Schrödinger equations with rough Hamiltonians

Elena Cordero; Fabio Nicola; Luigi Rodino

doi:10.3934/dcds.2015.35.4805

Article Contents

2015, Volume 35, Issue 10: 4805-4821. Doi: 10.3934/dcds.2015.35.4805

This issue Previous Article Lorentz-Morrey regularity for nonlinear elliptic problems with irregular obstacles over Reifenberg flat domains Next Article A class of mixing special flows over two--dimensional rotations

Schrödinger equations with rough Hamiltonians

1.
Department of Mathematics, University of Torino, via Carlo Alberto 10, 10123 Torino
2.
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino

Received: June 30, 2014

Revised: December 31, 2014

Published: September 2015

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Abstract

We consider a class of linear Schrödinger equations in $\mathbb{R}^d$ with rough Hamiltonian, namely with certain derivatives in the Sjöostrand class $M^{\infty,1}$. We prove that the corresponding propagator is bounded on modulation spaces. The present results improve several contributions recently appeared in the literature and can be regarded as the evolution counterpart of the fundamental result of Sjöstrand about the boundedness of pseudodifferential operators with symbols in that class.
Finally we consider nonlinear perturbations of real-analytic type and we prove local wellposedness of the corresponding initial value problem in certain modulation spaces.

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Mathematics Subject Classification: Primary: 35S10; Secondary: 35Q41, 42B35.

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