  We review the construction of ground states focusing on a real scalar field
whose dynamics is ruled by the Klein-Gordon equation on a large class of static
spacetimes. As in the analysis of the classical equations of motion, when
enough isometries are present, via a mode expansion the construction of
two-point correlation functions boils down to solving a second order, ordinary
differential equation on an interval of the real line. Using the language of
Sturm-Liouville theory, most compelling is the scenario when one endpoint of
such interval is classified as a limit circle, as it often happens when one is
working on globally hyperbolic spacetimes with a timelike boundary. In this
case, beyond initial data, one needs to specify a boundary condition both to
have a well-defined classical dynamics and to select a corresponding ground
state. Here, we take into account boundary conditions of Robin type by using
well-known results from Sturm-Liouville theory, but we go beyond the existing
literature by exploring an unnoticed freedom that emerges from the intrinsic
arbitrariness of secondary solutions at a limit circle endpoint. Accordingly,
we show that infinitely many one-parameter families of sensible dynamics are
admissible. In other words, we emphasize that physical constraints guaranteeing
the construction of full-fledged ground states do not, in general, fix one such
state unambiguously. In addition, we provide, in full detail, an example on $(1
+ 1)$-half Minkowski spacetime to spell out the rationale in a specific
scenario where analytic formulae can be obtained.
