NLS ground states on a hybrid plane

We study existence, nonexistence, and qualitative properties of ground states for a focusing, subcritical Nonlinear Schr\"odinger Equation on a hybrid plane, consisting of a half-line attached to a plane. Ground states are normalized minimizers of the associated energy, given by Nonlinear Schr\"odinger energies with contact interactions on the half-line and on the plane, plus a quadratic coupling term. At fixed mass, existence holds if the contact interaction on the half-line is not too repulsive, or the interaction on the plane is sufficiently attractive, or the coupling is strong enough. Nonexistence occurs when both interactions are sufficiently repulsive and the coupling is weak. Moreover, we discuss how the coupling affects the support and the symmetry properties of such ground states. These are the first results for a Nonlinear Schr\"odinger Equation on a mixed-dimensional manifold.