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**Analytic theory for the quadratic scattering wave front set and application to the Schrödinger equation.**
*(English)*
Zbl 1029.35001

Astérisque. 283. Paris: Société Mathématique de France. 128 p. EUR 23.00;$ 33.00 (2002).

It is well known that the smoothness (in \(x\in\mathbb{R}^n)\), for \(t> 0\), of the solution of the initial value problem for the Schrödinger equation is under the control of the behavior at infinity of the initial data (“propagation with infinite spead”). J. Wunsch [Duke Math. J. 98, 137-186 (1999; Zbl 0953.35121)] proposed to embed the two informations: behavior at infinity (decay, oscillations,…) and smoothness, in one unique object, which he called the \(C^\infty\) quadratic scattering (qsc) wave front set, in which the above phenomena of infinite speed propagation would appear as a propagation of singularities result.

The work of Wunsch relies on some geometrical point of view of R. B. Melrose [Lecture Notes Pure Appl. Math. 161, 85-130 (1994; Zbl 0837.35107)]. It begins by working on a compact manifold \(M\) with boundary \(\partial M\), which comes from a compactification of \(\mathbb{R}^n\). The second step is to define a quadratic scattering (qsc) cotangent bundle, \(^{\text{qsc}} T^*M\), where the canonical form is given by \(\alpha= \lambda \rho^{-3} d\rho+\mu \rho^{-2}dy\) if \((\rho,y)\) are local coordinates near the boundary. Local coordinates, near the boundary, in this qsc cotangent bundle are given by \((\rho,y,\lambda,\mu)\). Since only high frequencies are involved in the occuring of singularities, Melrose suggests to make a radial compactification in the fibers, that is to set, for large \(\lambda+|\mu |\), \[ \sigma= (\lambda^2+ |\mu |^2)^{-1/2},\overline \lambda=\sigma \lambda,\mu= \sigma\mu. \] Then we may define the extended qsc cotangent bundle \(^{\text{qsc}} \overline T^*M\) in which local coordinates, near the boundary of \(M\), are given by \((\rho,y,\sigma, (\overline\lambda, \overline\mu))\), where \(\rho\geq 0\), \(\sigma\geq 0\). Its boundary \(C\) is the union of two faces, \(^{\text{qsc}}\overline T^*_{\partial M}M=\{(\rho,y, \sigma,(\overline \lambda,\overline \mu)):\rho=0\}\) and \(^{\text{ qsc}}S^*M= \{(\rho,y, \sigma, (\overline \lambda, \overline\mu)): \sigma=0\}\). Wunsch defines the qsc wave front set as a subset of \(C\), by using Melrose’s theory of pseudo-differential operators on manifolds with corners. The authors, in the analytic case (but also in the \({\mathcal C}^\infty\) or Gevrey cases) use instead the Sjöstrand machinery of FBI transforms. Their analytic qsc wave front set \((^{\text{qsc}}WF_a)\) will be defined through a FBI transform with two scales \((h,k)\), instead of only one scale \(\lambda=1/k\) in the usual case. More precisely, they set for \(u\in L^2 (M)\), \[ Tu(\alpha, h,k)=\iint e^{ih^{-2}k^{-1}\varphi( \rho/h,y, \alpha,h)} a( \rho/h,y, \alpha,h,k) \chi(\rho/h,y) \overline{u(\rho,y)} d\rho dy. \] Here \(\varphi\) is a phase, \(a\) a symbol and \(\chi\) a cut-off function. The parameter \(h\) is used to decribe the behavior at infinity, while \(k\) is used to test the analytic smoothness. To show the invariance of the \(^{\text{qsc}}WF_a\), one has to make a careful study of the pseudo-differential operators in the complex domain, then in the real domain. The situation is complicated by the fact that the FBI phases have an imaginary part which goes to zero with \(h\).

Concerning the propagation theorems, the authors consider a Schrödinger equation with a Laplacian \(\Delta_g\) with respect to a scattering metric \(g\) in the sense of Melrose; this means that, near the boundary one can write \(g=\rho^{-4}d \rho^2+ \rho^{-2}h\), where \(h\) is a metric such that \(h|_{\partial M}\) is positive definite. This includes the flat metric \(h=d\omega^2\), but also the asymptotically flat metrics on \(\mathbb{R}^n\). They try to answer the following question. Let \(m_0\in C,u\) be a solution of the initial value problem for this Schrödinger equation and \(T>0\). On what condition on \(u_0:=u|_{t=0}\) do we have \(m_0\notin {^{\text{qsc}}WF_a}(u(T, \cdot))\)? The different statements, according to the position of \(m_0\in C\), will follow from four propagation results: propagation inside \(^{\text{qsc}\overline T^*M}\), inside \(^{\text{qsc}S^*M}\) or along the corner (for a uniform \(^{\text{qsc}}WF_a\) and fixed \(t)\), from the interior to the corner and finally from the boundary at infinity to the corner.

The work of Wunsch relies on some geometrical point of view of R. B. Melrose [Lecture Notes Pure Appl. Math. 161, 85-130 (1994; Zbl 0837.35107)]. It begins by working on a compact manifold \(M\) with boundary \(\partial M\), which comes from a compactification of \(\mathbb{R}^n\). The second step is to define a quadratic scattering (qsc) cotangent bundle, \(^{\text{qsc}} T^*M\), where the canonical form is given by \(\alpha= \lambda \rho^{-3} d\rho+\mu \rho^{-2}dy\) if \((\rho,y)\) are local coordinates near the boundary. Local coordinates, near the boundary, in this qsc cotangent bundle are given by \((\rho,y,\lambda,\mu)\). Since only high frequencies are involved in the occuring of singularities, Melrose suggests to make a radial compactification in the fibers, that is to set, for large \(\lambda+|\mu |\), \[ \sigma= (\lambda^2+ |\mu |^2)^{-1/2},\overline \lambda=\sigma \lambda,\mu= \sigma\mu. \] Then we may define the extended qsc cotangent bundle \(^{\text{qsc}} \overline T^*M\) in which local coordinates, near the boundary of \(M\), are given by \((\rho,y,\sigma, (\overline\lambda, \overline\mu))\), where \(\rho\geq 0\), \(\sigma\geq 0\). Its boundary \(C\) is the union of two faces, \(^{\text{qsc}}\overline T^*_{\partial M}M=\{(\rho,y, \sigma,(\overline \lambda,\overline \mu)):\rho=0\}\) and \(^{\text{ qsc}}S^*M= \{(\rho,y, \sigma, (\overline \lambda, \overline\mu)): \sigma=0\}\). Wunsch defines the qsc wave front set as a subset of \(C\), by using Melrose’s theory of pseudo-differential operators on manifolds with corners. The authors, in the analytic case (but also in the \({\mathcal C}^\infty\) or Gevrey cases) use instead the Sjöstrand machinery of FBI transforms. Their analytic qsc wave front set \((^{\text{qsc}}WF_a)\) will be defined through a FBI transform with two scales \((h,k)\), instead of only one scale \(\lambda=1/k\) in the usual case. More precisely, they set for \(u\in L^2 (M)\), \[ Tu(\alpha, h,k)=\iint e^{ih^{-2}k^{-1}\varphi( \rho/h,y, \alpha,h)} a( \rho/h,y, \alpha,h,k) \chi(\rho/h,y) \overline{u(\rho,y)} d\rho dy. \] Here \(\varphi\) is a phase, \(a\) a symbol and \(\chi\) a cut-off function. The parameter \(h\) is used to decribe the behavior at infinity, while \(k\) is used to test the analytic smoothness. To show the invariance of the \(^{\text{qsc}}WF_a\), one has to make a careful study of the pseudo-differential operators in the complex domain, then in the real domain. The situation is complicated by the fact that the FBI phases have an imaginary part which goes to zero with \(h\).

Concerning the propagation theorems, the authors consider a Schrödinger equation with a Laplacian \(\Delta_g\) with respect to a scattering metric \(g\) in the sense of Melrose; this means that, near the boundary one can write \(g=\rho^{-4}d \rho^2+ \rho^{-2}h\), where \(h\) is a metric such that \(h|_{\partial M}\) is positive definite. This includes the flat metric \(h=d\omega^2\), but also the asymptotically flat metrics on \(\mathbb{R}^n\). They try to answer the following question. Let \(m_0\in C,u\) be a solution of the initial value problem for this Schrödinger equation and \(T>0\). On what condition on \(u_0:=u|_{t=0}\) do we have \(m_0\notin {^{\text{qsc}}WF_a}(u(T, \cdot))\)? The different statements, according to the position of \(m_0\in C\), will follow from four propagation results: propagation inside \(^{\text{qsc}\overline T^*M}\), inside \(^{\text{qsc}S^*M}\) or along the corner (for a uniform \(^{\text{qsc}}WF_a\) and fixed \(t)\), from the interior to the corner and finally from the boundary at infinity to the corner.

Reviewer: Viorel Iftimie (Bucuresti)

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35A27 | Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs |

35J10 | Schrödinger operator, Schrödinger equation |

35A18 | Wave front sets in context of PDEs |

35A21 | Singularity in context of PDEs |

35A20 | Analyticity in context of PDEs |