In the 3rd volume of the Landau-Lifschitz text book ("Quantum mechanics") problem 3 after section 50 studies the Schroedinger equation on the line with symmetric double well potential, e.g. $U(x)=g(x^2-a^2)^2$ with $g,a>0$. In particular they estimate the difference between the first two lowest energy levels using the method of quasi-classical approximation. (The final result is not essential to my question, nevertheless see p. 184 in the 3rd edition of the book.)
As far as I understand, the essential point in the argument was that the potential is symmetric, i.e. $U(-x)=U(x)$. Apparently it is even more important that the potential $U$ has the same value in the two local minima (which are also the global minima).
I am wondering what happens in the case of the non-symmetric double well potential which has two local minima and assumes at them different values. For example I would be interested to see a proof of the following result I heard: if the wave function of a particle is localized near the local minimum where $U$ is larger (called a fake vacuum or metastable state) then after a long time the particle will be localized near the other local minimum (the true vacuum) due to tunneling.
I heard that metastable states are studied sometimes using eigen-functions of the Hamiltonian with non-real eigenvalues. A discussion of that would be of particular interest to me as well as a comparison with the standard approach using the usual spectrum of the Hamiltonian.