Cauchy Problem for Evolution Equations of Schrödinger Type

In (R. Agliardi, 1995, Internat. J. Math.6, 791–804) we proved the well-posedness of the Cauchy problem in H∞ for some p-evolution equations (p 1) with real characteristic roots. For this purpose some assumptions on the lower order terms are needed, which, in the special case p=1, recapture well-known results for hyperbolic operators. In (R. Agliardi, 1995, Internat. J. Math.6, 791–804) the leading coefficients are assumed to be constant. In this paper we allow them to be variable. Our result is applicable to 2-evolution differential operators with real characteristics, i.e., to Schrödinger type operators. This class of operators comprehends, for example, Schrödinger operator Dt−Δx or the plate operator D2t−Δ2x. The Cauchy problem in H∞ for such evolution operators has been studied extensively by Takeuchi when the coefficients in the principal part are constant and the characteristic roots are distinct