  The Schr\"odinger equation with a harmonic potential coupled to the Poisson
equation, called the Schr\"odinger-Newton-Hooke (SNH) system, has been
considered in a variety of physical contexts, ranging from quantum mechanics to
general relativity. Our work is directly motivated by the fact that the SNH
system describes the nonrelativistic limit of the Einstein-massive-scalar
system with negative cosmological constant. With this paper we begin the
investigations aiming at understanding solutions of the SNH system in the
energy supercritical spatial dimensions $d\geq 7$, where we expect to observe
interesting short wavelength behaviours due to the confinement of waves by the
trapping potential. Here we study stationary solutions and prove existence of
one-parameter families of nonlinear ground and excited states. The frequency of
the ground state as the function of the central density is shown to exhibit
different qualitative behaviours in dimensions $7\leq d\leq 15$ and $d\geq 16$,
which is expected to affect the stability properties of the ground states in
these dimensions. Our results bear many similarities to the analogous problem
that has been studied for the Gross-Pitaevskii equation.
