QUANTUM-MECHANICS AND PARTIAL-DIFFERENTIAL EQUATIONS Academic Article uri icon

abstract

  • This paper develops the basic theory of pseudo-differential operators on R n , through the Caldern-Vaillancourt (0, 0) L 2 -estimate, as a natural part of the harmonic analysis on the Heisenberg group, the group-theoretic embodiment of Heisenberg's Canonical Commutation Relations. The symbol mapping is given a group-theoretic interpretation consistent with the Kirillov method of orbits. By comparing different well-known realizations of the unique irreducible representation of the Heisenberg group, the Toeplitz operators on the complex n-ball are shown essentially to be pseudo-differential operators. The proof of the Caldern-Vaillancourt estimate is almost purely group-theoretic. Criteria for positivity, and for compactness are also given. 1980, All rights reserved.

published proceedings

  • JOURNAL OF FUNCTIONAL ANALYSIS

author list (cited authors)

  • HOWE, R.

citation count

  • 111

complete list of authors

  • HOWE, R

publication date

  • September 1980