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The Schr\"odinger equation with a harmonic potential coupled to the Poissonequation, called the Schr\"odinger-Newton-Hooke (SNH) system, has beenconsidered in a variety of physical contexts, ranging from quantum mechanics togeneral relativity. Our work is directly motivated by the fact that the SNHsystem describes the nonrelativistic limit of the Einstein-massive-scalarsystem with negative cosmological constant. With this paper we begin theinvestigations aiming at understanding solutions of the SNH system in theenergy supercritical spatial dimensions $d\geq 7$, where we expect to observeinteresting short wavelength behaviours due to the confinement of waves by thetrapping potential. Here we study stationary solutions and prove existence ofone-parameter families of nonlinear ground and excited states. The frequency ofthe ground state as the function of the central density is shown to exhibitdifferent 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 inthese dimensions. Our results bear many similarities to the analogous problemthat has been studied for the Gross-Pitaevskii equation.

Schr\"odinger-Newton-Hooke system in higher dimensions. Part I: Stationary states