Symbol Of Pseudodifferential Operator Research Articles

Asymptotic Series of General Symbol of Pseudo-Differential Operator Involving Fractional Fourier Transform

An asymptotic series of general symbol of pseudo-differential operator is obtained by using the theory of fractional Fourier transform.

THE FIRST COHOMOLOGY OF THE SUPERCONFORMAL ALGEBRA K(1|4)

The infinitesimal deformations of the embedding of the Lie superalgebra of contact vector fields on the supercircle S1|4 into the Poisson superalgebra of symbols of pseudodifferential operators on S1|2 are explicitly calculated.

Description of kernels and symbols of pseudo-differential operators of negative orders, associated to Gevrey classes

This work is denoted to a characterization of symbols and kernels of pseudodifferential operators of negative orders. It completes previous works obtained in [2], [3], [4] for the same operators of positive orders.

An accuracy improvement in Egorov’s Theorem

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 $S^m/S^{m-1}$ for the pseudodifferential operators, one uses refined principal symbols in $S^m/S^{m-2}$, which for classical operators correspond simply to the principal plus the subprincipal symbol, and can generally be regarded as the first two terms of its Weyl symbol expansion: we call it the principal Weyl symbol of the pseudodifferential operator. Particular unitary Fourier integral operators, associated to the graph of the canonical transformation, have to be used in the conjugation for the higher accuracy to hold, leading to microlocal representations by oscillatory integrals with specific symbols that are given explicitly in terms of the generating function that locally describes the graph of the transformation. The motivation for the result is based on the optimal symplectic invariance properties of the Weyl correspondence in ${\mathbb R}^n$ and its symmetry for real symbols.

Projectively Equivariant Quantization and Symbol Calculus: Noncommutative Hypergeometric Functions

We extend projectively equivariant quantization and symbol calculus to symbols of pseudo-differential operators. An explicit expression in terms of hypergeometric functions with noncommutative arguments is given. Some examples are worked out, one of them yielding a quantum length element on S3.

A coercive bilinear form for Maxwell's equations

When one wants to treat the time-harmonic Maxwell equations with variational methods, one has to face the problem that the natural bilinear form is not coercive on the whole Sobolev space H 1. One can, however, make it coercive by adding a certain bilinear form on the boundary of the domain. This addition causes a change in the natural boundary conditions. The additional bilinear form (see (2.7), (2.21), (3.3)) contains tangential derivatives of the normal and tangential components of the field on the boundary, and it vanishes on the subspaces of H 1 that consist of fields with either vanishing tangential components or vanishing normal components on the boundary. Thus the variational formulations of the “electric” or “magnetic” boundary value problems with homogeneous boundary conditions are not changed. A useful change is caused in the method of boundary integral equations for the boundary value problems and for transmission problems where one has to use nonzero boundary data. The idea of this change emerged from the desire to have strongly elliptic boundary integral equations for the “electric” boundary value problem that are suitable for numerical approximation. Subsequently, it was shown how to incorporate the “magnetic” boundary data and to apply the idea to transmission problems. In the present note we present this idea in full generality, also for the anisotropic case, and prove coercivity without using symbols of pseudo-differential operators on the boundary.

Functional integral and the Feynman-Kac formula in superspace

We consider the Cauchy problem for linear pseudodifferential equations in superspace. The solution is constructed in the form of series. It may be regarded as a definition of a chronological exponent of a pseudodifferential operator symbol and interpreted as a functional integral in superspace.