WKB $p$-forms are constructed as approximate solutions to boundary value problems associated with semiclassical Witten Laplacians. Naturally distorted Neumann or Dirichlet boundary conditions are considered.

We prove sharp Strichartz estimates for the semiclassical Schrödinger equation on a compact Riemannian manifold with a smooth, strictly geodesically concave boundary. We deduce classical Strichartz estimates for the Schrödinger equation outside a strictly convex obstacle, local existence for the $H^1$-critical (quintic) Schrödinger equation, and scattering for the subcritical Schrödinger equation in three dimensions.

A guiding principle in Kähler geometry is that the infinite-dimensional symmetric space $\mathcal{\mathscr{H}}$ of Kähler metrics in a fixed Kähler class on a polarized projective Kähler manifold $M$ should be well approximated by finite-dimensional submanifolds ${\mathcal{ℬ}}_k\subset \mathcal{\mathscr{H}}$ of Bergman metrics of height $k$ (Yau, Tian, Donaldson). The Bergman metric spaces are symmetric spaces of type $G_{ℂ}∕G$ where $G=U\left(d_k+1\right)$ for certain $d_k$. This article establishes some basic estimates for Bergman approximations for geometric families of toric Kähler manifolds.

The approximation results are applied to the endpoint problem for geodesics of $\mathcal{\mathscr{H}}$, which are solutions of a homogeneous complex Monge–Ampère equation in $A\times X$, where $A\subset ℂ$ is an annulus. Donaldson, Arezzo and Tian, and Phong and Sturm raised the question whether $\mathcal{\mathscr{H}}$-geodesics with fixed endpoints can be approximated by geodesics of ${\mathcal{ℬ}}_k$. Phong and Sturm proved weak $C^0$-convergence of Bergman to Monge–Ampère geodesics on a general Kähler manifold. Our approximation results show that one has $C^2\left(A\times X\right)$ convergence in the case of toric Kähler metrics, extending our earlier result on $ℂ{\mathbb{P}}^1$.