Properties of the Wavefunction for Singular Potentials

Abstract

For potentials more singular than the inverse square at the origin and with complex strength, the nature of the solutions to the Schrödinger equation is investigated. The difficulties occurring with attractive real potentials in the three-dimensional equation are discussed, and it is shown that no solutions exist in a domain including the origin. For complex or repulsive singular potentials with radial form varying as the inverse fourth or sixth power, a relatively simple series solution exists. This is also true for the singular Yukawa with inverse fourth power. These series are shown to be asymptotic by means of general theorems from the theory of ordinary differential equations. Other inverse powers have much more complicated solutions. In the Dirac case, all inverse powers and singular Yukawa forms have asymptotic series solutions, and the series is given explicitly for the singular powers. A form of the S-wave scattering length for the singular inverse fourth-power Yukawa V = g2e−iΔe−μr/r4 is derived which is valid for small values of μg.