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Given two spherically symmetric and short range potentials $V_0$ and V_1 forwhich the radial Schrodinger equation can be solved explicitely at zero energy,we show how to construct a new potential $V$ for which the radial equation canagain be solved explicitely at zero energy. The new potential and itscorresponding wave function are given explicitely in terms of V_0 and V_1, andtheir corresponding wave functions \phi_0 and \phi_1. V_0 must be such that itsustains no bound states (either repulsive, or attractive but weak). However,V_1 can sustain any (finite) number of bound states. The new potential V hasthe same number of bound states, by construction, but the corresponding(negative) energies are, of course, different. Once this is achieved, one canstart then from V_0 and V, and construct a new potential \bar{V} for which theradial equation is again solvable explicitely. And the process can be repeatedindefinitely. We exhibit first the construction, and the proof of its validity,for regular short range potentials, i.e. those for which rV_0(r) and rV_1(r)are L^1 at the origin. It is then seen that the construction extendsautomatically to potentials which are singular at r= 0. It can also be extendedto V_0 long range (Coulomb, etc.). We give finally several explicit examples.Comment: 26 pages, 3 figure

New classes of potentials for which the radial Schrödinger equation can be solved at zero energy