Observing Quantum Tunneling in Perturbation Series

We apply Borel resummation method to the conventional perturbation series of ground state energy in a metastable potential, V (x) = x/2− gx/4. We observe numerically that the discontinuity of Borel transform reproduces the imaginary part of energy eigenvalue, i.e., total decay width due to the quantum tunneling. The agreement with the exact numerical value is remarkable in the whole tunneling regime 0 < g < ̃ 0.7. ⋆ e-mail: [email protected] † e-mail: [email protected] The quantum tunneling is a purely non-perturbative phenomenon: This phrase has been widely accepted. Let us take a simple quantum mechanical example H = p 2 + 1 2 x − g 4 x. (1) Since the ground state is metastable in this potential, the eigenvalue is defined by the analytic continuation from g < 0. Equivalently, the Schrödinger equation is defined with a rotated boundary condition, ψ(x) → 0 for x = eπi/6t with t → ±∞ [1], provided that the “escape out” solution is taken. With this boundary condition, the semi-classical (WKB) calculation [1] gives the imaginary part of energy eigenvalue for g ≪ 1, ImE(g) ∼ − √ 8 πg exp ( − 4 3g )[ 1− 95 96 g +O(g) ] . (2) This is related to the total decay width Γ due to the quantum tunneling, ‡ Γ = −2ImE(g). The semi-classical result of the (2) vanishes to all order of the expansion on g, i.e., the tunneling effect is invisible in a simple expansion on the coupling constant. On the other hand, the conventional Rayleigh–Schrödinger perturbation series of ground state energy is given by