Strichartz and dispersive estimates for quantum bouncing ball model: exponential sums and Van der Corput methods in 1d semi-classical Schr\"odinger equations

We analyze the one-dimensional semi-classical Schr\"odinger equation on the half-line with a linear potential and Dirichlet boundary conditions. Our main focus is on establishing improved dispersive and Strichartz estimates for this model, which govern the space-time behavior of solutions. We prove refined Strichartz bounds using Van der Corput-type derivative tests, beating previous known results where Strichartz estimates incur 1/4 losses. Moreover, assuming sharp bounds for certain exponential sums, our results indicate the possibility to reduce these losses further to $1/6 + \epsilon$ for all $\epsilon>0$, which would be sharp. We further expect that analogous Strichartz bounds should hold within the Friedlander model domain in higher dimensions.