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Pseudo-differential operators and a canonical operator in general symplectic manifolds. (Russian) Zbl 0538.58035
In this work a global calculus of pseudo-differential operators (p.d.o.) is developed under the approximation \(O(h^ 2)\), by giving also the complete proof of some previously announced results. §1 is the introduction which provides a general account of the subject, describing its links with the quantisation theory. In §2 some formulas and definitions on p.d.o. and canonical operators (c.o.) in \(R^{2n}\) are given. Let \(S(R^ n)\) be the Schwartz space of \(C^{\infty}\)-fast growing functions on \(R^ n\). One considers on \(R^ n\)-nonhomogeneous h-p.d.o. with symbols in \(S(R^ n)\). For \(h>0\), \(\Psi \in S=S(R^ n), f\in S(R^{2n}), f=f(q,p),\) let \(\hat f:S\to S\) be the Weyl p.d.o. given by \[ (\hat f\Psi)(q)=(2\pi)^{-n}\int \exp(iqp)f(q+q'/2,hp)(\int \exp(- iq'p)\Psi(q')dq)dp. \] Some properties of this operator are recalled. Then let \(\Lambda\) be an arbitrary Lagrangian submanifold in \(R^ n_ q\oplus R^ n_ p\) and let \(\Sigma\subset \Lambda\) be the set of singular projections of \(\Lambda\) on \(R^ n_ q\). In general position \(\Sigma\) is an n-1 dimensional cycle. By taking a fixed point \(m\in \Lambda \backslash \Sigma\) as origin, one denotes by \(\nu [m\to m']\) the index of the intersection with \(\Sigma\) of the path \(\Gamma m\to m'\) on \(\Lambda\) joining m with m’. By definition this is the index of \(\Gamma\) on \(\Lambda\). Let also \(\theta =pdq|_{\Lambda}\). One assumes that for every 1-cycle \(\Gamma\) on \(\Lambda\) the following quantisation condition holds \[ 1/(2\pi h)\int_{\Gamma}\theta -\nu [\Gamma]/4\in Z. \] The canonical operator (c.o) is defined by a map \({\mathcal K}\equiv {\mathcal K}^ h_{\Lambda,\sigma,m}:C^{\infty}_ 0(\Lambda)\to S(R^ n_ q)\) via a regular and positive measure \(\sigma\) on \(\Lambda\). A basic notion is the oscillation support mod \(O(h^ n)\) of \(\Psi \in CL^ 2(R^ n)\) defined as \(osc^ h(\Psi)=R^{2n}\backslash \{z^ 0\in R^{2n},\quad \exists U\in \vartheta_{z^ 0}\) such that \(<\phi\Psi,f>= O(h^{2k})\) for every \(f\in C^{\infty}_ 0(U)\},\) where \(\phi\Psi\) is given by \(<\phi \Psi,f>=(\hat f\Psi,\Psi)_{L^ 2}.\) To a canonical transformation \(\gamma:D\to \gamma(D)\) of a domain \(D\subseteq R^{2n}\) a sheaf homomorphism \(\Pi(D)\to \Pi(\gamma(D))\) is associated. Several lemmas in §3 give the main properties of these sheaves and of the integral operators \(T(\gamma,z^ 0,\phi)\) defined on \(L^ 2(R^ n)\) with the kernel \[ (2\pi ih)^{-n/2}({\mathcal K}^ h_{\Gamma p(\gamma),\gamma_*can,(\gamma(z^ 0),z^ 0)}[\phi](q,q')) \] where \(\Gamma p(\gamma)=\{(z,z'),\quad z=\gamma(z'),\quad z'\in \bar D\}, \gamma_*can=dz'\) is the natural measure on it, and \(\bar D\) is the closure of D. In §4 the sheaf of germs of oscillant functions on a symplectic manifold \({\mathcal X}\) are considered, by computing the h-p.d.o. mod \(O(h^ 2)\). For a \(C^{\infty}\)-manifold \({\mathcal X}\) with the symplectic form \(\omega\) there is an atlas \(\{(\bar U_{\alpha},\zeta_{\alpha})\}\) of \({\mathcal X}\), given by \(\zeta_{\alpha}:\bar U_{\alpha}\to \bar D_{\alpha}\subseteq R^ n_ q\oplus R^ n_ p\) such that \(\zeta^*_{\alpha}(dp\wedge dq)=\omega |_{\bar U_{\alpha}}.\) One considers the sheaves \(\Pi_{\alpha}= \Pi(D_{\alpha}).\) The transition sheaf homomorphism \(T_{\alpha \beta}:\Pi_{\alpha}|_{U_{\alpha}\cap U_{\beta}}\to \Pi_{\beta}|_{U_{\alpha}\cap U_{\beta}}\) defines a two dimensional Čech cocycle and so a cohomological class \(\kappa \in H^ 2({\mathcal X},Z).\) So one has Theorem 1. The obstructions to the coherence of the global sheaf \(\Pi\) (\({\mathcal X})\) whose restrictions on \(U_{\alpha}\) are \(\Pi_{\alpha}\) are two cohomological classes, namely \([\omega]\in H^ 2({\mathcal X},R)\) and \(\kappa /2(mod 2)\in H^ 2({\mathcal X},Z_ 2).\) Theorem 2 furnishes the existence of a sheaf \(\Pi\) (\({\mathcal X})\) with desired properties on a closed manifold \({\mathcal X}\) endowed with a family of symplectic forms \(\omega =\{\omega_{\lambda}\}\), with \(\lambda =\lambda(h)\), all of them satisfying the announced quantisation condition. In §5 the c.o. on a Lagrangian submanifold of the generalized symplectic manifold are considered. Theorem 3 gives a commutation formula for the operators \({\mathcal K}_{\Lambda,\sigma}\) and some related properties. Theorem 4 claims that in the case \([\omega]=0\), \(\kappa =0(mod Z_ 4)\) the different non-equivalent sheaves \(\Pi\) (\({\mathcal X})\) are in 1-1 correspondence with \(H^ 1({\mathcal X},R)\times H^ 1({\mathcal X},Z_ 4).\) Theorem 5 gives the cohomology class \(t(\Lambda)=\exp((i/h)\theta^{\Lambda}-(\pi /2)\nu^{\Lambda})\) in the case \(\omega =d\theta\), \(\kappa =0\) (or \(\kappa =0(mod Z_ 4))\), where \(\theta^{\Lambda}\in H^ 1({\mathcal X},R)\) is the class of the restriction of \(\theta\) on \(\Lambda\) and \(\nu^{\Lambda}\in H^ 1(\Lambda,{\mathbb{Z}})\) (or respectively \(\nu^{\Lambda}\in H^ 1(\Lambda,Z_ 4)).\) In §6 the projective representations of the groups of canonical transformations in symplectic manifolds are considered. The main result is formulated in Theorem 6 which gives under some conditions an exact operator \(T(\gamma,z^ 0)\) on \(\Gamma_ 0({\mathcal X})\) independently of \(\phi \in C_ 0^{\infty}({\mathcal X}).\) In §7 the construction of homogeneous p.d.o. is considered assuming that there is a free action of the real group \(R_+\) on \({\mathcal X}\). Theorem 7 extends in this case some results in §2. The final remarks in §8 outline that the constructed sheaf \(\Pi\) (\({\mathcal X})\) didn’t coincide with the sheaf of germs of the cross sections for a line bundle (or an SU(1)-bundle) on \({\mathcal X}\). The quantisation construction can also be performed for the \(C^{\infty}\)-functions.
Reviewer: M.Tarina

58J40 Pseudodifferential and Fourier integral operators on manifolds
55N30 Sheaf cohomology in algebraic topology
55S35 Obstruction theory in algebraic topology
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)