## Spherically-symmetric solutions of the Schrödinger-Newton equations.(English)Zbl 0936.83037

The goal of the authors is to understand the quantum state reduction as a gravitational phenomenon in the framework of the Newtonian gravitation. For this they consider the Schrödinger-Newton equations. This is a coupled system consisting of the Schrödinger equation for a particle moving in its own gravitational field, where this is generated by its own probability density via the Poisson equation. Using both numerical and analytic investigations, the authors find a discrete family of spherically-symmetric solutions of these equations, everywhere regular, and with normalizable wavefunctions. The solutions are labelled by the non-negative integers, the $$n$$th solution having $$n$$ zeros in the wavefunction. These solutions are the only globally defined spherically-symmetric solutions.

### MSC:

 83C99 General relativity 81T20 Quantum field theory on curved space or space-time backgrounds 83E05 Geometrodynamics and the holographic principle
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