Tunneling for spatially cut-off P(ϕ)2-Hamiltonians

We study the asymptotic behavior of low-lying eigenvalues of spatially cut-off P(ϕ)2-Hamiltonian in the semi-classical limit. We determine the semi-classical limit of the lowest eigenvalue of the Hamiltonian in terms of the Hessian of the potential function of the corresponding classical equation. Moreover, we prove that the gap of the lowest two eigenvalues goes to 0 exponentially fast in the semi-classical limit when the potential function is double well type. In fact, we prove that the exponential decay rate is greater than or equal to the Agmon distance between two zero points of the symmetric double well potential function. Also we study basic properties of the Agmon distance and instanton.