Singular Degree of a Rational Matrix Pseudodifferential Operator

The Perturbed Maxwell Operator as Pseudodifferential Operator

As a first step to deriving effective dynamics and ray optics, we prove that the perturbed periodic Maxwell operator in d = 3 can be seen as a pseudodifferential operator. This necessitates a better understanding of the periodic Maxwell operator M0. In particular, we characterize the behavior of M0 and the physical initial states at small crystal momenta k and small frequencies. Among other thi...

The pseudodifferential operator A(x, D)

The pseudodifferential operator (p.d.o.)A(x,D), associated with the Bessel operator d2/dx2 + (1− 4μ2)/4x2, is defined. Symbol class Hm ρ,δ is introduced. It is shown that the p.d.o. associated with a symbol belonging to this class is a continuous linear mapping of the Zemanian spaceHμ into itself. An integral representation of p.d.o. is obtained. Using Hankel convolution Lσ,α-norm continuity of...

Pseudodifferential Energy Estimates of Singular Perturbations

Singular pseudodifferential calculus for wavetrains and pulses

We generalize the analysis of [12] and develop a singular pseudodifferential calculus. The symbols that we consider do not satisfy the standard decay with respect to the frequency variables. We thus adopt a strategy based on the Calderón-Vaillancourt Theorem. The remainders in the symbolic calculus are bounded operators on L, whose norm is measured with respect to some small parameter. Our main...

Pseudodifferential Operators, Corners and Singular Limits

In the first part of my talk I shall describe some of the properties one should expect of a calculus of pseudodifferential operators which corresponds to the microlocalization of a Lie algebra of vector fields. This is not intended to be a formal axiomatic program but it leads one to consider conditions on the Lie algebra for such microlocalization to be possible. The symbolic structure of the ...