×

Energy methods via coherent states and advanced pseudo-differential calculus. (English) Zbl 0885.35152

Cordaro, Paulo D. (ed.) et al., Multidimensional complex analysis and partial differential equations. A collection of papers in honor of François Treves. Proceedings of the Brazil-USA conference, June 12–16, 1995, São Carlos, Brazil. Providence, RI: American Mathematical Society. Contemp. Math. 205, 177-201 (1997).
The author finds a sufficient condition in terms of its symbols, in order that a selfadjoint operator \(Q(t)= q(t,x,D_x)\) satisfies an a priori estimate of the type: \[ |(D_t+ iQ(t))u|_{L^2(\mathbb{R},{\mathcal H})}\geq {1\over c} |u|_{L^2(\mathbb{R},{\mathcal H})}\tag{1} \] for some \(c>0\). Moreover, this sufficient condition can be viewed as a strengthened version of Nirenberg-Trèves’ condition \((\Psi)\), which is known to be necessary for (1). The proof is based on the use of the Wick quantization.
For the entire collection see [Zbl 0865.00044].

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
PDFBibTeX XMLCite