Consider the Schroedinger equation on the line $$i\frac{\partial \Psi(x,t)}{\partial t}=[-\frac{d^2}{dx^2}+V(x)]\Psi(x,t),$$ where one assumes that $V(x)\to +\infty$ as $|x|\to +\infty$, and $V$ has two local minima.

The case of symmetric potential, i.e. $V(-x)=V(x)$, seems to be discussed in physics textbooks using the WKB method (see e.g. the 3rd volume of the Landau-Lifshitz textbook on theoretical physics).

I am interested in the situation of non-symmetric potential $V$ especially when it takes different values at the two local minima (the global minimum is called the true vacuum, and the other one the fake vacuum). This case seems to be very different from the symmetric one. In physics literature it is often stated that if a wave function $\Psi(x,0)$ is localized near the fake vacuum, then after a long time $\Psi(x,t)$ will be localized near the true vacuum.

Is there a mathematically rigorous treatment of this problem?

REMARK. It seems that the above statement is not always true and it is very sensitive to the shape of $V$. In the paper "Resonances in quantum mechanical tunneling" by M.M.Nieto et al. in Phys.Lett.B (1985) what they claim is the following:

"In asymmetric double-well potentials, it can be tacitly assumed that a wave function in the higher-energy well (false vacuum) will always tunnel to the lower well, given enough time. However, in general this is not true. Whether a state can significantly tunnel to the true vacuum is a very sensitive function of the shape of the potential. We illustrate this with analytic and numerical examples. Thus, if there is not dissipation or coupling to other modes, a wave function may not tunnel."

The method of the paper seems to be not mathematically rigorous and to large extend numerical.

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