The unifying theme of the research project is the analysis of partial differential equations ruling on relevant classical or quantum models in domains where spatial singularities can be present. Singularities can develop as a consequence of the dynamics (blow-up), or they can be present from the beginning in the domain where the model is set. In this second case singularities can describe spatial inhomogeneities of the propagation medium (described for example by point or delta interactions with various co-dimension or by quantum graphs) or they can serve as effective approximations (zero range models). The plan of the project includes the following main research lines:
1. Systems of quantum particles with zero-range interactions
2. The semiclassical limit in presence of singular interactions
3. Schrödinger operators and singular interactions in acoustics: scattering theory
4. Nonlinear Schrödinger equation in domains with spatial singularities: stability, instability and asymptotic properties
The project has produced a number of significant and original results contained in 25 papers. These results extend and deepen the theoretical understanding of the stationary behavior and time evolution of systems governed by the wave, Schrödinger, or Dirac equations, while also laying a solid foundation for future investigations. Furthermore, new scientific interactions have been established between the participants in the project and external researchers, thereby expanding the participants’ scientific network and enhancing their presence in the international scientific community.
1. Systems of quantum particles with zero-range interactions
2. The semiclassical limit in presence of singular interactions
3. Schrödinger operators and singular interactions in acoustics: scattering theory
4. Nonlinear Schrödinger equation in domains with spatial singularities: stability, instability and asymptotic properties
The project has produced a number of significant and original results contained in 25 papers. These results extend and deepen the theoretical understanding of the stationary behavior and time evolution of systems governed by the wave, Schrödinger, or Dirac equations, while also laying a solid foundation for future investigations. Furthermore, new scientific interactions have been established between the participants in the project and external researchers, thereby expanding the participants’ scientific network and enhancing their presence in the international scientific community.