Pseudo-differential operators and a canonical operator in general symplectic manifolds.

*(Russian)*Zbl 0538.58035In 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

##### MSC:

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.) |