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

Details Tunneling for spatially cut-off $P(φ)_2$-Hamiltonians Tunneling for spatially cut-off $P(φ)_2$-Hamiltonians

2011-04-04

arxiv.org/abs/1104.0486v5

Physics

Mathematical Physics

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Scientific paper

We study the asymptotic behavior of low-lying spectrum of spatially cut-off $P(\phi)_2$-Hamiltonian under semi-classical limit. The corresponding classical equation of the $P(\phi)_2$-field is a nonlinear Klein-Gordon equation. We determine the semi-classical limit of the lowest eigenvalue of the spatially cut-off $P(\phi)_2$-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. We give an upper bound on the convergence rate by an approximate Agmon distance between zero points of the potential function. When the space is a finite interval, we prove that the approximate Agmon distance is equal to the infinite dimensional analogue of the Agmon distance in the case of Schr\"odinger operators.

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