  The familiar $c\rightarrow \infty$ nonrelativistic limit converts the
Klein-Gordon equation in Minkowski spacetime to the free Schroedinger equation,
and the Einstein-massive-scalar system without a cosmological constant to the
Schroedinger-Newton (SN) equation. In this paper, motivated by the problem of
stability of Anti-de Sitter (AdS) spacetime, we examine how this limit is
affected by the presence of a negative cosmological constant $\Lambda$.
Assuming for consistency that the product $\Lambda c^2$ tends to a negative
constant as $c\rightarrow \infty$, we show that the corresponding
nonrelativistic limit is given by the SN system with an external harmonic
potential which we call the Schrodinger-Newton-Hooke (SNH) system. We then
derive the resonant approximation which captures the dynamics of small
amplitude spherically symmetric solutions of the SNH system. This resonant
system turns out to be much simpler than its general-relativistic version,
which makes it amenable to analytic methods. Specifically, in four spatial
dimensions, we show that the resonant system possesses a three-dimensional
invariant subspace on which the dynamics is completely integrable and hence can
be solved analytically. The evolution of the two-lowest-mode initial data (an
extensively studied case for the original general-relativistic system), in
particular, is described by this family of solutions.
