A counterexample to dispersive estimates for Schrödinger operators in higher dimensions

Abstract

In dimension $n>3$ we show the existence of a compactly supported potential in the differentiability class $C^{\alpha }$, $\alpha <\frac{n-3}{2}$, for which the solutions to the linear Schrödinger equation in ${\mathbb{R}}^n$, $$-i{\partial }_{t}u=-\Delta u+Vu,u(0)=f,$$ do not obey the usual $L^1\to L^{\infty }$ dispersive estimate. This contrasts with known results in dimensions $n\le 3$, where a pointwise decay condition on $V$ is generally sufficient to imply dispersive bounds.

References in corpus (2)

• Dispersive estimates for Schrodinger operators in dimensions one and three
• Dispersive estimates for Schroedinger operators in dimension two

Cited by in corpus (13)

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• Dispersive Estimates for higher dimensional Schrödinger Operators with threshold eigenvalues I: The odd dimensional case
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• The $L^p$ boundedness of wave operators for Schrödinger Operators with threshold singularities
• A weighted dispersive estimate for Schrödinger operators in dimension two
• On pointwise decay of waves
• The $L^p$-continuity of wave operators for higher order Schrödinger operators
• On the $L^p$ boundedness of the Wave Operators for fourth order Schrödinger operators
• A note on endpoint $L^p$-continuity of wave operators for classical and higher order Schrödinger operators
• Counterexamples to $L^p$ boundedness of wave operators for classical and higher order Schrödinger operators
• Dispersive estimates for matrix Schrödinger operators in dimension two
• The Massless Dirac Equation in Three Dimensions: Dispersive estimates and zero energy obstructions