On the existence of resonances in the transmission probability for interactions arising from derivatives of Dirac’s delta function. (English) Zbl 1047.81567

Summary: 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 \(\delta(x)\), are studied in the zero-range limit. Particularly, for a countable set of interaction strength values, a non-zero transmission through the point potential \(\delta'(x)\), 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 \(\delta''(x)\), 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.


81U05 \(2\)-body potential quantum scattering theory
35B34 Resonance in context of PDEs
46F10 Operations with distributions and generalized functions
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