On regular Fréchet-Lie group of invertible inhomogeneous Fourier integral operators on \(\mathbf R^ n\) (Q1922153)

The Schrödinger equation \(\partial \phi/\partial t ={i\over\hbar} H\phi\) with time-dependent Hamiltonian is interpreted as an equation for integral curves of a time-dependent right-invariant vector field \(X\) on a Fréchet-Lie group, then the solution is represented by a product integral; see \textit{H. Omori, Y. Maeda, A. Yoshioka} [ibid. 3, 353-390 (1980; Zbl 0461.58003); 4, 221-253 (1981; Zbl 0486.58002)] and the same authors together with \textit{O. Kobayashi} [ibid. 4, 255-277 (1981; Zbl 0486.58003); 5, 365-398 (1982; Zbl 0515.58004); 6, 217-246 (1983; Zbl 0537.58005); 7, 315-336 (1984; Zbl 0565.58037); 8, 1-47 (1985; Zbl 0582.58034)] for the general theory. In more detail, the Fréchet-Lie group mentioned above is generated by the Fourier integral operators \[ \iint a(\overline{x},\xi)e^{i/\hbar\cdot (S(\overline{x},\xi)+f(\overline{x},\xi)-\xi x)}u(x)dxd\xi, \] where \(S\) and \(f\) are homogeneous functions on \(\mathbb{R}^{2n}-0\) of degrees 2 and 1, respectively, and \[ a(x,\xi)\sim a_0(\omega)+a_{-1}(\omega) \rho^{-1}+\dots\quad (\omega\in S^{2n-1},\rho=(|x|^2 +|\xi|^2)^{1/2}). \] The relevant Lie algebra consists of smooth functions with asymptotics \[ X=i(a_2(\omega)\rho^2+a_1(\omega)\rho) + a_0(\omega) + a_{-1}(\omega)\rho^{-1}+ \dots, \] where \(a_2\), \(a_1\) are real-valued and \(a_0,a_{-1},\dots\) complex-valued smooth functions. The fundamental solution of the Schrödinger equation is represented by the product integral \(\Phi=\Pi \text{ exp }Xd\tau.\) The proofs are presented with all necessary details.