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

6.3. Bohr-Sommerfeld approximation

1. Bohr-Sommerfeld approximation
2. Quasiegenvalues and quasimodes

Bohr-Sommerfeld approximation

Consider Schrödinger 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.

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}).
2. Furthermore, spacing between eigenvalues i.e. $E_{n+1}-E_n$ is $2\pi h F'(E) +O(h^2)$ where $F'(E)=\partial_E F(E)$.
3. Eigenfunctions are $e^{ih^{-1}S(x)}A_0(x)$ nodulo $O(h)$ uniformly on $[x_E^-+\varepsilon, x_E^+-\varepsilon]$ for any $\varepsilon >0$.
4. 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.

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 $E$.
2. All this holds for more general $1$-dimensional Hamiltonians. In particular, for $H'=F(H)$ we have similar results albeit with $F(E)=E$.
3. 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).
4. If $V'(x_E^+)=0$ (or/and $V'(x_E^-)=0$) then eigenvalue are more dense ear $E$.
5. 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 $h^{-\frac{1}{4}} \He_n (h^{-\frac{1}{2}}x)($ where $\He_n$ 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).

Quasieigenvalues and quasimodes

Question: Do quasieigenvalues approximate eigenfunctions and do quasimodes approximate eigenfunctions?

Sometimes yes, sometimes no...

Example 2. Consider 1D Schrödinger operator with potential like this (try to formulate precise assumptions)

Simple well

Then near level $E^*$ Bohr-Sommerfeld eigenvalues $e_n(h)$ are simple and also true eigenvalues are simple, distance between neighbouring eigenvalues is $E_{n+1}(h)-E_n(h) \asymp h$, and distance between Bohr-Sommerfeld eigenvalues is $e_{n+1}(h)-e_n(h) \asymp h$ and therefore Bohr-Sommerfeld quasieigenvalues provide a good approximation for eigenvalues \begin{gather} E_n(h)= e_n(h)+O(h^\infty). \label{eq-6.3.7} \end{gather} The same is true for corresponding eigenfumctions \begin{gather} \Psi_n(h)= \psi_n(h)+O(h^\infty). \label{eq-6.3.8} \end{gather}

Example 3. Consider 1D Schrödinger operator with potential like this (try to formulate precise assumptions)

Symmetric double well well

Then near level $E^*$ Bohr-Sommerfeld eigenvalues $e_n(h)$ are double and corresponding quasimodes are supported in the left or right segments.

However true eigenvalues are simple but paired $E'_n(h)$ and $e''_n(h)$, corresponding to functions $\Psi'_n(x)$ and $\Psi''_n(x)$ which are even and odd with respect to $x$, distance between neighbouring eigenvalues is $E'_{n+1}(h)-E'_n(h) \asymp h$, $E''_{n+1}(h)-E''_n(h) \asymp h$ but distance between eigenvalues in the same pair is $E''_{n}(h)-E'_n(h) = O(h^\infty)$ (more precisely $O(e^{-\kappa h^{-1}})$ and there is method to calculate $\kappa>0$. Connection between left and right segments is due to tunnelling.

Distance between Bohr-Sommerfeld eigenvalues is $e_{n+1}(h)-e_n(h) \asymp h$ and therefore Bohr-Sommerfeld quasieigenvalues provide a good approximation for eigenvalues \begin{gather} E'_n(h)= e_n(h)+O(h^\infty), \qquad E''_n(h)= e_n(h)+O(h^\infty). \label{eq-6.3.9} \end{gather} However quasimodes do not provide a good approximation for eigenfunctions.

Example 4. Consider 1D Schrödinger operator with potential like this (try to formulate precise assumptions)

Well on the island

Then near level $E^*$ Bohr-Sommerfeld eigenvalues $e_n(h)$ are simple.

But there are not true eigenvalues close to $E^*$, the spectrum here is continuous. However $e_n(h)$ are approximating resonances $E_n(h)$, $\Im E_n(h)>0$, which are studied in the Scattering theory and they correspond to quasistable states: if we sovle non-Stationary Schrödinger equation with initial data $\psi_n(h)$ then the solution $\Psi_n (t,h)$ decays albeit very slowly, "escaping to infinity" due to tunnelling.

\begin{gather} E_n(h)= e_n(h)+O(h^\infty). \label{eq-6.3.10} \end{gather}

Remark 2. One can combine these examples: f.e. in Example 4 consider well deeper than the "sea level"; then quasieigenvalues below sea level would approximate eigenvalues and quasieigenvalues above it will approximate resonances.

$\Leftarrow$  $\Uparrow$  $\Rightarrow$