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In this paper, we derive the $L^p$-boundedness of the variation operatorsassociated with the heat semigroup which is generated by the high orderSchr\"odinger type operator $(-\Delta)^2+V^2$. Further more, we prove theboundedness of the variation operators on Morrey spaces. In the proof of themain results, we always make use of the variation inequalities associated withthe heat semigroup generated by the biharmonic operator $(-\Delta)^2.$

Boundedness of variation operators associated with the heat semigroup generated by high order Schr\"odinger type operators