WKB in dimension 1. 3

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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 $h^{-\frac{1}{4}} e_n (h^{-\frac{1}{2}}($ 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).

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