Semiclassical analysis of low lying eigenvalues. II. Tunnelling


We discuss the leading asymptotics of eigenvalue splittings of $-\frac{1}{2}\Delta + \lambda^2V$ in the limit as $\lambda \to\infty$, and where $V$ is a non-negative potential with several zeros. For example, if $E_0(\lambda)$, $E_1(\lambda)$ are the two lowest eigenvalues in a situation where $V$ has precisely two zeros, $a$ and $b$, related by a symmetry, then $\lim_{\lambda\to \infty} – (\lambda)^{-1} \mathrm{ln}[E_1(\lambda)-E_0(\lambda)]$ is given as the distance from $a$ to $b$ in a certain Riemann metric.


Barry Simon