Consider the (quantum) Hamiltonian on the real line
$$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x).$$


Let us assume that the potential $V$ is an even smooth functions with exactly two non-degenerate minima,
and $\lim\_{|x|\to +\infty}V(x)=+\infty$. Such a $V$ is called double well potential.


Let $E\_1,E\_2$ be the first two minimal eigenvalues of $H$. It is known in physics literature (see problem 3 after $\S$ 50 in Landau-Lifshitz, vol. 3) that under some extra assumption which are not quite specified there one has $$E\_2-E\_1 \approx \frac{\omega\hbar}{\pi} \exp(-\frac{1}{\hbar} C), $$
where $\omega, C$ are positive constant which can be written down explicitly, and the result is understood asymptotically as $\hbar\to 0$.


**I am looking for a mathematically more rigorous discussion of this result where, in particular, the assumptions are formulated more explicitly.**



Energy levels of double well potential
This is either


Helffer-Sjostrand


<https://www.tandfonline.com/doi/abs/10.1080/03605308408820335>


or Barry Simon


<https://www.jstor.org/stable/pdf/2007072.pdf?refreqid=excelsior%3A258084917fff9e0c10088abbb2679c55>


PS: I cannot resist pointing out that the last paper is in Annals of Mathematics proving, as you notice yourself, an exercise in Landau/Lifshitz.


