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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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    $\begingroup$ On my opinion, this question (and other questions of the type "how to state the problem in mathematical language") belong to the physicsSE. $\endgroup$ May 20, 2018 at 15:05
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    $\begingroup$ @AlexandreEremenko: It is unlikely to get there a reference to a mathematically rigorous treatment. Moreover a few days ago I asked there a similar question in a slightly different form, but did not get a satisfactory answer. The answer by Carlo Beenakker below I got here is interesting and relevant (but probably not final). $\endgroup$
    – asv
    May 20, 2018 at 15:13
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    $\begingroup$ The problem is that your QUESTION is not stated in mathematical language. If you can state it as a mathematical question, you can probably get an answer in this site. $\endgroup$ May 20, 2018 at 15:17
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    $\begingroup$ This is a part of my question - how to state it mathematically correctly. It might be a non-trivial problem. I am unable to do that, and it is unlikely to get such a statement on a purely physical site. $\endgroup$
    – asv
    May 20, 2018 at 15:27
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    $\begingroup$ There are of course gazillions of (rigorous) mathematical papers on these general topics, but your question is much too vague in this form. For example, if the initial $\psi$ is an eigenfunction, then the time evolution becomes trivial and obviously none of the phenomena you describe can occur. $\endgroup$ May 20, 2018 at 19:40

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Tunnel splitting of the spectrum and bilocalization of eigenfunctions in an asymmetric double well has a "theorem" on the bilocalization phenomenon (wave function localized in both asymmetric wells); The theorem might satisfy some requirements of rigor...

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