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RESEARCH PRODUCT

On Scattering and Bound States for a Singular Potential

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To understand the origin of the difficulties in the determination of the physical wavefunc­ tion for an attractive inverse square potential, we study a model in which the singularity at the origin is substituted by a repulsive core. The structure of the spectrum of energy levels is discussed in some detail. The physical interpretation of the solutions of the Schrodinger equation for a potential of the form - (-h 2 /2m) 11/ r 2 presents difficulties, which occur for 11 larger than (l + 1/2)\ where l is the angular momentum. The difficulties are due to the fact that the condition of square integrability usually imposed on the wavefunction is not sufficient in this case to determine phase shifts or energies of bound states. For strongly attractive singular potentials, as in the case of the attractive inverse square potential, both solutions of the wave equation are square integrable, so that Von Neumann's criterium fails to determine a unique solution. A solution can be built which is finite for r~oo, but it will be finite for every value of the energy from zero to minus infinity, so that no discrete energy spectrum can be determined. The problem of the determination of the energies of the bound states for the inverse square potential has been discussed by several authors. Some of these authors postulate additional conditions on the wavefunctions representing physical states so as to obtain a discrete energy spectrum. Shortley 1 ) discussed the quantum mechanical problem as related to the classical case, and did not try to eliminate the absence of quantization of the energy levels. Landau and Lif­ shitz2) showed how the lack of quantization is related to the singularity of the potential at the origin. Recently Guggenheim, 3 ) based on simple arguments of dimensional analysis, showed how it is impossible to define uniquely energy levels for the attractive inverse square potential if no extra parameter is pro­ vided. Case, 4 ) besides assuming the square integrability of the wavefunctions, introduced the extra condition that the wavefunctions representing physical states form an orthogonal set. This assumption is sufficient to define the spacing of