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We study mirror symmetry (A-side vs B-side) in the framework of quantumdifferential systems. We focuse on the logarithmic and non-resonant case, whichdescribes the geometric situation. We show that quantum differential systemsprovide a good framework in order to generalize the construction of therational structure on the A-side given by Katzarkov, Kontsevitch and Pantev forthe projective spaces. As an application, we compute the rational structureobtained in this way on the orbifold cohomology of weighted projective spaces.As an example we also calculate, using quantum differential systems, a mirrorpartner of the Hirzebruch surface F2.

Quantum differential systems and some applications to mirror symmetry