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

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.