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A necessary condition of local solvability for pseudo-differential equations with double characteristics

Fernando Cardoso, François Trèves (1974)

Annales de l'institut Fourier

Pseudodifferential operators P ( x , D ) ∼ ∑ j = 0 + ∞ P m - j ( x , D ) are studied, from the viewpoint of local solvability and under the assumption that, micro-locally, the principal symbol factorizes as P m = Q L 2 with Q elliptic, homogeneous of degree m - 2 , and L homogeneous of degree one, satisfying the following condition : there is a point ( x 0 , ξ 0 ) in the characteristic variety L = 0 and a complex number z such that d ξ Re ( z L ) ≠ 0 at ( x 0 , ξ 0 ) and such that the restriction of Im ( z L ) to the bicharacteristic strip of Re ( z L ) vanishes of order k < + ∞ at ( x 0 , ξ 0 ) , changing sign there from minus to...

A parametrix construction for wave equations with C 1 , 1 coefficients

Hart F. Smith (1998)

Annales de l'institut Fourier

In this article we give a construction of the wave group for variable coefficient, time dependent wave equations, under the hypothesis that the coefficients of the principal term possess two bounded derivatives in the spatial variables, and one bounded derivative in the time variable. We use this construction to establish the Strichartz and Pecher estimates for solutions to the Cauchy problem for such equations, in space dimensions n = 2 and n = 3 .

An accuracy improvement in Egorov's theorem.

Jorge Drumond Silva (2007)

Publicacions Matemàtiques

We prove that the theorem of Egorov, on the canonical transformation of symbols of pseudodifferential operators conjugated by Fourier integral operators, can be sharpened. The main result is that the statement of Egorov's theorem remains true if, instead of just considering the principal symbols in Sm/Sm-1 for the pseudodifferential operators, one uses refined principal symbols in Sm/Sm-2, which for classical operators correspond simply to the principal plus the subprincipal symbol, and can generally...

An analysis of quantum Fokker-Planck models: a Wigner function approach.

Anton Arnold, José L. López, Peter A. Markowich, Juan Soler (2004)

Revista Matemática Iberoamericana

The analysis of dissipative transport equations within the framework of open quantum systems with Fokker-Planck-type scattering is carried out from the perspective of a Wigner function approach. In particular, the well-posedness of the self-consistent whole-space problem in 3D is analyzed: existence of solutions, uniqueness and asymptotic behavior in time, where we adopt the viewpoint of mild solutions in this paper. Also, the admissibility of a density matrix formulation in Lindblad form with Fokker-Planck...

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