  In this paper we present sets of linear integral equations which make
possible to compute the finite volume expectation values of the trace of the
stress energy tensor ($\Theta$) and the $U(1)$ current ($J_\mu$) in any
eigenstate of the Hamiltonian of the sine-Gordon model. The solution of these
equations in the large volume limit allows one to get exact analytical formulas
for the expectation values in the Bethe-Yang limit. These analytical formulas
are used to test an earlier conjecture for the Bethe-Yang limit of expectation
values in non-diagonally scattering theories. The analytical tests have been
carried out upto three particle states and gave agreement with the conjectured
formula, provided the definition of polarized symmetric diagonal form-factors
is modified appropriately. Nevertheless, we point out that our results provide
only a partial confirmation of the conjecture and further investigations are
necessary to fully determine its validity. The most important missing piece in
the confirmation is the mathematical proof of the finiteness of the symmetric
diagonal limit of form-factors in a non-diagonally scattering theory.
