Geodesic
Splitting of the lowest energy levels of the Schr\"odinger equation and asymptotic behavior of the fundamental solution of the equation $hu_t=h^2\Delta u/2-V(x)u$
Teoretičeskaâ i matematičeskaâ fizika, Tome 87 (1991) no. 3, pp. 323-375

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For the equation $h\partial u/\partial t=h^2\Delta u/2-V(x)u$ with positive potential $V(x)$, global exponential asymptotic behavior of the fundamental solution is obtained by the method of the tunnel canonical operator. In the case of a potential with degenerate points of global minimum, the behavior of the solutions to the Cauchy problem is investigated at times of order $t=h^{-(1+\varkappa)}$, $\varkappa>0$. The developed theory is used to obtain exponential asymptotics of the lowest eigenfunctions of the Schrödinger operator $-h^2\Delta/2-V(x)$ and to estimate the tunnel effect. 
@article{TMF_1991_87_3_a0,
     author = {S. Yu. Dobrokhotov and V. N. Kolokoltsov and V. P. Maslov},
     title = {Splitting of the lowest energy levels of the {Schr\"odinger} equation and asymptotic behavior of the fundamental solution of the equation $hu_t=h^2\Delta u/2-V(x)u$},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {323--375},
     publisher = {mathdoc},
     volume = {87},
     number = {3},
     year = {1991},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1991_87_3_a0/}
}
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S. Yu. Dobrokhotov; V. N. Kolokoltsov; V. P. Maslov. Splitting of the lowest energy levels of the Schr\"odinger equation and asymptotic behavior of the fundamental solution of the equation $hu_t=h^2\Delta u/2-V(x)u$. Teoretičeskaâ i matematičeskaâ fizika, Tome 87 (1991) no. 3, pp. 323-375. http://geodesic.mathdoc.fr/item/TMF_1991_87_3_a0/