Tunneling for spatially cut-off $P(\varphi {)}_2$-Hamiltonians

We study the asymptotic behavior of low-lying eigenvalues of spatially cut-off $P(\varphi {)}_2$-Hamiltonian under semi-classical limit. The corresponding classical equation of the $P(\varphi {)}_2$-field is a nonlinear Klein-Gordon equation which is an infinite dimensional Newton’s equation. We determine the semi-classical limit of the lowest eigenvalue of the spatially cut-off $P(\varphi {)}_2$-Hamiltonian in terms of the Hessian of the potential function of the Klein-Gordon equation. Moreover, we prove that the gap of the lowest two eigenvalues goes to 0 exponentially fast under 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. The Agmon distance is a Riemannian distance on the Sobolev space $H^{1\mathrm{/}2}(\text{RR})$ defined by a Riemannian metric which is formally conformal to $L^2$-metric. Also we study basic properties of the Agmon distance and instanton.