Abstract
For an exponentially decaying potential, analytic structure of the s-wave S-matrix can be determined up to the slightest detail, including position of all its poles and their residues. Beautiful hidden structures can be revealed by its domain coloring. A fundamental property of the S-matrix is that any bound state corresponds to a pole of the S-matrix on the physical sheet of the complex energy plane. For a repulsive exponentially decaying potential, none of infinite number of poles of the s-wave S-matrix on the physical sheet corresponds to any physical state. On the second sheet of the complex energy plane, the S-matrix has infinite number of poles corresponding to virtual states and a finite number of poles corresponding to complementary pairs of resonances and anti-resonances. The origin of redundant poles and zeros is confirmed to be related to peculiarities of analytic continuation of a parameter of two linearly independent analytic functions. The overall contribution of redundant poles to the asymptotic completeness relation, provided that the residue theorem can be applied, is determined to be an oscillating function.
Highlights
There have been known for long time many exactly solvable models [1], yet it has remained rare that one can determine complete analytic structure of the S-matrix
It is rather that the exponentially decaying potential has been at the cornerstone of non-relativistic quantum scattering [2, 8,9,10,11,12,13], because it enables to analytically determine the S-matrix. Thereby it provides a window into the class of potentials of infinite range
Analytic structure of the S-matrix can be determined up to the finest detail, including position of all its poles and their residues. (Note in passing that even if analytic form of the S-matrix is known, a complete determination of the position of all its poles and their residues is a nontrivial task, as is the case here.) Beautiful hidden structures can be revealed by its domain coloring
Summary
There have been known for long time many exactly solvable models [1], yet it has remained rare that one can determine complete analytic structure of the S-matrix. The redundant poles and zeros do not correspond to any bound state, half-bound state, (anti)resonance, or a virtual state [5,6,7] This bears important consequences for relating analytic properties of the S-matrix to physical states. It is rather that the exponentially decaying potential has been at the cornerstone of non-relativistic quantum scattering [2, 8,9,10,11,12,13], because it enables to analytically determine the S-matrix Thereby it provides a window into the class of potentials of infinite range. The repulsive example turns out to be rather extreme example in that the resulting S-matrix (12) will be shown to have infinite number of redundant poles on the physical sheet in the complex energy plane without a single bound state. We end up with discussion of a number of important issues and conclusions
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