A Poisson Structure on Spaces of Symbols.
Abstract
A Poisson structure (anti-symmetric bilinear local operator on functionals, obeying the Jacobi identify) is established on certain function spaces (spaces of symbols of pseudodifferential operators on R to the nth power). The spaces of functionals thus become (infinite-dimensional) Lie Algebras. This type of Lie algebra structure has been established previously for functionals of functions of a single variable (n = 1) only. For n = 1, the theorem of Gardner, as generalized by Gel'fand, Dikii, and others, is proved: that is, the residues of the zeta-function of the elliptic symbol are in involution with respect to the appropriate Poisson bracket. In contrast, it is shown by explicit example that the residues of the zeta functions of higher-dimensional elliptic symbols are generally not in involution.
Document Details
- Document Type
- Technical Report
- Publication Date
- Apr 01, 1978
- Accession Number
- ADA060660
Entities
People
- W. Symes
Organizations
- University of Wisconsin–Madison