On the elementary Schrödinger bound states and their multiplets

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Abstract
The problem of the existence of elementary bound states is discussed. A−trivial−observation that every elementary wave function ψ[i](r) is an exact bound state for an appropriate potential, V(r)=V [i][ψ(r),r], is shown to lead to a very transparent form of the ‘‘quasiexact’’ (QE) solvability condition V [i]=V [j] for doublets and multiplets of the ψ’s. In this sense, the particular class of elementary ansätze, ψ[i](r)=r λpolynomial(r 2) ×exp[r μpolynomial(r 2)], also defines the particular class of QE‐solvable potentials. They have an elementary nonpolynomial (rational) form, possibly also with a strongly singular−repulsive−core at the origin. The properties of these forces are discussed in detail.