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

We discuss the energy splitting ∆ǫ of nearly degenerate eigenstates which classically correspond to symmetrically distributed tori in phase space. We find that the naive expectation that the tunneling-induced splitting vanishes faster than any power of h̄ in semi-classical limit actually relies on certain smoothness assumption of the Hamiltonian. The quantum transition between the semi-classical eigenstates will be greatly enhanced when the corresponding degenerate tori in phase space are connected by line(s) where the Hamiltonian is not smooth. The leading term in semi-classical expansion of ∆ǫ is derived under the assumption that the the nonsmoothness depends only upon xor p-coordinate, which shows that ∆ǫ decays as h̄ when h̄ → 0 with k being the order of non-smoothness. We conjecture that the non-smoothness-enhanced transition and the resulting power-law decay of ∆ǫ are typical in non-smooth systems.