A counterexample to dispersive estimates for Schr\"odinger operators in higher dimensions
Goldberg, M. ; Visan, M.
arXiv, 0508206 / Harvested from arXiv
Text on arXiv
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 Schr\"odinger equation in $\R^n$, $$ -i\partial_t u = - \Delta u + Vu, \quad u(0)=f, $$ 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.
Publié le : 2005-08-11
Classification:  Mathematics - Analysis of PDEs,  Mathematical Physics 
@article{0508206,
     author = {Goldberg, M. and Visan, M.},
     title = {A counterexample to dispersive estimates for Schr\"odinger operators in
  higher dimensions},
     journal = {arXiv},
     volume = {2005},
     number = {0},
     year = {2005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0508206}
}

Goldberg, M.; Visan, M. A counterexample to dispersive estimates for Schr\"odinger operators in
  higher dimensions. arXiv, Tome 2005 (2005) no. 0, . http://gdmltest.u-ga.fr/item/0508206/