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Some remarks on the Schrödinger equation with a potential in $L^{r}_{t}L^{s}_{x}$

Piero D'Ancona, Vittoria Pierfelice, Nicola Visciglia

TL;DR

We study the linear Schrödinger equation with a time-dependent potential V(t,x) in the class L^r_t L^s_x and show that a bounded L^r_t L^s_x norm yields the full set of Strichartz estimates under the scaling condition $\frac{1}{r} + \frac{n}{2s} = 1$, with counterexamples proving optimality for both local and global bounds. Our elementary, integrability-based method treats V as a small perturbation on short time intervals via a fixed-point argument in a Strichartz-based Banach space, then patches local solutions to obtain a global result and shows energy conservation when F=0. We also develop a local-well-posedness framework by viewing V as a small perturbation of V(0,x) on short intervals and extending via standard continuation to maximal intervals, thereby preserving Strichartz bounds. The results extend dispersive theory to large, sign-changing time-dependent potentials and provide a practical route to Strichartz estimates when only $L^r_t L^s_x$ control of the potential is available.

Abstract

We study the dispersive properties of the linear Schr\"odinger equation with a time-dependent potential $V(t,x)$. We show that an appropriate integrability condition in space and time on $V$, i.e. the boundedness of a suitable $L^{r}_{t}L^{s}_{x}$ norm, is sufficient to prove the full set of Strichartz estimates. We also construct several counterexamples which show that our assumptions are optimal, both for local and for global Strichartz estimates, in the class of large unsigned potentials $V\in L^{r}_tL^{s}_x$.

Some remarks on the Schrödinger equation with a potential in $L^{r}_{t}L^{s}_{x}$

TL;DR

We study the linear Schrödinger equation with a time-dependent potential V(t,x) in the class L^r_t L^s_x and show that a bounded L^r_t L^s_x norm yields the full set of Strichartz estimates under the scaling condition , with counterexamples proving optimality for both local and global bounds. Our elementary, integrability-based method treats V as a small perturbation on short time intervals via a fixed-point argument in a Strichartz-based Banach space, then patches local solutions to obtain a global result and shows energy conservation when F=0. We also develop a local-well-posedness framework by viewing V as a small perturbation of V(0,x) on short intervals and extending via standard continuation to maximal intervals, thereby preserving Strichartz bounds. The results extend dispersive theory to large, sign-changing time-dependent potentials and provide a practical route to Strichartz estimates when only control of the potential is available.

Abstract

We study the dispersive properties of the linear Schr\"odinger equation with a time-dependent potential . We show that an appropriate integrability condition in space and time on , i.e. the boundedness of a suitable norm, is sufficient to prove the full set of Strichartz estimates. We also construct several counterexamples which show that our assumptions are optimal, both for local and for global Strichartz estimates, in the class of large unsigned potentials .

Paper Structure

This paper contains 10 sections, 4 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['th.1']}
  3. Case A: $r\in[2,\infty[$
  4. Case B: $r\in[1,2]$
  5. Proof of Theorem \ref{['th.2']}
  6. Proof of the counterexamples
  7. An eigenvalue problem.
  8. Proof of Theorem \ref{['th.3']}, case $1/r+n/(2s)<1$, $r\neq\infty$
  9. Proof of Theorem \ref{['th.3']}, case $1/r+n/(2s)>1$, $r\neq\infty$
  10. Proof of Proposition \ref{['prop.4']}

Key Result

Theorem 1.1

Let n\geq2, let I be either the interval [0,T] or [0,+\infty[, and assume V(t,x) is a real valued potential belonging to for some fixed r\in[1,\infty[ and s\in ]n/2,\infty]. Let u_{0}\in L^{2} and F\in L_{I}^{\tilde{p}'}L_{\space}^{\tilde{q}'} for some admissible pair ({\tilde{p}'},{\tilde{q}'}). Then the integral equation eq.schrI has a unique solution u\in C_{I}L^{2} which belongs to L_{I}^{p}L

Theorems & Definitions (14)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.2
  • Theorem 1.2
  • Remark 1.4
  • Example 1.1
  • Example 1.2
  • ...and 4 more