Hypercontractive semigroups and two dimensional self-coupled Bose fields

Abstract

We present an abstract perturbation theory for operators of the form H0 + V obeying four properties: (1) H0 is a positive self-adjoint operator on L2(M, μ) with μ a probability measure so that e-tH0 is a contraction on L1 for each t > 0; (2) e-TH0 is a bounded map of L2 to L4 for some T; (3) V ε{lunate} Lp(M, μ) for some p > 2; (4) e-tV ε{lunate} L1 for all t > 0. We then show that spatially cutoff Bose fields in two-dimensional space-time fit into this framework. Finally, we discuss some details of two-dimensional Bose fields in the abstract including coupling constant analyticity in the spatially cutoff case. © 1972.