NLS ground states on the half-line with point interactions

TL;DR

This paper studies the existence and uniqueness of nonlinear Schrödinger (NLS) ground states on the half-line with point interactions, analyzing how attraction or repulsion and the power of nonlinearity affect their existence across different regimes.

Contribution

It provides a comprehensive analysis of ground state existence and properties for NLS with point interactions on the half-line, covering subcritical and critical regimes, and distinguishes effects of attractive versus repulsive interactions.

Findings

Ground states exist for all masses with attractive interactions in subcritical case.

Ground states only exist for large masses with repulsive interactions in subcritical case.

In the critical case, ground states exist below a critical mass in the attractive case, but not in the repulsive case.

Abstract

We investigate the existence and the uniqueness of NLS ground states of fixed mass on the half-line in the presence of a point interaction at the origin. The nonlinearity is of power type, and the regime is either $L^2$-subcritical or $L^{2}$-critical, while the point interaction is either attractive or repulsive. In the $L^{2}$-subcritical case, we prove that ground states exist for every mass value if the interaction is attractive, while ground states exist only for sufficiently large masses if the interaction is repulsive. In the latter case, if the power is less or equal to four, ground states coincide with the only bound state. If instead, the power is greater than four, then there are values of the mass for which two bound states exist, and neither of the two is a ground state, and values of the mass for which two bound states exist, and one of them is a ground state. In the $L^{2}$-critical case, we prove that ground states exist for masses strictly below a critical mass value in the attractive case, while ground states never exist in the repulsive case.