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In this work we introduce a new family of twelve-step linear multistepmethods for the integration of the Schr\"odinger equation. The new methods areconstructed by adopting a new methodology which improves the phase lagcharacteristics by vanishing both the phase lag function and its firstderivatives at a specific frequency. This results in decreasing the sensitivityof the integration method on the estimated frequency of the problem. Theefficiency of the new family of methods is proved via error analysis andnumerical applications.

High Order Multistep Methods with Improved Phase-Lag Characteristics for the Integration of the Schr\"odinger Equation