The pseudodifferential operator and Wilson bases (Q1354980)

The author considers pseudodifferential operators of the form \[ \sigma(x,D)= (2\pi)^{-n} \iint \sigma((x+ y)/2,\xi) e^{i\xi(x-y)} f(y)dy d\xi, \] where \(f\in S\), and the Weyl symbol \(\sigma(z,\xi)\) satisfies several growth conditions, in particular \(\sigma(z,\xi)\to \infty\) as \(| x|+|\xi|\to \infty\). A model for the operators under consideration is the differential operator \(-\Delta+| x|^2\). The main result in this paper is an approximate diagonalization of these operators using the Wilson basis. This is an orthogonal basis for the space \(L^2\), constructed by \textit{I. Daubechies}, \textit{S. Jaffard} and \textit{J.-L. Journé} [SIAM J. Math. Anal. 22, 554-572 (1991; Zbl 0754.46016)]. The author uses his approximate diagonalization result to obtain a priori estimates in some modulation spaces. The author also proves approximate estimates for large eigenvalues of \(\sigma(x,D)\) by the values of \(\sigma(z,\xi)\) on a lattice in the phase space. As a consequence of these estimates, the author describes the asymptotic behavior of the occurrence of large eigenvalues.