Trapping by repulsion: the NLS with a delta-prime

TL;DR

This paper analyzes the existence and explicit form of stationary states in the one-dimensional Schrödinger equation with a repulsive delta-prime potential and focusing nonlinearity, revealing novel ground state phenomena.

Contribution

It provides explicit expressions for stationary states and demonstrates the existence of nonlinear ground states arising from a repulsive potential, due to an emergent energy space dimension.

Findings

Ground states exist for any strength of the delta-prime interaction in the subcritical case.

Explicit formulas for ground states are derived across all regimes.

Repulsive potentials can produce nonlinear ground states, contrary to typical expectations.

Abstract

We establish the existence and provide explicit expressions for the stationary states of the one-dimensional Schr\"odinger equation with a repulsive delta-prime potential and a focusing nonlinearity of power type. Furthermore, we prove that, if the nonlinearity is subcritical, then ground states exist for any strength of the delta-prime interaction and for every positive value of the mass. This result supplies an example of ground states arising from a repulsive potential, a counterintuitive phenomenon explained by the emergence of an additional dimension in the energy space, induced by the delta-prime interaction. This new dimension contains states of lower energy and is thus responsible for the existence of nonlinear ground states that do not originate from linear eigenfunctions. The explicit form of the ground states is derived by addressing the ancillary problem of minimizing the action functional on the Nehari manifold. We solve such problem in the subcritical, critical, and supercritical regimes.