First, we will explore 1D, 2D, and 3D Schrödinger operators with delta interactions at discrete points, focusing on their rigorous definitions and spectral properties. Historically, the seminal work on one-dimensional linear models with delta interactions was by Kronig and Penney in 1931, describing a nonrelativistic electron in a fixed crystal lattice. Later, Bethe, Peierls, and Thomas discussed three-dimensional models for nonrelativistic quantum particles interacting with "very short range" potentials.

Next, we will discuss the Nonlinear Schrödinger Equation (NLSE) with delta interaction on a line and on metric graphs, emphasizing their variational and stability properties. The standard NLSE with point interaction has been proposed as an effective model for Bose-Einstein Condensates (BEC) with defects or impurities. Applications extend to graph-like structures, such as planar self-focusing waveguides and various fiber optics devices.

In the third part of the talk, we will examine the NLSE with point-concentrated nonlinearity (nonlinear delta potential), focusing on its variational properties. Interest in this model has grown due to its applications in solid-state and condensed matter physics, such as charge accumulation in semiconductor interfaces, nonlinear propagation in Kerr-type media with localized defects, and BECs in optical lattices with isolated defects created by focused laser beams.

Finally, we will highlight open problems and future directions in the field.