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 $\lim_{|x|\to +\infty}\Psi(x)=+\infty$. 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.

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.