EXACT, E=0, QUANTUM SOLUTIONS FOR GENERAL POWER-LAW POTENTIALS

Abstract

For zero energy, E=0, we derive exact, quantum solutions for all power-law potentials, V(r)=−γ/rν, with γ>0 and −∞<ν<∞. The solutions are, in general, Bessel functions of powers of r. For ν>2 and l≥1 the solutions are normalizable. Surprisingly, the solutions for ν<−2, which correspond to highly repulsive potentials, are also normalizable, for all l≥0. For these |ν|>2 the partial-wave Hamiltonians, Hl, have overcomplete sets of normalizable eigensolutions. We discuss how to obtain self-adjoint extensions of Hl such that the above E=0 solutions become included in their domains. When 2>ν≥−2 the E=0 solutions are not square-integrable. The ν=2 solutions are also unnormalizable, but are exceptional solutions. We also find that, by increasing the dimension of the Schrödinger equation beyond 4, an effective centrifugal barrier is created which is sufficient to cause binding when E=0 and ν>2, even for l=0. We discuss the physics of the above solutions and compare them to the corresponding classical solutions, which are derived elsewhere.