  We consider the k-folded sine-Gordon model, obtained from the usual version
by identifying the scalar field after k periods of the cosine potential. We
examine (1) the ground state energy split, (2) the lowest lying multi-particle
state spectrum and (3) vacuum expectation values of local fields in finite
spatial volume, combining the Truncated Conformal Space Approach, the method of
the Destri-de Vega nonlinear integral equation (NLIE) and semiclassical
instanton calculations. We show that the predictions of all these different
methods are consistent with each other and in particular provide further
support for the NLIE method in the presence of a twist parameter. It turns out
that the model provides an optimal laboratory for examining instanton
contributions beyond the dilute instanton gas approximation. We also provide
evidence for the exact formula for the vacuum expectation values conjectured by
Lukyanov and Zamolodchikov.
