Power-Law Energy Splitting Generated By Tunneling Between Non-smooth Tori

We discuss the energy level splitting $\Delta\epsilon$ due to quantum tunneling between congruent tori in phase space. In analytic cases, it is well known that $\Delta\epsilon$ decays faster than power of $\hbar$ in the semi-classical limit. This is not true in non-smooth cases, specifically, when the tori are connected by line on which the Hamiltonian is not smooth. Under the assumption that the non-smoothness depends only upon the x- or p-coordinate, the leading term in the semi-classical expansion of $\Delta\epsilon$ is derived, which shows that $\Delta \epsilon$ decays as $\hbar^{k+1}$ when $\hbar\to 0$ with k being the order of non-smoothness.