Canonical Operator on Punctured Lagrangian Manifolds

Abstract

A version of Maslov’s canonical operator on Lagrangian manifolds with singularities of special form (the so-called punctured manifolds) is described. These manifolds naturally arise in the construction of asymptotic solutions for a wide class differential and pseudodifferential equations; in particular, for Petrovsky hyperbolic systems or pseudo-differential equations of water waves.

Appendix

A. Proofs of auxiliary assertions

A.1. Proof of Proposition 1

For the proof, it is convenient to use “polar coordinates” by representing the variables $p$ in the form

$$p=r \omega ,\qquad r= \lvert p\rvert \equiv(p_1^2+\dots +p_n^2)^{1/2} \in\overline{ \mathbb{R} }_+=[0,\infty), \quad \omega =\frac{p}{ \lvert p\rvert }\in \mathbb{S} ^{n-1}.$$

The new variables $(x,r, \omega )$ (where $\omega$ is constrained by the condition $\omega _1^2+\dots+ \omega _n^2=1$) are coordinates on the manifold with boundary

$$ W= \mathbb{R} ^n\times[0,\infty)\times \mathbb{S} ^{n-1},$$

(A.1)

which is obtained by adding the value $r=0$. In these coordinates, the mapping $\jmath\colon \Lambda \to \mathbb{R} ^{2n}$ is represented by the functions

$$ x=X( \alpha ), \qquad r=R( \alpha ):= \lvert P( \alpha )\rvert ,\qquad \omega = \Omega ( \alpha ):=\frac{P( \alpha )}{ \lvert P( \alpha )\rvert }.$$

(A.2)

Further, it follows from condition (2.5) that $P_\rho(\phi,0)\ne0$, and hence

$$P(\phi,\rho)=\rho Q(\phi,\rho), \qquad Q(\phi,\rho)\ne0\quad \text{is a smooth function.}$$

Therefore, the functions

$$R(\phi,\rho)=\rho \lvert Q(\phi,\rho)\rvert ,\qquad \Omega (\phi,\rho)=\frac{Q(\phi,\rho)}{ \lvert Q(\phi,\rho)\rvert }$$

can smoothly be extended to the points at which $\rho=0$, and thus a smooth mapping $\imath: \Lambda \to W$ is well defined. Further, the Hamiltonian vector field corresponding to system (2.6) can be written out in these coordinates in the form

$$ \begin{aligned} V_W( \lambda )&=\sum_{j=1}^{n} \biggl \{ \biggl [ \biggl (F+r \frac{ \partial\, F}{ \partial\, r} -\sum_{k=1}^{n} \omega _k \frac{ \partial\, F}{ \partial\, \omega _k} \biggr ) \omega _j+ \frac{ \partial\, F}{ \partial\, \omega _j} \biggr ] \frac{ \partial\, }{ \partial\, x_j} \\ &\qquad{}+ \biggl [ \omega _j\sum_{k=1}^{n} \omega _k \frac{ \partial\, F}{ \partial\, x_k} - \frac{ \partial\, F}{ \partial\, x_j} \biggr ] \frac{ \partial\, }{ \partial\, \omega _j} \biggr \} -r \biggl [\sum_{k=1}^{n} \omega _k \frac{ \partial\, F}{ \partial\, x_k} \biggr ] \frac{ \partial\, }{ \partial\, r} \end{aligned}$$

(A.3)

and can be extended to a smooth vector field on $W$ which is everywhere tangent to $\partial\, W$. The trajectories of this vector field with initial data (A.2) for $\alpha \in \Lambda ^\circ$ are defined for $t\in[0,T]$; passing to the limit, we see that the same holds for $\alpha \in \partial\, \Lambda$. This implies that the functions $(X( \alpha ,t),P( \alpha ,t))$ are well defined as smooth functions on $\Lambda \times[0,T]$, and $P( \alpha ,t)=0$ for $\alpha \in \partial\, \Lambda$. The conditions (2.3), (2.4) for $t\in[0,T]$ and $\alpha \notin \partial\, \Lambda$ follow from the Hamiltonian property of system (3.1); the validity of condition (2.5) for $t\in[0,T]$ can be proved as follows. This condition for $t=0$ is equivalent to the fact that the mapping

$$ \mathbf{ \Lambda }:= \partial\, \Lambda \times \mathbb{R} _+\ni (\phi,\tau)\longmapsto (\mathbf{X}(\phi,\tau), \mathbf{P}(\phi,\tau)) := \biggl (X(\phi,0),\lim_{ \varepsilon \to0} \frac1 \varepsilon P(\phi, \varepsilon \tau) \biggr )$$

(A.4)

is an immersion, and therefore defines a homogeneous Lagrangian manifold. The evolution of this mapping, as can readily be calculated, is determined by the Hamiltonian system with the Hamiltonian $\lambda ^{(1)}(x,p)$, which is the homogeneous part of degree $1$ of the Hamiltonian $\lambda (x,p)$; therefore, it remains an immersion for all $t\in[0,T]$, which means that condition (2.5) is also satisfied. This completes the proof of Proposition 1.

A.2. Proof of Proposition 2

It suffices to prove that $N^*Y$ is either a simple manifold or, for $\operatorname{codim} Y=1$, the union of two such manifolds, since the shift along the trajectories a Hamiltonian vector field preserves all conditions in the definition of a simple punctured Lagrangian manifold.

(i) Let $(x_0,p_0)\in N^*Y$ be an arbitrary point. Then, identifying $T \mathbb{R} ^n_x$ and $T^* \mathbb{R} _x^n$ using Euclidean metric, we obtain

$$T_{(x_0,p_0)}(N^*Y)\simeq T_{(x_0,p_0)}(NY)\simeq T_{x_0}Y\oplus (T_{x_0}Y)^\perp=T_{x_0} \mathbb{R} ^n.$$

This isomorphism enables us to lift the Euclidean volume form $dx\equiv dx_1\wedge\dots\wedge dx_n$ given on $T_{x_0} \mathbb{R} ^n$ to the space $T_{(x_0,p_0)}(N^*Y)$. Since the point $(x_0,p_0)$ is arbitrary, we obtain a volume form $d \sigma$ on $N^*Y$. Thus, $N^*Y$ is orientable (and is oriented by $d \sigma$) even if the submanifold $Y$ itself is nonorientable (for example, the Möbius strip).

(ii) The form $P\,dX$ is identically zero on $N^*Y$. Therefore, its integral over any cycle is zero.

(iii) The triviality of the Maslov index is equivalent to the condition that there is a continuous branch of the argument of the Jacobian (3.7) on $N^*Y$, where we can take for $d \sigma$ an arbitrary real volume form on $\Lambda$; for example, the form mentioned in part (i) (see [15]). Let us write out such a branch explicitly. Let $Y$ be a submanifold of codimension $k\in\{0,\dots,n\}$, and let $x_0\in Y$ be an arbitrary point. Replacing $Y$ by its tangent plane at this point and taking into account the fact that the Jacobian ${\mathscr{J}} ( \alpha )$ does not change when the variables $x_1,\dots,x_n$ are permuted (the Jacobian is preserved for any odd permutation of variables, since the sign of the form $d \sigma$, which is included in the definition of the Jacobian, changes as well), we can assume that $Y$ is given by equations $x'=Ax''$, where

$$x'=\{x_1,\dots,x_k\}\quad\text{and}\quad x''=\{x_{k+1},\dots,x_n\}$$

(one of the groups of variables can be empty; then one should make an obvious redesignation). Accordingly, $N^*Y$ is given by the equations $x'=Ax''$ and $p''=-A^*p'$ and, for the coordinates on $N^*Y$, one can take $(p',x'')$. Now simple calculations show that

$$\begin{aligned} \, d \sigma &=d(p'+Ax'')\wedge d(x''-A^*p')= \det\begin{pmatrix} E_k & A \\ -A^* & E_{n-k} \\ \end{pmatrix}dp'\wedge dx'',\\ d(X-iP)&=d(Ax''-ip')\wedge d(x''+iA^*p') =\det\begin{pmatrix} -iE_k & A \\ iA^* & E_{n-k} \\ \end{pmatrix}dp'\wedge dx'', \end{aligned}$$

where $E_m$ stands for the identity matrix of the size $m\times m$, and we use reductions of the type

$$dp'\wedge dx''=dp_1\wedge\dots\wedge dp_k\wedge dx_{k+1}\wedge\dots\wedge dx_n.$$

Thus,

$${\mathscr{J}} ( \alpha )=\frac{d(X-iP)}{d \sigma _0}=(-i)^k, \;\text{and one can take}\;\arg {\mathscr{J}} ( \alpha ) =-\frac{\pi k}{2}.$$

A.3. Proof of Proposition 3

(a) It follows from condition (3.11) (iii) that the following condition holds on the set $C_\Phi$ for all sufficiently small $\rho>0$:

$$\operatorname{rank} \begin{pmatrix} \Psi_{\theta x}(x,\theta,\rho) & \Psi_{\theta\theta}(x,\theta,\rho) \\ \Psi_{x}(x,\theta,\rho)+\rho\Psi_{\rho x}(x,\theta,\rho) & \Psi_\theta(x,\theta,\rho) +\rho\Psi_{\rho\theta}(x,\theta,\rho) \\ \end{pmatrix}=m$$

(recall that $\Psi_\theta=0$ on $C_\Phi$), and this means, by the implicit function theorem, that one can locally express, from the equations defining the set (3.10), $m$ variables among

$$(x_1,\dots,x_n,\theta_1,\dots,\theta_{m-1})$$

via the remaining variables (which we denote by $\phi=(\phi_1,\dots,\phi_{n-1})$) and $\rho$, and thus, in a neighborhood of the hyperplane $\rho=0$, the set $C_\Phi$ is locally defined by parametric equations of the form

$$ x=X(\phi,\rho),\quad \theta=\Theta(\phi,\rho),\qquad \rho=\rho,$$

(A.5)

and, among equations (A.5), there are $n$ tautological equations $\phi=\phi$, $\rho=\rho$. Thus, in a neighborhood of the specified hyperplane, the set $C_\Phi$ is a smooth manifold with the boundary $\partial\, C_\Phi$ distinguished by the equation $\rho=0$. For $\rho>0$, the set $C_\Phi$ is also a smooth manifold by condition (3.11) (ii).

We claim that the mapping (3.12) satisfies the conditions of Definition 1 and therefore defines a punctured Lagrangian manifold. The condition $P( \alpha )\ne0$ for $\alpha \in \Lambda ^\circ$ follows from (3.11) (i). Further, by (3.11) (ii), for $\rho>0$, the function $\Phi(x,\theta,\rho)$ is a nondegenerate phase function in the usual sense of the term, and thus conditions (2.3) and (2.4) are satisfied by the standard construction of Lagrangian manifolds using phase functions. By continuity, condition (2.4) is satisfied for $\rho=0$ as well. Thus, it remains to show that relation (2.5) holds. Taking into account what was said at the end of Sec. A, it is sufficient to establish that the limit mapping of the form (A.4) obtained from the mapping (3.12) is an immersion. This mapping has the form (locally on $\partial\, C_\Phi$)

$$ x=X_0(\phi),\quad p=P_0(\phi,\tau),\qquad\tau>0,$$

(A.6)

where $X_0(\phi)=X(\phi,0)$ and

$$P_0(\phi,\tau)=\frac1 \varepsilon P(\phi, \varepsilon \tau) =\lim_{ \varepsilon \to0}\frac{ \varepsilon \tau}{ \varepsilon } \Psi_x(X(\phi, \varepsilon \tau),\Theta(\phi, \varepsilon \tau), \varepsilon \tau) =\tau\Psi_x(X(\phi,0),\Theta(\phi,0),0).$$

Consider the phase function

$$\Phi_0(x,\theta,\tau)=\tau\Phi(x,\theta,0),$$

defined on the set $B\times \mathbb{R} _+$, where

$$B=\{(x,\theta)\colon(x,\theta,0)\in V\}.$$

A straightforward calculation shows that this phase function is nondegenerate (this follows from condition (3.11) (iii)) and that

$$C_{\Phi_0}=(C_\Phi\cap\{\rho=0\})\times \mathbb{R} _+.$$

Thus, the mapping (A.6) is precisely the mapping $j_{\Phi_0}$ associated with the nondegenerate phase function $\Phi_0$ and therefore defines a uniform Lagrangian immersion, as desired.

(b) Note that, if $\Phi(x,\xi)$ is a nondegenerate phase function in the sense of Definition 2 and $x=x(y)$ is a smooth change of variables, then the function

$$\widetilde \Phi(y,\xi)=\Phi(x(y),\xi)$$

is a nondegenerate phase function as well, and the punctured Lagrangian manifold defined by this function is related to the punctured Lagrangian manifold defined by the function $\Phi(x,\xi)$ by the canonical transformation

$$x=x(y),\qquad p= \biggl ( \biggl ( \frac{ \partial\, x}{ \partial\, y} (y) \biggr )^T \biggr )^{-1}q$$

generated by the specified change of variables. Therefore, before proving the existence of a phase function defining a punctured Lagrangian manifold, one can first make a change of variables which is convenient for reasoning.

Let a punctured Lagrangian manifold be given. Let us show that, in a neighborhood of a given point $\alpha _*=(\phi_*,0)$, this manifold can be defined by a nondegenerate phase function. (The representability in a neighborhood of any point not lying on the boundary $\partial\, \Lambda$ follows from the standard theory.) Consider the limit homogeneous manifold A.4. As is known [18], by a change of the coordinates, it is possible to achieve the situation in which the variables $p$ form a coordinate system on $\Lambda _0$ in a neighborhood of the ray

$$\{\phi_*\}\times \mathbb{R} _{+\tau}.$$

Assume that such a change of coordinates has been made. Then

$$ \operatorname{rank} \begin{pmatrix} P_{\phi\rho}(\phi,0) & P_{\rho}(\phi,0) \\ \end{pmatrix}=n$$

(A.7)

for $\rho$ close to $\rho_*$. We claim that the functions

$$r=R( \alpha )\equiv \lvert P( \alpha )\rvert$$

and

$$\omega =\Omega( \alpha )\equiv P( \alpha )/ \lvert P( \alpha )\rvert \in \mathbb{S} ^{n-1}$$

can be chosen as coordinates on $\Lambda$ in a neighborhood of the point $\alpha _*$. To this end, we represent $P$ in the form

$$P(\phi,\rho)=\rho Q(\phi,\rho), \qquad \text{at the points where $Q(\phi,\rho)$ does not vanish. }$$

Then

$$P_\rho(\phi,0)=Q(\phi,0),\qquad P_{\phi\rho}(\phi,0)=Q_\phi(\phi,0),$$

and condition (A.7) can be rewritten in the form

$$ \operatorname{rank} \begin{pmatrix} Q_\phi(\phi,0) & Q(\phi,0) \\ \end{pmatrix}=n.$$

(A.8)

Further,

$$\frac{ \partial\, ( \Omega ,R)}{ \partial\, (\phi,\rho)} (\phi,0)= \begin{pmatrix} \Pi Q_\phi & \Pi Q_\rho \\ 0 & \lvert Q\rvert \\ \end{pmatrix},\qquad\text{where} \qquad \Pi u=\frac{1}{ \lvert Q\rvert } \biggl (u-\frac{\langle Q,u\rangle}{ \lvert Q\rvert ^2}Q \biggr )$$

(we omit the arguments $(\phi,0)$ of the function $Q$ and its derivatives). We claim that the rank of the $(n\times(n-1))$ matrix $\Pi Q_\phi$ is equal to $n-1$. Suppose that this is not the case, i.e., there is a nonzero vector $a\in \mathbb{C} ^{n-1}$ such that $\Pi Q_\phi a=0$. Then

$$Q_\phi a=\frac{\langle Q,Q_\phi a\rangle}{ \lvert Q\rvert ^2}Q,\quad \text{or}\quad Q_\phi a+Qb=0,\quad\text{where }b=-\frac{\langle Q,Q_\phi a\rangle}{ \lvert Q\rvert ^2},$$

which contradicts condition (A.8). Thus, in a neighborhood of the point $\alpha _*$, the punctured Lagrangian manifold $\Lambda$ can be defined by the formulas

$$X=X( \omega ,r),\qquad P=P( \omega ,r)\equiv \omega r,\qquad \omega \in U\subset \mathbb{S} ^{n-1},\quad r\in[0,r_0),$$

for some domain $U$ and for a sufficiently small number $r_0>0$.

Without loss of generality, we may assume that the functions

$$\theta:=( \omega _1,\dots, \omega _{n-1})$$

form a system of coordinates in $U$; then

$$\omega _n=\pm\sqrt{1-\theta^2}$$

(to be definite, we choose the sign “plus”). Redenote $r$ by $\rho$ ($\rho$ is no longer needed in its old meaning) and write

$$\Phi(x,\theta,\rho)=\rho\langle \omega ,x\rangle+S( \omega ,\rho),\qquad \text{where}\quad S( \omega ,\rho)=-\int_{ \alpha _*}^{( \omega ,\rho)}X\,dP$$

(the integral does not depend on the path of integration by the Lagrangian property). Let us now note that

$$\Phi(x,\theta,0)=0$$

and

$$\Phi_x= \omega \rho\ne0\quad\text{for}\quad \rho\ne0.$$

Further,

$$\Psi(x,\theta,\rho)=\langle \omega ,x\rangle+\frac1\rho S( \omega ,\rho).$$

Since

$$dS( \omega ,\rho)=-\langle X( \omega ,\rho), \omega \rangle\,d\rho- \rho\langle X( \omega ,\rho),d \omega \rangle,$$

it follows that equations (3.10) for the manifold $C_\Phi$ acquire the form

$$0 =\Psi_{\theta_j}=x_j-X_j( \omega ,\rho) -\frac{ \omega _j}{ \omega _n}(x_n-X_n( \omega ,\rho)),\qquad j=1,\dots,n-1,$$

(A.9)

$$0 =\Psi+\rho\Psi_\rho=\langle x-X( \omega ,\rho), \omega \rangle.$$

(A.10)

We can represent equations (A.9) in the form

$$x_j-X_j( \omega ,\rho)=\frac{ \omega _j}{ \omega _n}(x_n-X_n( \omega ,\rho)),\qquad j=1,\dots,n$$

(we have added the $n$-th equation, which is satisfied automatically). Applying now equation (A.10), we obtain $(x_n-X_n)/ \omega _n=0$, i.e., $x_n=X_n$; substituting this into (A.9), we see that the equations of the manifold $C_\Phi$ acquire the form

$$x=X( \omega ,r).$$

The mapping $\jmath_\Phi$ is defined by the formulas

$$x=X( \omega ,r),\qquad p=\Phi_x= \omega r=P( \omega ,r),$$

i.e., this phase function defines our punctured manifold indeed. We should only verify the nondegeneracy conditions of the phase function, i.e., conditions (3.11) (ii) and (iii). The first of them relates to the interior of the set $C_\Phi$ and follows from the classical results on phase functions, and, to verify the other condition, it suffices to note that

$$\begin{pmatrix} \Psi_{\theta x}(x,\theta,0)\\ \Psi_{x}(x,\theta,0) \end{pmatrix} =\begin{pmatrix} 1 & 0 & \dots & 0 & -\frac{ \omega _1}{ \omega _n} \\ 0 & 1 &\dots &0& -\frac{ \omega _2}{ \omega _n} \\ \vdots & \vdots & \ddots & \vdots &\vdots\\ 0 & 0 & \dots & 1 & -\frac{ \omega _{n-1}}{ \omega _n}\\ \omega _1 & \omega _2 & \dots & \omega _{n-1} & \omega _n \\ \end{pmatrix};$$

in this matrix, the first $n-1$ rows are obviously linearly independent, the last row is nonzero and orthogonal to each of the previous rows, and thus the matrix is nonsingular.

This completes the proof of the proposition.