Consider a model for a spin chain. I somehow am able to find a general formula for the expectation value of some observable in both periodic and open boundary conditions. ie., under PBC, I have
$\langle O \rangle_{PBC} = f(\{\alpha\},L)$
and under OBC, I have
$\langle O \rangle_{OBC} = g(\{\alpha\},L)$
where $\{\alpha\}$ is the set of other parameters of the model. Now if I am interested in the thermodynamic limit, is it always the case that the limits of the above two functions are the same?
$\lim_{L\rightarrow\infty}f(\{\alpha\},L) = lim_{L\rightarrow\infty}g(\{\alpha\},L)$?
Furthermore, is there any general way to relate the two functions $f,g$ to eachother?
