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A counterexample to dispersive estimates for Schrödinger operators in higher dimensions

M. Goldberg·M. Visan
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Abstract

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

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M. Goldberg, M. Visan. 2005-08-11. A counterexample to dispersive estimates for Schrödinger operators in higher dimensions. https://doi.org/10.1007/s00220-006-0013-5

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