Gain of regularity for semilinear Schrödinger equations

We discuss local existence and gain of regularity for semilinear Schrödinger equations which generally cause loss of derivatives. We prove our results by advanced energy estimates. More precisely, block diagonalization and Doi's transformation, together with symbol smoothing for pseudodifferential operators with nonsmooth coefficients, apply to systems of Schrödinger-type equations. In particular, the sharp Gårding inequality for pseudodifferential operators whose coefficients are twice continuously differentiable, plays a crucial role in our proof.