The problem of existence of elementary (= quasi-exact) multiplets of bound states is discussed.
Background: It is easy to assign a potential V(f) to any given elementary wavefunction f.
Idea: For the doublet of solutions f and g the two potentials must coincide, V(f)=V(g).
Conclusion: Our doublet- (or multiplet-) solvable class {V} will depend on our "solution form" {f}.
We discuss the feasibility of this constuction and its algebraic background and analytic applications.
You may click on the ps formatted full text description of these results as published in
J. Math. Phys. 33 (1992) 2785 - 94.
An alternative Riccati-like development of the same idea may be found in the Ushveridze's book (1994).