<p>We introduce a family of smooth functions which are &quot;less regu-lar&quot; than the Gevrey functions, and study its basic properties. In particular&nbsp;we prove the standard results concerning algebra property and stability under finite order derivation. Moreover, we &nbsp;construct infnite order operators&nbsp;which leads us to the definition of class with ultradifferentiable property. We&nbsp;also prove that our classes are inverse-closed, and this result is the essential&nbsp;part in the proof of our main result presented in the final Chapter. Moreover,&nbsp;using the techniques of microlocal analysis, we introduce and investigate the<br />corresponding wave front sets, and the prove the results related to singular&nbsp;support of a distribution. Our main results shows how the singularities of&nbsp;solutions to partial differential equations (PDE&#39;s in short) propagate in the&nbsp;framework of our regularity.</p> / <p>U ovoj tezi defini&scaron;emo novu klasu glatkih funkcija i izučavamo njihove osnovne osobine. Pokazujemo da na&scaron;e klase imaju svojsto algebre kao i da su zatvorene u odnosu na delovanje operatora izvoda konačnog reda.Sta vi&scaron;e, konstrui&scaron;emo diferencijalne operatore beskonačnog reda i to nas dovodi do definicije ultradiferencijabilnih klasa funkcija. Takode dokazujemo osobinu zatvorenosti u odnosu na inverze, i taj rezultat je najvažniji deo u dokazu glavne teoreme koja je formulisana u poslednjoj glavi. Koristeći tehnike mikrolokalne analize, uvodimo i izučavamo odgovarajuće talasne frontove, i pokazujemo odgovarajuća tvrdjenja vezana za singularni nosač distribucije. Na&scaron; glavni rezultat pokazuje kako se prostiru singulariteti re&scaron;enja linearnih parcijalnih diferencijalnih jednačina u okviru na&scaron;e regularnosti.</p>

In the first part of this thesis the spectrum of a matrix operator is determined. For this the coefficients of the matrix operator are assumed to satisfy rather general properties which combine smoothness and decay. With this the asymptotics of the eigenfunctions can be determined. This in turn leads to properties of the spectra with the aid of the M-matrix. In the second part it will be shown that if a discrete canonical system has absolutely continuous spectrum of a certain multiplicity, then there is a corresponding number of linearly independent solutions y which are bounded in a weak sense.

[pt] Este trabalho apresenta uma abordagem inovadora para calcular autopares de operadores diferenciais (OD), utilizando programação genética (PG) e computação simbólica. Na literatura atual, o Método dos Elementos Finitos (MEF) e o Método das Diferenças Finitas (MDF) são os mais utilizados. Tais métodos usam discretização para converter o OD em uma matriz finita e, por isso, apresentam limitações como perda de acurácia e presença de soluções espúrias. Além disso, se o domínio do OD fosse alterado, os autopares precisariam ser calculados novamente, pois a representação matricial do operador depende dos parâmetros do problema. Nesse contexto, este trabalho propõe evoluir autofunções analiticamente usando PG, sem discretização do domínio. Com isso, é possível incorporar parâmetros, o que torna a solução obtida válida para uma classe de problemas. Este texto descreve o modelo para OD normais, aplicando conceitos de indivíduos multi-árvore e diferenciação simbólica. O modelo evolui auto-funções e, a partir delas, calcula os autovalores empregando a razão de Rayleigh. Experimentos baseados em aplicações da Física mostram que a técnica proposta é capaz de encontrar as autofunções analíticas com a acurácia igual ou melhor que as técnicas numéricas supracitadas. Finalmente, a técnica proposta é aplicada ao problema de propagação de ondas eletromagnéticas em poços de petróleo em ULF e UHF. As soluções analíticas são dadas em função do diâmetro e do percentual de incrustações no poço. Os resultados mostram que, para um conjunto suficientemente grande de valores distintos dos parâmetros, a técnica apresenta tempo de execução inferior às técnicas clássicas, mantendo a acurácia destas. / [en] This work presents an innovative approach to calculate the eigenpairs of linear differential operators (LDO), employing genetic programming (GP) and symbolic computation. In the current literature, the Finite Element Method (FEM) and the Finite Difference Method (FDM) are more commonly used. Such methods use discretization to convert the LDO to a finite matrix, therefore causing loss of accuracy and presence of spurious solutions. Additionally, if the domain of the LDO was changed, the eigenpairs would need to be recalculated, since the matrix representation of the LDO depends on the parameters of the problem. In this context, this work proposes to evolve eigenfunctions analytically using GP, without domain discretization. Hence, it is possible to incorporate the parameter, which makes a obtained solution valid for a class of problems. This text describes the model for normal LDO, applying concepts of multi-tree individuals and symbolic differentiation. The presented model evolves eigenfunctions and, then, calculates the eigenvalues using the Rayleigh quotient. Experiments based on Physics problems show that the proposed technique is able to find the analytical eigenfunctions with the same accuracy of the numerical techniques mentioned above. Finally, the proposed technique is applied to the problem of propagation of electromagnetic waves in oil wells in ULF and UHF. The analytical solutions are given as a function of the diameter and percentage of CaCO in the well. The results show that, for a sufficiently large set of distinct values of the parameters, the technique presents execution time inferior to the FEM, while maintaining its accuracy.

L'anneau $\Diff(n)$ des opérateurs différentiels $\h$-déformés apparaît dans la théorie des algèbres de réduction.Dans cette thèse, nous construisons les anneaux des opérateurs différentiels généralisés sur les espaces vectoriels $\h$-déformés de type $\gl$. Contrairement aux espaces vectoriels $q$-déformés pour lequel l'anneau des opérateurs différentiels est unique \`a isomorphisme pr\`es, l'anneau généralisé des opérateurs différentiels $\h$-déformés $\Diffs(n)$ est indexée par une fonction rationnelle $\sigma$ en $n$ variables, solution d'un syst\`eme d\'eg\'en\'er\'e d'\'equations aux diff\'erences finies. Nous obtenons la solution g\'en\'erale de ce syst\`eme. Nous montrons que le centre de $\Diffs(n)$ est un anneau des polynômes en $n$ variables. Nous construisons un isomorphisme entre des localisations de l'anneau $\Diffs(n)$ et de l’algèbre de Weyl $\text{W}_n$ l’étendue par $n$ indéterminés. Nous présentons des conditions irréductibilité des modules de dimension fini de $\Diffs(n)$. Finalement, nous discutons des difficultés a trouver les constructions analogues pour l'anneau $\Diff(n,N)$ correspondant \`a $N$ copies de $\Diff(n)$. / The ring $\Diff(n)$ of $\h$-deformed differential operators appears in the theory of reduction algebras. In this thesis, we construct the rings of generalized differential operators on the $\h$-deformed vector spaces of $\gl$-type. In contrast to the $q$-deformed vector spaces for which the ring of differential operators is unique up to an isomorphism, the general ring of $\h$-deformed differential operators $\Diffs(n)$ is labeled by a rational function $\sigma$ in $n$ variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system. We show that the center of $\Diffs(n)$ is a ring of polynomials in $n$ variables. We construct an isomorphism between certain localizations of $\Diffs(n)$ and the Weyl algebra $\W_n$ extended by $n$ indeterminates. We present some conditions for the irreducibility of the finite dimensional $\Diffs(n)$-modules. Finally, we discuss difficulties for finding analogous constructions for the ring $\Diff(n, N)$ formed by several copies of $\Diff(n)$.

Dans cette thèse, nous étudions les propriétés de concentration des fonctions propres du laplacien discret sur des graphes réguliers de degré fixé dont le nombre de sommets tend vers l'infini. Cette étude s'inspire de la théorie de l'ergodicité quantique sur les variétés. Par analogie avec cette dernière, nous développons un calcul pseudo-différentiel sur les arbres réguliers : nous définissons des classes de symboles et des opérateurs associés, et nous prouvons un certain nombre de propriétés de ces classes de symboles et opérateurs. Nous montrons notamment que les opérateurs sont bornés dans L², et nous donnons des formules de l'adjoint et du produit. Nous nous servons ensuite de cette théorie pour montrer un théorème d'ergodicité quantique pour des suites de graphes réguliers dont le nombre de sommets tend vers l'infini. Il s'agit d'un résultat de délocalisation de la plupart des fonctions propres dans la limite des grands graphes réguliers. Les graphes vérifient une hypothèse d'expansion et ne comportent pas trop de cycles courts, deux hypothèses vérifiées presque sûrement par des suites de graphes réguliers aléatoires. / N this thesis, we study concentration properties of eigenfunctions of the discrete Laplacian on regular graphs of fixed degree, when the number of vertices tend to infinity. This study is made in analogy with the Quantum Ergodicity theory on manifolds. We construct a pseudo-differential calculus on regular trees by defining symbol classes and associated operators and proving some properties of these classes of symbols and operators. In particular we prove that the operators are bounded on L² and give adjoint and product formulas. We then use this theory to prove a Quantum Ergodicity theorem on large regular graphs. This is a property of delocalization of most eigenfunctions in the large scale limit. We consider expander graphs with few short cycles (for instance random large regular graphs). These hypothesis are almost surely satisfied by sequences of random regular graphs.

Konforme Potenzen des Dirac Operators einer semi Riemannschen Spin-Mannigfaltigkeit werden untersucht. Wir präsentieren einen neuen Beweis, basierend auf dem Traktor Kalkül, für die Existenz von konformen ungeraden Potenzen des Dirac Operators auf semi Riemannschen Spin-Mannigfaltigkeiten. Desweiteren konstruieren wir eine neue Familie von konform kovarianten linearen Differentialoperatoren auf dem standard spin Traktor Bündel. Weiterhin verallgemeinern wir den Existenzbeweis für konforme ungerade Potenzen des Dirac Operators auf semi Riemannsche Spin-Mannigfaltigkeiten. Da die Existenzbeweise konstruktive sind, erhalten wir explizite Formeln für die konforme dritte und fünfte Potenz des Dirac Operators. Basierend auf den expliziten Formeln zeigen wir, dass die konforme dritte und fünfte Potenz des Dirac Operators formal selbstadjungiert (anti selbstadjungiert) bezüglich des L2-Skalarproduktes auf dem Spinorbündel ist. Abschliessend präsentieren wir neue Strukturen der konformen ersten, dritten und fünften Potenz des Dirac Operators: Es existieren lineare Differentialoperatoren auf dem Spinorbündel der Ordnung kleiner gleich eins, so dass die konforme erste, dritte und fünfte Potenz des Dirac Operators ein Polynom in jenen Operatoren ist. / Conformal powers of the Dirac operator on semi Riemannian spin manifolds are investigated. We give a new proof of the existence of conformal odd powers of the Dirac operator on semi Riemannian spin manifolds using the tractor machinery. We will also present a new family of conformally covariant linear differential operators on the standard spin tractor bundle. Furthermore, we generalize the known existence proof of conformal power of the Dirac operator on Riemannian spin manifolds to semi Riemannian spin manifolds. Both proofs concering the existence of conformal odd powers of the Dirac operator are constructive, hence we also derive an explicit formula for a conformal third- and fifth power of the Dirac operator. Due to explicit formulas, we show that the conformal third- and fifth power of the Dirac operator is formally self-adjoint (anti self-adjoint), with respect to the L2-scalar product on the spinor bundle. Finally, we present a new structure of the conformal first-, third- and fifth power of the Dirac operator: There exist linear differential operators on the spinor bundle of order less or equal one, such that the conformal first-, third- and fifth power of the Dirac operator is a polynomial in these operators.

Wir betrachten Optimalsteuerungsprobleme, die von partiellen Differentialgleichungen beziehungsweise Variationsungleichungen mit Differentialoperatoren erster Ordnung abhängen. Wir führen die Reformulierung eines Tagebauplanungsproblems, das auf stetigen Funktionen beruht, ein. Das Resultat ist ein Optimalsteuerungsproblem für Viskositätslösungen einer Eikonalgleichung. Die Existenz von Lösungen dieses und bestimmter Hilfsprobleme, die von semilinearen PDG‘s mit künstlicher Viskosität abhängen, wird bewiesen, Stationaritätsbedingungen hergeleitet und ein schwaches Konsistenzresultat für stationäre Punkte präsentiert. Des Weiteren betrachten wir Optimalsteuerungsprobleme, die von stationären Variationsungleichungen erster Art mit linearen Differentialoperatoren erster Ordnung abhängen. Wir diskutieren Lösbarkeit und Stationaritätskonzepte für diese Probleme. Für letzteres vergleichen wir Ergebnisse, die entweder durch die Anwendung von Penalisierungs- und Regularisierungsansätzen direkt auf Ebene von Differentialoperatoren erster Ordnung oder als Grenzwertprozess von Stationaritätssystemen für viskositätsregularisierte Optimalsteuerungsprobleme unter passenden Annahmen erhalten werden. Um die Konsistenz von ursprünglichem und regularisierten Problemen zu sichern, wird ein bekanntes Ergebnis für Lösungen von VU’s mit degeneriertem Differentialoperator erweitert. In beiden Fällen ist die erhaltene Stationarität schwächer als W-stationarität. Die theoretischen Ergebnisse werden anhand numerischer Beispiele verifiziert. Wir erweitern diese Ergebnisse auf Optimalsteuerungsprobleme bezüglich zeitabhängiger VU’s mit Differentialoperatoren erster Ordnung. Hierfür wird die Existenz von Lösungen bewiesen und erneut ein Stationaritätssystem mit Hilfe verschwindender Viskositäten unter bestimmten Beschränktheitsannahmen hergeleitet. Die erhaltenen Ergebnisse werden anhand von numerischen Beispielen verifiziert. / We consider problems of optimal control subject to partial differential equations and variational inequality problems with first order differential operators. We introduce a reformulation of an open pit mine planning problem that is based on continuous functions. The resulting formulation is a problem of optimal control subject to viscosity solutions of a partial differential equation of Eikonal Type. The existence of solutions to this problem and auxiliary problems of optimal control subject to regularized, semilinear PDE’s with artificial viscosity is proven. For the latter a first order optimality condition is established and a mild consistency result for the stationary points is proven. Further we study certain problems of optimal control subject to time-independent variational inequalities of the first kind with linear first order differential operators. We discuss solvability and stationarity concepts for such problems. In the latter case, we compare the results obtained by either utilizing penalization-regularization strategies directly on the first order level or considering the limit of systems for viscosity-regularized problems under suitable assumptions. To guarantee the consistency of the original and viscosity-regularized problems of optimal control, we extend known results for solutions to variational inequalities with degenerated differential operators. In both cases, the resulting stationarity concepts are weaker than W-stationarity. We validate the theoretical findings by numerical experiments for several examples. Finally, we extend the results from the time-independent to the case of problems of optimal control subject to VI’s with linear first order differential operators that are time-dependent. After establishing the existence of solutions to the problem of optimal control, a stationarity system is derived by a vanishing viscosity approach under certain boundedness assumptions and the theoretical findings are validated by numerical experiments.

In dieser Dissertation untersuchen wir das Verhalten von Schrödinger-Operatoren unter Riemannschen Überlagerungen. Wir betrachten folgende Situation: Sei eine Riemannsche Überlagerung und ein Schrödinger-Operator S mit glattem, von unten beschränktem Potential auf der Basismannigfaltigkeit gegeben. Sei S‘ der Lift von S auf die Überlagerungsmannigfaltigkeit. Man sieht leicht, dass das Minimum des Spektrums von S nicht größer als das Minimum des Spektrums von S‘ ist. R. Brooks hat als erster untersucht, wann die Gleichheit gilt. Er bewies insbesondere, dass eine normale Riemannsche Überlagerung einer geschlossenen Mannigfaltigkeit genau dann amenabel ist, wenn sie das Minimum des Spektrums des Laplace-Operators unverändert lässt. Zusammen mit W. Ballmann und H. Matthiesen bewiesen wir, dass amenable Riemannsche Überlagerungen immer das Minimum des Spektrums von Schrödinger-Operatoren erhalten; dies verallgemeinert Resultate von R. Brooks sowie von P. Bérard und Ph. Castillon. In dieser Dissertation beweisen wir, dass im Fall vollständiger Mannigfaltigkeiten das Spektrum von S im Spektrum von S‘ enthalten ist. Tatsächlich beweisen wir diese Beziehung sogar für eine deutlich größere Klasse von Differentialoperatoren. Obwohl Amenabilität eine natürliche Bedingung für die Gleichheit der Minima der Spektren ist, ist es unklar, inwieweit diese Bedingung optimal ist. In dieser Dissertation beweisen wir: Wenn eine Riemannsche Überlagerung das Minimum des Spektrums eines Schrödinger-Operators erhält, und wenn dieses zum diskreten Spektrum des Operators auf der Basismannigfaltigkeit gehört, dann ist die Überlagerung amenabel. Man beachte, dass wir keinerlei geometrische oder topologische Bedingungen an die Mannigfaltigkeiten stellen. Dies verallgemeinert sowohl frühere Resultate von R. Brooks, T. Roblin und S. Tapie als auch ein kürzliches Resultat aus einer gemeinsamen Arbeit mit W. Ballmann und H. Matthiesen. / In this thesis, we investigate the behavior of the spectrum of Schrödinger operators under Riemannian coverings. To set the stage, consider a Riemannian covering and a Schrödinger operator S on the base manifold, with smooth potential bounded from below potential. Let S’ be the lift of S on the covering space. It is easy to see that the bottom (that is, the minimum) of the spectrum of S is no greater than the bottom of the spectrum of S’. R. Brooks was the first one to examine when the equality holds. In particular, he proved that a normal Riemannian covering of a closed manifold is amenable if and only if it preserves the bottom of the spectrum of the Laplacian. Generalizing former results of R. Brooks, and P. Berard and Ph. Castillon, in a joint work with W. Ballmann and H. Matthiesen, we proved that amenable Riemannian coverings preserve the bottom of the spectrum of Schrödinger operators. In this thesis, we prove that if, in addition, the manifolds are complete, then the spectrum of S is contained in the spectrum of S’. As a matter of fact, we establish this result for a quite wide class of differential operators. Although amenability is a natural assumption for the preservation of the bottom of the spectrum, it is not clear to what extent it is optimal. In this thesis, we prove that if a Riemannian covering preserves the bottom of the spectrum of a Schrödinger operator, which belongs to the discrete spectrum of the operator on the base manifold, then the covering is amenable. It is worth to point out that we do not impose any geometric or topological assumptions on the manifolds. This generalizes former results by R. Brooks, T. Roblin and S. Tapie, and a recent result of a joint work with W. Ballmann and H. Matthiesen.

<p>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.</p><p>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.</p><p>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.</p><p>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).</p><p>In this thesis we consider wave-front sets with respect to Fourier Banach function spaces. Roughly speaking, we take <em>B</em> 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 <em>FB</em> 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.</p> / <p>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.</p><p>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.</p><p>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.</p><p>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).</p><p>I denna avhandling betraktar vi vågfrontsmängder med avseende på Fourier Banach funktionsrum. Detta kan ses som att vi låter <em>B</em> 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 <em>FB</em> 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.</p>

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. / 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.