Journal article
On the existence of resonances in the transmission probability for interactions arising from derivatives of Dirac's delta function
The scattering properties of regularizing finite-range potentials constructed in the form of squeezed rectangles, which approximate the first and second derivatives of the Dirac delta function δ(x), are studied in the zero-range limit. Particularly, for a countable set of interaction strength values, a non-zero transmission through the point potential δ′(x), defined as the weak limit (in the standard sense of distributions) of a special dipole-like sequence of rectangles, is shown to exist when the rectangles are squeezed to zero width.
A tripole sequence of rectangles, which gives in the weak limit the distribution δ′′(x), is demonstrated to exhibit the total transmission for a countable sequence of the rectangle's width that tends to zero. However, this tripole sequence does not admit a well-defined point interaction in the zero-range limit, making sense only for a finite range of the regularizing rectangular-like potentials.
Language: | English |
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Year: | 2003 |
Pages: | 7589-7600 |
ISSN: | 13616447 and 03054470 |
Types: | Journal article |
DOI: | 10.1088/0305-4470/36/27/311 |
ORCIDs: | Christiansen, Peter Leth |