# Energy levels of double well potential

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.

• I take it you mean $\lim_{|x|\to\infty}V(x)=+\infty$ in your definition of a double-well potential, right?
– gmvh
Apr 25, 2021 at 10:42
• Oops... Corrected. Thanks.
– makt
Apr 25, 2021 at 11:07
• I presume you mean the WKB approximation for the level splitting? Apr 25, 2021 at 13:22
• I don't understand the term "nondegenerate". That usually means that there are two things that are not equal (possibly the values of $V$ at the two minima in the present context). Since $V$ is an even function, I suspect it's more likely that something is degenerate here as opposed to nondegenerate? Apr 25, 2021 at 15:01
• @MichaelEngelhardt : By non-degenerate minima I mean that the second derivatives at the minima is positive.
– makt
Apr 25, 2021 at 18:02

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.

Double wells, by Evans M. Harrell, Comm. Math. Phys. 75, 239 (1980), should be sufficiently rigorous. The result for the tunnel splitting, theorem 2.22, applies to a symmetric double-well potential in arbitrary number $$n$$ of dimensions, constructed as the sum $$V(x-f)+V(x+f)$$ of two single-well potentials $$V(x)$$, bounded and of compact support, after translation by $$\pm f$$.

The tunnel splitting is given for large $$f$$ in terms of the eigenfunction $$\Phi(x)$$ for the lowest eigenstate in the potential $$V(x)$$, $$E_1-E_2=\left(\int\Phi(x-f)[V(x-f)+V(x+f)]\Phi(x+f)\,dx\right)(1+o(f^{-n})).$$

• In my question the potential is symmetric (even). By non-degenerate I meant that the second derivative at the minima is positive.
– makt
Apr 25, 2021 at 18:06
• I think from a formal point of view, the problem here is that $h\to 0$ in the OP doesn't in an obvious way correspond to pushing the wells apart by sending $f\to\infty$. Rather, it seems equivalent to a rescaling $V(x)\to V(hx)$ of a potential (which does push them out to infinity, but also changes their shape). Apr 25, 2021 at 22:12
• @ChristianRemling --- true, thanks for pointing this out; perhaps from the physics point of view a scaling with the separation of the wells is somewhat more natural than a scaling with Planck's constant... Apr 26, 2021 at 8:15