Localization of the eigenfunctions of a Bloch-Torrey operator on the half-plane
Résumé
We consider a non-self adjoint operator of the form -h 2 ∆ + i(V (x) + α(x)y) on the upper half plane y > 0 with Dirichlet boundary conditions on {y = 0} with V ⩾ 0, V admitting a non-degenerate minimum at x = 0 and α ′ (0) = 0. We study its eigenfunctions associated to the smallest eigenvalues in magnitude in the semiclassical limit h → 0. Elementary variational estimates show that these eigenfunctions are localized near the point (0, 0) at the scales O(h 1/3 ) in x and O(h 2/3 ) in y. In this paper, we show that the O(h 1/3 ) localization in x is not optimal; more precisely, we establish that the eigenfunctions are concentrated in a neighborhood of size O(h 1/2 ) of the axis {x = 0}, and this scale is shown to be sharp. The proof relies on the symbolic calculus of operator-valued pseudodifferential operators.
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