Quantum mechanics and partial differential equations
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This paper develops the basic theory of pseudo-differential operators on R n , through the Calderón-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 Calderón-Vaillancourt estimate is almost purely group-theoretic. Criteria for positivity, and for compactness are also given. © 1980, All rights reserved.
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