  We develop refined Strichartz estimates at $L^2$ regularity for a class of
time-dependent Schr\"{o}dinger operators. Such refinements begin to
characterize the near-optimizers of the Strichartz estimate, and play a pivotal
part in the global theory of mass-critical NLS. On one hand, the harmonic
analysis is quite subtle in the $L^2$-critical setting due to an enormous group
of symmetries, while on the other hand, the spacetime Fourier analysis employed
by the existing approaches to the constant-coefficient equation are not adapted
to nontranslation-invariant situations, especially with potentials as large as
those considered in this article.
  Using phase space techniques, we reduce to proving certain analogues of
(adjoint) bilinear Fourier restriction estimates. Then we extend Tao's bilinear
restriction estimate for paraboloids to more general Schr\"{o}dinger operators.
As a particular application, the resulting inverse Strichartz theorem and
profile decompositions constitute a key harmonic analysis input for studying
large data solutions to the $L^2$-critical NLS with a harmonic oscillator
potential in dimensions $\ge 2$. This article builds on recent work of Killip,
Visan, and the author in one space dimension.
