Strichartz estimates for a class of Schrödinger equations with a drift

TL;DR

This work studies Strichartz-type estimates for a class of Schrödinger equations with a real drift, allowing degenerate diffusion matrices $Q$ and drift matrices $B$ and introducing a local homogeneous dimension $D$ via a canonical form. The authors develop an intrinsic framework based on the controllability Gramian $Q(t)$ and a volume function $V(t)=\det Q(t)$, obtaining weighted mixed-norm Strichartz estimates (Theorem A) under a growth condition (A) and complementary results (Theorem B) under a weaker growth condition (B). The approach blends the Ginibre–Velo $T^*T$ method, conformal self-similar coordinates, and a detailed analysis of $V(t)$, including large-time asymptotics and various canonical forms, with connections to restriction phenomena and twisted Laplacians. These results clarify how degeneracy and drift influence dispersive behavior and extend Strichartz-type theory to anisotropic, degenerate settings relevant to nonlinear Schrödinger equations and quantum-wire models.

Abstract

We establish new intrinsic Strichartz estimates for solutions of the Cauchy problem for a class of possibly degenerate Schrödinger equations with a real drift.

Theorems & Definitions (44)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 3.1
• Theorem 3.2
• proof
• Corollary 3.3
• Proposition 3.4
• Theorem 3.5