Upper bound for the partition function of $P(\varphi)_2$ Euclidean field theory with Dirichlet boundary conditions

The $P(\varphi)_2$ Euclidean (quantum) field theory on a bounded interval with zero-value boundary conditions is considered. An asymptotic representation of the partition function in terms of the partition function of the free field and a factor that depends on the interaction is discussed. The hypothesis is partly justified. Namely, an upper bound is obtained for the partition function in terms of the right-hand side of the asymptotic expression.