  We consider the Klein-Gordon equation on analytic spacetimes with an analytic
Cauchy surface. In this setting, we prove the existence of pure analytic
Hadamard states. The proof is based on considering an elliptic operator
obtained by Wick rotating the Klein-Gordon operator in a neighborhood of a
Cauchy hypersurface. The Cauchy data of Hadamard two-point functions are
constructed as Calder\'{o}n projectors (suitably generalized if the
hypersurface is non-compact) for the elliptic operator.
