Many-Level Formula for Scattering from Extremely Strong Square Potentials

Abstract

A version of the many-level formula for scattering is introduced, considering the potential strength as the fundamental variable. It is applied to extremely strong square potentials. Under such circumstances, we must deal with a sum over a large number of nonresonant terms. The mathematical technique developed here will be directly applicable in the discussion of a singular potential scattering. Furthermore, McVoy, Heller, and Bolsterli have pointed out that a deep square well and a high barrier produce similar scattering amplitudes, in spite of the difference in sign. This intriguing effect is analyzed in the light of the many-level formula. We can understand it clearly in terms of the dominance of the orthogonality effect. It refers to such an effect as is a consequence of the fact that the wave function is orthogonal to any normalizable function introduced within the range of the square potential, which occurs if the potential strength tends to infinity. In the case of attraction, the depth of the square well must be outside the "width of a resonance" in order that this effect may dominate. We may define the relative probability of encountering a resonance with the ratio of the width of a resonance to the distance between two adjacent resonances. It tends to zero as the depth increases without limit.