Strichartz and dispersive estimates for quantum bouncing ball model: Exponential sums and Van der Corput methods in 1D semi-classical Schrödinger equations

Abstract

We analyze the one-dimensional semi-classical Schrödinger 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 $\frac{1}{4}$ losses. Moreover, assuming sharp bounds for certain exponential sums, our results indicate the possibility to reduce these losses further to $\frac{1}{6}+ϵ$ for all $ϵ>0$, which would be sharp. We further expect that analogous Strichartz bounds should hold within the Friedlander model domain in higher dimensions.