Quantum cohomology of projective bundles

We construct an I-function of the projective bundle P(V) associated with a not necessarily split vector bundle V\to B as a Fourier transform of the S^1-equivariant J-function of the total space of V and show that it lies on the Givental Lagrangian cone of P(V). Using this result, we show that the quantum cohomology D-module of P(V) splits into the direct sum of copies of the quantum cohomology D-module of the base space B. This has applications to the semisimplicity of big quantum cohomology.