The attractive nonlinear delta-function potential

We solve the one-dimensional nonlinear Schrodinger equation for an attractive delta-function potential at the origin, [(p(2)/2m)- Omega delta (x)\phi (x)\(alpha)]phi (x)=E phi (x), and obtain the bound state in closed form as a function of the nonlinear exponent a. The bound state probability profile decays exponentially away from the origin, with a profile width that increases monotonically with alpha, becoming an almost completely extended state when alpha-->2(-). At alpha =2, the bound state suffers a discontinuous change to a delta function-like profile. A further increase of a increases the width of the probability profile, although the bound state is stable only for alpha <2. The transmission of plane waves across the potential increases monotonically with <alpha> and is insensitive to the sign of the opacity Omega. (C) 2002 American Association of Physics Teachers.