A partial answer to a conjecture of B. Simon concerning unique continuation (Q915959)

This work concerns the unique continuation property for the Schrödinger operator \(H=-\Delta +V\). One of the main results is the following. Under suitable hypotheses, it is proved that: if for some constants A, \(\alpha >0\) it turns out \[ \int_{B_ r}u^ 2 dx=O(\exp [-A/r^{\alpha \epsilon (r_ 0)}]) \] as \(r\to 0\), then the solution u of the Schrödinger equations is identically zero. Note that the proof of the above result does not use Carleman estimates.