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We study the asymptotic behavior of low-lying eigenvalues of spatiallycut-off $P(\phi)_2$-Hamiltonian under semi-classical limit. The correspondingclassical equation of the $P(\phi)_2$-field is a nonlinear Klein-Gordonequation which is an infinite dimensional Newton's equation. We determine thesemi-classical limit of the lowest eigenvalue of the spatially cut-off$P(\phi)_2$-Hamiltonian in terms of the Hessian of the potential function ofthe Klein-Gordon equation. Moreover, we prove that the gap of the lowest twoeigenvalues goes to 0 exponentially fast under semi-classical limit when thepotential function is double well type. In fact, we prove that the exponentialdecay rate is greater than or equal to the Agmon distance between two zeropoints of the symmetric double well potential function. The Agmon distance is aRiemannian distance on the Sobolev space $H^{1/2}(\RR)$ defined by a Riemannianmetric which is formally conformal to $L^2$-metric. Also we study basicproperties of the Agmon distance and instanton.

Tunneling for spatially cut-off $P(\phi)_2$-Hamiltonians