Nonperturbative vacuum energy density in two-dimensional scalar models

An upper bound that is uniform with respect to the coupling constant $g$ and the field is obtained for the effective potential for a two-dimensional scalar field theory with arbitrary self-interaction. The “nonexistence” of the :$\cos\alpha\varphi$: and :$\varphi^{2N}\exp\alpha\varphi$: models for $\alpha^2\geq 8\pi$ is proved. Exact asymptotic behaviors with respect to $g$ are found for the vacuum energy density for the $P(\varphi)_2$ and Hoegh-Krohn :$\exp\alpha\varphi$: models, and also for the total propagator at zero momentum.