Schrödinger equation with Coulomb potential admits no exact solutions

Analytical solutions of Schrödinger equation for systems including three particles or more are strongly desirable. However, here we show by mathematical induction that the exact solutions do not exist if the potential energy is expressed by the Coulomb potential. This is currently the most universally used potential. In an ongoing project we are working to find a mathematically derived potential for which solutions exist. [Display omitted] •Exact solutions for Schrödinger equation are known only for few simple cases.•Coulomb potential is most commonly used to represent potential energy.•We prove that solutions do not exist for three bodies with Coulomb potential.•This opens the way for other potentials to be used. We formally prove that Schrödinger equation with Coulomb potential has no nontrivial exact solutions for three or more unrestrained particles. New potentials are needed to solve this issue. This is the subject of an ongoing project. The proof technique is to logically embed three loops of mathematical induction into each other. Our work is an expansion of a work by Bartlett. It follows that the solutions proposed by Fock are not exact, although accurate to 40 digits.