Strict condition for the $L^2$-wellposedness of fifth and sixth order dispersive equations

We provide a set of conditions that is necessary and sufficient for the $L^2$-wellposedness of the Cauchy problem for fifth and sixth order variable-coefficient linear dispersive equations. The necessity of these conditions had been presented by Tarama, and we scrutinized their proof to split the conditions into several parts so that an inductive argument is applicable. This inductive argument simplifies the engineering process of the appropriate pseudodifferential operator needed for the proof of $L^2$-wellposedness.