NLS ground states on the half-line with point interactions

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.

arXiv comment17 pages

References in corpus (13)

• Theory of Bose-Einstein condensation in trapped gases
• The Nonlinear Schrödinger Equation on the Interval
• Fast soliton scattering by delta impurities
• A Free Particle on a Circle with Point Interaction
• Variational properties and orbital stability of standing waves for NLS equation on a star graph
• Soliton solutions of nonlinear Schroedinger equation on simple networks
• Ground state and orbital stability for the NLS equation on a general starlike graph with potentials
• Ground states for the planar NLSE with a point defect as minimizers of the constrained energy
• Sine-Gordon solitons in networks: Scattering and transmission at vertices
• Competing nonlinearities in NLS equations as source of threshold phenomena on star graphs
• Existence, structure, and robustness of ground states of a NLSE in 3D with a point defect
• Solving the quantum non-linear Schrodinger equation with delta-type impurity
• Existence of the ground state for the NLS with potential on graphs

Cited by in corpus (4)

• A general review on the NLS equation with point-concentrated nonlinearity
• Normalized solutions of one-dimensional defocusing NLS equations with nonlinear point interactions
• Nonlinear scalar field equation with point interaction
• The search for NLS ground states on a hybrid domain: Motivations, methods, and results