Propagation of singularities; wave front sets

1974 ◽  
pp. 120-138
Author(s):  
Michael Taylor
Keyword(s):  
Wave Front ◽  
Wave Front Sets ◽  
Propagation Of Singularities

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.

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 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.

scholarly journals Characterization of Wave Front Sets in Fourier-Lebesgue Spaces and Its Application

2013 ◽  
Vol 56 (1) ◽  
pp. 1-17
Author(s):  
Keiichi Kato ◽  
Masaharu Kobayashi ◽  
Shingo Ito
Keyword(s):  
Wave Front ◽  
Lebesgue Spaces ◽  
Wave Front Sets
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