Palestras e Seminários



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Palestrante: Nataliia Goloshchapova

Responsável: Estefani Moraes Moreira (Este endereço de email está sendo protegido de spambots. Você precisa do JavaScript ativado para vê-lo.)

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Resumo:  In this talk, we will introduce linear and nonlinear Schrödinger models with point interactions.

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.

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