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The paper discusses nonlinear singular perturbations of delta type of thefractional Schr\"odinger equation$\imath\partial_t\psi=\left(-\triangle\right)^s\psi$, with$s\in(\frac{1}{2},1]$, in dimension one. Precisely, we investigate local andglobal well posedness (in a strong sense), conservations laws and existence ofblow-up solutions and standing waves.

Nonlinear singular perturbations of the fractional Schr\"odinger equation in dimension one