Springer Online Journal Archives 1860-2000
Abstract We study the real, Euclidean, classical field equation $$(\mu ^2 + \Delta )\varphi + \lambda F(\varphi ) = f,\mu ^2 〉 0$$ where φ: ℝ d →ℝ is suitably small at infinity. We study existence and regularity assuming that λ≧0,F∈C ∞(ℝ), andaF(a)≧0∀a∈∝. These hypotheses allow strongly nonlinearF and nonunique solutions forf≠0. WhenF′≧0, we prove uniqueness, various contractivity properties, analytic dependence on the coupling constant λ, and differentiability in the external sourcef. For applications in the loop expansion in quantum field theory, it is useful to know that φ is in the Schwartz classL wheneverf is, and we provide a proof of this fact. The technical innovations of the problem lie in treating the noncompactness of R d , the strong nonlinearity ofF, and the polynomial weights in the seminorms definingL.
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