Stationary phase and microlocal analysis

2017 ◽  
Author(s):  
Christopher D. Sogge
Keyword(s):  
Stationary Phase ◽  
Wave Front ◽  
Pseudodifferential Operators ◽  
Microlocal Analysis ◽  
Equivalent Definition ◽  
Method Of Stationary Phase ◽  
Wave Front Sets ◽  
Propagation Of Singularities ◽  
Definition Of

This chapter discusses basic techniques from the theory of stationary phase. After giving an overview of the method of stationary phase, the chapter moves on to a discussion of pseudodifferential operators, by going over the basics from the calculus of pseudodifferential operators and their various microlocal properties, in the process obtaining an equivalent definition of wave front sets, before defining pseudodifferential operators on manifolds and going over some of their properties. The chapter then lays out the propagation of singularities as well as Egorov's theorem, which involves conjugating pseudodifferential operators. Finally, this chapter describes the Friedrichs quantization, and differentiates it from the Kohn-Nirenberg quantization presented earlier in the chapter.

scholarly journals Beyond gevrey regularity: Superposition and propagation of singularities

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2763-2782 ◽  
Author(s):  
Stevan Pilipovic ◽  
Nenad Teofanov ◽  
Filip Tomic
Keyword(s):  
Wave Front ◽  
Regularity Condition ◽  
Differential Operators ◽  
Ultradifferentiable Functions ◽  
Gevrey Regularity ◽  
Gevrey Classes ◽  
Partial Differential Operators ◽  
Wave Front Sets ◽  
Propagation Of Singularities ◽  
Two Parameters

We propose the relaxation of Gevrey regularity condition by using sequences which depend on two parameters, and define spaces of ultradifferentiable functions which contain Gevrey classes. It is shown that such a space is closed under superposition, and therefore inverse closed as well. Furthermore, we study partial differential operators whose coefficients are less regular then Gevrey-type ultradifferentiable functions. To that aim we introduce appropriate wave front sets and prove a theorem on propagation of singularities. This extends related known results in the sense that assumptions on the regularity of the coefficients are weakened.

scholarly journals On duality theory and pseudodifferential techniques for Colombeau algebras: generalized delta functionals, kernels and wave front sets

2006 ◽  
Vol 133 (31) ◽  
pp. 115-136 ◽  
Author(s):  
Claudia Garetto ◽  
G. Hormann
Keyword(s):  
Wave Front ◽  
Pseudodifferential Operators ◽  
Symbol Calculus ◽  
Wave Front Set ◽  
Colombeau Algebras ◽  
Generalized Complex Numbers ◽  
Topological Modules ◽  
Wave Front Sets ◽  
Basic Facts

Summarizing basic facts from abstract topological modules over Colombeau generalized complex numbers we discuss duality of Colombeau algebras. In particular, we focus on generalized delta functional and operator kernels as elements of dual spaces. A large class of examples is provided by pseudodifferential operators acting on Colombeau algebras. By a refinement of symbol calculus we review a new characterization of the wave front set for generalized functions with applications to microlocal analysis. AMS Mathematics Subject Classification (2000): 46F30, 46A20, 47G30.

scholarly journals MICROLOCAL ANALYSIS OF GENERALIZED FUNCTIONS: PSEUDODIFFERENTIAL TECHNIQUES AND PROPAGATION OF SINGULARITIES

2005 ◽  
Vol 48 (3) ◽  
pp. 603-629 ◽  
Author(s):  
Claudia Garetto ◽  
Günther Hörmann
Keyword(s):  
Pseudodifferential Operators ◽  
Hyperbolic Equations ◽  
Regularity Theory ◽  
Regularity Result ◽  
Generalized Functions ◽  
Generalized Solutions ◽  
Original Definition ◽  
Characteristic Sets ◽  
Propagation Of Singularities ◽  
Definition Of

AbstractWe characterize microlocal regularity, in the $\mathcal{G}^{\infty}$-sense, of Colombeau generalized functions by an appropriate extension of the classical notion of micro-ellipticity to pseudodifferential operators with slow-scale generalized symbols. Thus we obtain an alternative, yet equivalent, way of determining generalized wavefront sets that is analogous to the original definition of the wavefront set of distributions via intersections over characteristic sets. The new methods are then applied to regularity theory of generalized solutions of (pseudo)differential equations, where we extend the general non-characteristic regularity result for distributional solutions and consider propagation of $\mathcal{G}^{\infty}$-singularities for homogeneous first-order hyperbolic equations.

scholarly journals Propagation of the Gabor wave front set for Schrödinger equations with non-smooth potentials

2015 ◽  
Vol 27 (01) ◽  
pp. 1550001 ◽  
Author(s):  
Elena Cordero ◽  
Fabio Nicola ◽  
Luigi Rodino
Keyword(s):  
Wave Front ◽  
Modulation Spaces ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Wave Front Set ◽  
Local Propagation ◽  
The Cauchy Problem ◽  
Wave Front Sets ◽  
Propagation Of Singularities ◽  
Gabor Wave Front Set

We consider Schrödinger equations with real-valued smooth Hamiltonians, and non-smooth bounded pseudo-differential potentials, whose symbols may not even be differentiable. The well-posedness of the Cauchy problem is proved in the frame of the modulation spaces, and results of micro-local propagation of singularities are given in terms of Gabor wave front sets.

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