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We establish dispersive estimates and local decay estimates for the timeevolution of non-self-adjoint matrix Schr\"odinger operators with thresholdresonances in one space dimension. In particular, we show that the decay ratesin the weighted setting are the same as in the regular case after subtracting afinite rank operator corresponding to the threshold resonances. Such matrixSchr\"odinger operators naturally arise from linearizing a focusing nonlinearSchr\"odinger equation around a solitary wave. It is known that the linearizedoperator for the 1D focusing cubic NLS equation exhibits a threshold resonance.We also include an observation of a favorable structure in the quadraticnonlinearity of the evolution equation for perturbations of solitary waves ofthe 1D focusing cubic NLS equation.

Dispersive estimates for 1D matrix Schr\"odinger operators with threshold resonance