The authors consider the problem of existence of stationary solutions to a one-dimensional nonlinear Schrödinger equation of the form
with a bounded smooth potential V, , h small. They show that for every critical point of V there is an such that for all one has a nontrivial stationary solution to (1) which lies in a neighborhood of a translate around the critical point of the ground state solution of the free equation (that is equation (1) without potential); this means that the solution is concentrated near and in some sense nonspreading. The method the authors use in their proof is inspired by procedures to find instanton solutions for the Yang-Mills equation, namely a Lyapunov-Schmidt reduction to finite dimensions combined with a crucial argument using the nondegeneracy of the ground state (i.e. the kernel of the Frechet derivative at of the nonlinear operator belonging to (1) should be spanned by the components of grad .