WKB in dimension ≥ 2. 1


### Preliminaries

#### Introduction.

Recall that construction of Chapter 5 works as long as $S(x,t)$1 exists; in other words as long as projection $\pi_x:\Lambda_t\ni (x,p)\to x \in \bR^d$ is a diffeomorphism.

In the previous Chapter 6 we considered $1$-dimensional case. Now we need a bit more of sophistication.

Definition 1. Lagrangian manifold is a smooth $d$-dimensional manifold $\Lambda \subset \bR^{2d}$ on which symplectic form vanishes: $$\sigma :=\sum_{1\le j\le d} dx_j\wedge dp_j =0 \label{eq-7.1.1}$$

The following statements could be proven:

Lemma 1.

1. $\Lambda = \{(x,p): \, p= \nabla S(x)\}$ is a Lagrangian manifold such that $\pi_x$ is a local diffeomorphism.
2. If $\Lambda$ is a Lagrangian manifold such that $\pi_x$ is a local diffeomorphism then $\Lambda = \{(x,p): \, p= \nabla S(x)\}$ for some function $S(x)$.

Recall that $\Lambda_t$ is a Lagrangian manifold constructed in the following way:

• $\Lambda_0=\{ (x,S_{0\, x} \}$ is defined as $t=0$.
• $\Lambda_t$ is an evolution of $\Lambda_0$ along Hamiltonian trajectories \begin{align} &\frac{dx}{dt}=H_{p},\label{eq-7.1.2}\\ &\frac{dp}{dt}=-H_{x}.\label{eq-7.1.3} \end{align} Recall that $S(x,t)$ is defined from \begin{align} &\frac{dS}{dt}=p\cdot x-H(x,p),\label{eq-7.1.4}\\ &S|_{t=0}=S_0.\label{eq-7.1.5} \end{align} We skip subscript $_0$ at $H$.

Lemma 2. Hamiltonian flow (\ref{eq-7.1.2})--(\ref{eq-7.1.3}) preserves symplectic form and therefore $\Lambda_t$ is a Lagrangian manifold.

Lemma 3. At each point $(x,p)$ there exists a partition $(I,J)$ of the set $\{1,\ldots, d\}$ such that $\pi_I:\Lambda \ni (x,p)\to (x_I, p_J)$ is a local diffeomorphism.