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Semi-classical limit of the lowest eigenvalue of $$P(\phi)_2$$ Hamiltonian on a finite interval. (English) Zbl 1218.81048
Hara, Takashi (ed.) et al., Mathematical quantum field theory and renormalization theory. The Nishijin Plaza of Kyushu University, Fukuoka, Japan, November 26–29, 2009. Dedicated to Izumi Ojima and Kei-ichi Ito on the occasion of their 60th birthday. Fukuoka: Kyushu University, Faculty of Mathematics. COE Lecture Note 30, 114-126 (2011).
Introduction: Spatially cut-off $$P(\phi)_2$$-Hamiltonian is used to construct a non-trivial scalar quantum field [B. Simon, The $$P(\phi )_{2}$$ Euclidean (quantum) field theory. Princeton, NJ: Princeton University Press (1974; Zbl 1175.81146); J. Dereziński and C. Gérard, Commun. Math. Phys. 213, No. 1, 39–125 (2000; Zbl 1082.81518)]. The Hamiltonian contains a small physical parameter which is called the Planck constant $$\hbar$$. The classical field equation which is associated with the $$P(\phi)_2$$ quantum field is a non-linear Klein-Gordon equation. It is natural to guess that the asymptotics of spectrum of the spatially cut-off $$P(\phi)_2$$-Hamiltonian is determined by the corresponding classical system. In this paper, we discuss the semi-classical limit of the lowest eigenvalue of the $$P(\phi)_2$$-Hamiltonian in the case where the space is $$[-l/2,l/2]$$, where $$l>0$$. The contents of this report is based on [the author, J. Funct. Anal. 256, No. 10, 3342–3367 (2009; Zbl 1179.81075)].
For the entire collection see [Zbl 1207.81005].
##### MSC:
 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory 81T10 Model quantum field theories 81T08 Constructive quantum field theory