The attractive nonlinear delta-function potential

Details The attractive nonlinear delta-function potential The attractive nonlinear delta-function potential

2001-02-16

arxiv.org/abs/physics/0102053v1

Physics

Physics Education

submitted to Am. J. of Phys., sixteen pages, three figures

Scientific paper

10.1119/1.1417529

We solve the continuous one-dimensional Schr\"{o}dinger equation for the case of an inverted {\em nonlinear} delta-function potential located at the origin, obtaining the bound state in closed form as a function of the nonlinear exponent. The bound state probability profile decays exponentially away from the origin, with a profile width that increases monotonically with the nonlinear exponent, becoming an almost completely extended state when this approaches two. At an exponent value of two, the bound state suffers a discontinuous change to a delta-like profile. Further increase of the exponent increases again the width of the probability profile, although the bound state is proven to be stable only for exponents below two. The transmission of plane waves across the nonlinear delta potential increases monotonically with the nonlinearity exponent and is insensitive to the sign of its opacity.

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