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JOURNALS // Teoreticheskaya i Matematicheskaya Fizika // Archive

TMF, 1991 Volume 87, Number 3, Pages 323–375 (Mi tmf5495)

This article is cited in 17 papers

Splitting of the lowest energy levels of the Schrödinger equation and asymptotic behavior of the fundamental solution of the equation $hu_t=h^2\Delta u/2-V(x)u$

S. Yu. Dobrokhotov, V. N. Kolokoltsov, V. P. Maslov


Abstract: 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.

Received: 29.12.1990


 English version:
Theoretical and Mathematical Physics, 1991, 87:3, 561–599

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