lnu.sePublications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Local wave-front sets of Banach and Fréchet types, with applications to pseudo-differential operators
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
University of Torino, Department of Mathematics.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Let ω, ω0 be appropriate weight functions and B be an invariant BF-space. We introduce the wave-front set WFFB(ω)(f) with respect to weighted Fourier Banach space FB(ω). We prove the usual mapping properties for pseudo-differential operators Opt(a) with symbols a inS^{ω0}_{ρ,0} hold for such wave-front sets.

Keywords [en]
Wave-front, Fourier, Banach, modulation, micro-local
National Category
Mathematical Analysis
Research subject
Natural Science, Mathematics
Identifiers
URN: urn:nbn:se:lnu:diva-2437OAI: oai:DiVA.org:lnu-2437DiVA, id: diva2:310410
Note
Pre-print in ArXiv:0911.1867Available from: 2010-04-14 Created: 2010-04-13 Last updated: 2012-01-03Bibliographically approved
In thesis
1. Propagation of singularities for pseudo-differential operators and generalized Schrödinger propagators
Open this publication in new window or tab >>Propagation of singularities for pseudo-differential operators and generalized Schrödinger propagators
2010 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we discuss different types of regularity for distributions which appear in the theory of pseudo-differential operators and partial differential equations. Partial differential equations often appear in science and technology. For example the Schrödinger equation can be used to describe the change in time of quantum states of physical systems. Pseudo-differential operators can be used to solve partial differential equations.  They are also appropriate to use when modeling different types of problems within physics and engineering. For example, there is a natural connection between pseudo-differential operators and stationary and non-stationary filters in signal processing. Furthermore, the correspondence between symbols and operators when passing from classical mechanics to quantum mechanics essentially agrees with symbols and operators in the Weyl calculus of pseudo-differential operators.

In this thesis we concentrate on investigating how regularity properties for solutions of partial differential equations are affected under the mapping of pseudo-differential operators, and in particular of the free time-dependent Schrödinger operators.

The solution of the free time-dependent Schrödinger equation can be expressed as a pseudo-differential operator, with non-smooth symbol, acting on the initial condition. We generalize a result about non-tangential convergence, which was obtained by Sjögren and Sjölin (1989) for the free time-dependent Schrödinger equation.

Another way to describe regularity for a distribution is to use wave-front sets. They do not only describe where the singularities are, but also the directions in which these singularities appear. The first types of wave-front sets (analytical wave-front sets) were introduced by Sato (1969, 1970). Later on Hörmander introduced ``classical'' wave-front sets (with respect to smoothness) and showed results in the context of pseudo-differential operators with smooth symbols, cf. Hörmander (1985).

In this thesis we consider wave-front sets with respect to Fourier Banach function spaces. Roughly speaking, we take B as a Banach space, which is invariant under translations and embedded between the space of Schwartz functions and the space of temperated distributions. Then we say that the wave-front set of a distribution contains all points (x0, ξ0) such that no localization of the distribution at x0, belongs to FB in the direction ξ0. We prove that pseudo-differential operators with smooth symbols shrink the wave-front set and we obtain opposite embeddings by using sets of characteristic points of the operator symbols.

Abstract [sv]

I denna avhandling diskuterar vi olika typer av regularitet för distributioner som uppkommer i teorin för pseudodifferentialoperatorer och partiella differentialekvationer. Partiella differentialekvationer förekommer inom naturvetenskap och teknik. Exempelvis kan Schrödingerekvationen användas för att beskriva förändringen med tiden av kvanttillstånd i fysikaliska system. Pseudodifferentialoperatorer kan användas för att lösa partiella differential\-ekvationer. De användas också för att modellera olika typer av problem inom fysik och teknik. Det finns till exempel en naturlig koppling mellan pseudodifferentialoperatorer och stationära och icke-stationära filter i signalbehandling. Vidare gäller att relationen mellan symboler och operatorer vid övergången från klassisk mekanik till kvantmekanik i huvudsak överensstämmer med symboler och operatorer inom Weylkalkylen för pseudodifferentialoperatorer.

I den här avhandlingen koncentrerar vi oss på att undersöka hur regularitetsegenskaper för lösningar till partiella differentialekvationer påverkas under verkan av pseudodifferentialoperatorer, och speciellt för de fria tidsberoende Schrödingeroperatorerna.

Lösningen av den fria tidsberoende Schrödingerekvationen kan uttryckas som en pseudodifferentialoperator, med icke-slät symbol, verkande på begynnelsevillkoret. Vi generaliserar ett resultat om icke-tangentiell konvergens av Sjögren och Sjölin (1989) för den fria tidsberoende Schrödingerekvationen.

Ett annat sätt att beskriva regularitet hos en distribution är med hjälp av vågfrontsmängder. De beskriver inte bara var singulariteterna finns, utan också i vilka riktningar dessa singulariteter förekommer. De första typerna av vågfrontsmängder (analytiska vågfrontsmängder) introducerades av Sato (1969, 1970). Senare introducerade Hörmander ''klassiska'' vågfrontsmängder (med avseende på släthet) och visade resultat för verkan av pseudodifferentialoperatorer med släta symboler, se  Hörmander (1985).

I denna avhandling betraktar vi vågfrontsmängder med avseende på Fourier Banach funktionsrum. Detta kan ses som att vi låter B vara ett Banachrum, som är invariant under translationer och är inbäddat mellan rummet av Schwartzfunktioner och rummet av tempererade distributioner. Vågfrontsmängden av en distribution innehåller alla punkter (x0, ξ0) så att ingen lokalisering av distributionen kring x0, tillhör FB i riktningen ξ0. Vi visar att pseudodifferentialoperatorer med släta symboler krymper vågfrontsmängden och vi får motsatta inbäddningar med hjälp mängder av karakteristiska punkter till operatorernas symboler.

Publisher
p. 70
Keywords
Banach spaces, Fourier, Generalized time-dependent Schrödinger equation, Micro-local, Modulation spaces, Non-tangential convergence, Pseudo-differential operators, Regularity, Wave-front sets, Banachrum, Fourier, Generaliserad tidsberoende Schrödinger ekvation, Icke-tangentiell konvergens, Mikrolokal, Modulationsrum, Pseudodifferentialoperatorer, Regularitet, Vågfrontsmängder
National Category
Mathematical Analysis
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-2447 (URN)
Presentation
2010-03-26, Weber, Universitetsplatsen 1, Växjö, 13:15 (Swedish)
Opponent
Supervisors
Available from: 2010-04-20 Created: 2010-04-14 Last updated: 2010-08-23Bibliographically approved
2. Properties of wave-front sets and non-tangential convergence
Open this publication in new window or tab >>Properties of wave-front sets and non-tangential convergence
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we consider regularity properties for solutions to partial differential equations and pseudo-differential equations. The thesis mainly concerns wave-front sets and micro-local properties. Regularity properties are also viewed in terms of nontangential convergence for the generalized free time-dependent Schrödinger equations, where the Laplace operator is replaced by more general functions.

Wave-front sets describe location of singularities and the directions of their propagation. We establish usual and convenient mapping properties for such wave-front sets under action of pseudodifferential operators with smooth symbols.

We define three components of wave-front sets with respect to appropriate Banach and Fréchet spaces, in order to describe local properties as well as behavior far away, including heavy oscillations. The union of these components is called the global wavefront set. For these wave-front sets, we establish micro-local and micro-ellipticity properties for pseudo-differential operators in appropriate symbol classes. We obtain the classical wave-front sets as special cases (cf. Hörmander [9]). For the type of wave-front sets which describe local properties we also prove equivalence between wave-front sets of Fourier Banach function and modulation space types.

To open up for numerical computations we introduce admissible lattices and Gabor pairs to define discrete versions of wave-front sets with respect to Fourier Lebesgue and modulation spaces. Furthermore, we prove that these wave-front sets agree with each other and with the corresponding wave-front sets of continuous type. We also consider the link between analytic functions and temperate distributions in terms of such wave-front sets.

The last part of this thesis concerns counter examples of nontangential convergence for the generalized time-dependent Schrödinger equation with initial data in Sobolev spaces.

Place, publisher, year, edition, pages
Växjö, Kalmar: Linnaeus University Press, 2011. p. 161
Series
Linnaeus University Dissertations ; 58/2011
Keywords
Fourier Banach spaces, Generalized free time-dependent Schrödinger equation, Micro-local, Modulation spaces, Non-tangential convergence, Pseudo-differential operators, Regularity, Wave-front sets.
National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-12963 (URN)978-91-86491-96-3 (ISBN)
Public defence
2011-10-20, Weber, Universitetsplatsen 1, Växjö, 13:15 (English)
Opponent
Supervisors
Available from: 2011-08-09 Created: 2011-06-23 Last updated: 2017-01-11Bibliographically approved

Open Access in DiVA

No full text in DiVA

Authority records BETA

Johansson, Karoline

Search in DiVA

By author/editor
Johansson, Karoline
By organisation
School of Computer Science, Physics and Mathematics
Mathematical Analysis

Search outside of DiVA

GoogleGoogle Scholar

urn-nbn

Altmetric score

urn-nbn
Total: 97 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf