$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\const}{\operatorname{const}}$ $\newcommand{\sgn}{\mathrm{sgn}}$

Consider SchrÃ¶dinger operator \begin{equation} H=h^2D^2 +V(x) \label{eq-6.3.1} \end{equation} and an energy level $E$ such that \begin{align} &V(x_E^-)=V(x_E^+)=E, && V(x)< E \iff \ x_E^-< x < x_E^+, \label{eq-6.3.2}\\[3pt] &V'(x_E^-)< 0, \quad V'(x_E^+) > 0. \label{eq-6.3.3} \end{align}

We are interested in the *energy levels* (i.e. eigenvalues) of $H$ close to $E$. To do this we employ the *stationary theory*. Consider Lagrangian manifold
$\Lambda_E=\{(x,p): p^2+V(x)=E\}$ and construct function $S$ on it which everywhere except the end points $(x_E^\pm)$ can be locally expressed as function of $x$ satisfying $S_x^2 + V(x)=E$.

However globally $S(x)$ is not defined uniquely: as point $(x,p)$ circles once counterclockwise $\Lambda$ its increment is $\Delta S= \int_{\Lambda_E} p\,dx$. On the other hand,

**Exercise 1.**
Prove that Maslov index of this path is $2\mod 4$.

Therefore argument of amplitude $u_h(x)$ is increased by $h^{-1}\int_{\Lambda_E} p\,dx +\pi$ where $\pi$ comes from the increment of amplitude $A(x)$.

Since $u_h(x)$ must be a function of $x$ we conclude that this increment is $\equiv 0\mod 2\pi\bZ$: \begin{equation} F(E):= -\frac{1}{2\pi h} \int_{\Lambda_E} p\,dx = n +\frac{1}{2}\qquad n\in \mathbb{Z}. \label{eq-6.3.4} \end{equation}

This is *Bohr-Sommerfeld formula*. One can prove rigorously

**Theorem 1.**

- In the framework of (\ref{eq-6.3.2})--(\ref{eq-6.3.3}) eigenvalues of $H$ close to $E$ are $E_n + O(h^2)$ where $E_n$ are obtained from equation (\ref{eq-6.3.4}).
- Furthermore, spacing between eigenvalues i.e. $E_{n+1}-E_n$ is $2\pi h F'(E) +O(h^2)$ where $F'(E)=\partial_E F(E)$.
- Eigenfunctions are $e^{ih^{-1}S(x)}A_0(x)$ nodulo $O(h)$ uniformly on $[x_E^-+\varepsilon, x_E^+-\varepsilon]$ for any $\varepsilon >0$.
- On the other hand near $x_E^-\pm$ in $p$-representations eigenfunctions are $e^{-ih^{-1}\tilde{S}(x)}\tilde{A}_0(x)$ nodulo $O(h)$.

**Remark 1.**

- $F(E)=\iint _{\{ H(x,p)< E\}}\,dxdp$ is an area of $\{ H(x,p)< E\}$. In our particular case \begin{equation} F(E)=2\int_{x_E^-}^{x_E^+} \sqrt{E-V(x)}\,dx \label{eq-6.3.5} \end{equation} and \begin{equation} F'(E)=\int_{x_E^-}^{x_E^+} \frac{dx}{\sqrt{E-V(x)}}= T(E) \label{eq-6.3.6} \end{equation} is a period of Hamiltonian trajectory on energy level $E$.
- All this holds for more general $1$-dimensional Hamiltonians. In particular, for $H'=F(H)$ we have similar results albeit with $F(E)=E$.
- If there are several
*potential wells*$\{x_{k,E}< x < x_{k,E}^+\}$, $k= 1,\ldots, K$ then one needs to calculate $E_{k,n}$ near $E$ (with different $n\in \bZ$) and take union (perturbation would be exponentially small). - If $V'(x_E^+)=0$ (or/and $V'(x_E^-)=0$ then eigenvalue are more dense ear $E$.
- This is essentially $1$-dimensional results. In higher dimensions eigenvalues are much more dense and we can talk only about average spacings and not about eigenfunctions but rather
*quasimodes*which in fact are linear combinations of the eigenfunctions (with near the same eigenvalues).

**Example 1.**
As $V(x)=x^2$ (harmonic oscillator) then $F(E)=\pi E$ and $E_n= (2n+1)h$ precisely. Eigenfunctions are $h^{-\frac{1}{4}} e_n (h^{-\frac{1}{2}}($ where $e_n$ are *Hermite functions*.

**Remark 2.**
What we denote by "$h$"' physicists denote by "$\hbar$", and "their"
$h=2\pi \hbar$ is the minimal possible action (according to N. Bohr).