FIELD HAMILTONIAN FOR COMPLEX COUPLING CONSTANT(l)

We consider hamiltonians HB=Ho+ ,3HI(g),where Ho is the hamiltonian of a free Bose field $(x) of mass m> 0 in two-dimensional space-time, H I ( g ) = jg(x): P($(x)): dx where geO is a spatial cutoff and P is an arbitrary polynomial which is bounded below, and the coupling constant ,3 is in the cut plane, i.e. /3# negative real. We show that HB generates a semigroup with hypercontractive properties and satisfies higher order estimates of the form I H o N r R ; <co, where N is the number operator, R,=(H,+b)-l , r a positive integer, and ,3, s, and b are suitably chosen. For any 0 5 @ < P, RB converges in norm to Ro as +0 with larg ,3I (= 0.Finally we discuss applications of these results and establish asymptotic series and Borel summability for various objects in the real ,3 theory.