1. The Schrödinger equation on a star-shaped graph under general coupling conditions
- Author
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Andreea Grecu and Liviu I. Ignat
- Subjects
Statistics and Probability ,Physics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,General Physics and Astronomy ,Propagator ,Statistical and Nonlinear Physics ,Coupling (probability) ,01 natural sciences ,Schrödinger equation ,010101 applied mathematics ,Linear map ,symbols.namesake ,Kernel (algebra) ,Modeling and Simulation ,Quantum graph ,symbols ,0101 mathematics ,Laplace operator ,Mathematical Physics ,Mathematical physics ,Resolvent - Abstract
We investigate dispersive and Strichartz estimates for the Schr\"{o}dinger time evolution propagator $\mathrm{e}^{-\mathrm{i}tH}$ on a star-shaped metric graph. The linear operator, $H$, taken into consideration is the self-adjoint extension of the Laplacian, subject to a wide class of coupling conditions. The study relies on an explicit spectral representation of the solution in terms of the resolvent kernel which is further analyzed using results from oscillatory integrals. As an application, we obtain the global well-posedness for a class of semilinear Schr\"{o}dinger equations.
- Published
- 2018
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