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The recently observed quantum states of neutrons bound in a gravitationalfield are analyzed in the framework of one-parameter isospectral hamiltonians.Potentials isospectral to the usual Newton potential are explicitly constructedfor the first time, then constrained using measured properties of the neutrongravitational states. The corresponding wave functions and the neutron fluxesare also calculated and analyzed in a simple model, including the ground stateand the excited state contributions. The constructed isospectral potentials arediscussed as candidates for a possible modification of Newton's law at asubmillimetre scale. Our results indicate that significant deviations from theNewtonian gravity at submillimetre distances could be compatible withexperiment.

Interpretation of the neutron quantum gravitational states in terms of isospectral potentials