Splitting of the low-energy levels of the Schroedinger equation and the fundamental solution asymptotics of the equation hut = 1/2h2Δu - V(x)u

For the equation hδu/δt = h2Δu/2 - V(x)u with a positive potential V(x) a global exponential asymptotics of the fundamental solution is obtained by using the tunnel canonical operator method. For a potential with simple global minima the solution behaviour of the Cauchy problem at large times t = h-(1=x), x > 0 is studied. These results are used to get the exponential asymptotics for low eigenfunctions of the Schroedinger operator - 1/2h2Δu - V(x)u and estimates for tunneling effect