quantum mirage. We present a quantum mechanical theory for the quantum mirage inside an ideal quantum corral and predict that the mirage will occur in corrals with shapes other than elliptical.