  We develop an algorithm which can be used to exclude the existence of
classical breathers (periodic finite energy solutions) in scalar field
theories, and apply it to several cases of interest. In particular, the
technique is used to show that a pair of potentially periodic solutions of the
3+1 Sine-Gordon Lagrangian, found numerically in earlier work, are not
breathers. These ``pseudo-breather states'' do have a signature in our method,
which we suggest can be used to find similar quasi-bound state configurations
in other theories. We also discuss the results of our algorithm when applied to
the 1+1 Sine-Gordon model (which exhibits a well-known set of breathers), and
$\phi ^4$ theory.
