SCALE-FREE AND QUANTITATIVE UNIQUE CONTINUATION FOR INFINITE DIMENSIONAL SPECTRAL SUBSPACES OF SCHRODINGER OPERATORS

We prove a quantitative unique continuation principle for infinite dimensional spectral subspaces of Schrodinger operators. Let Lambda(L) - (-L/2, L/2)(d) and H-L = -Delta(L) + V-L be a Schrodinger operators on L-2(Lambda(L)) with a bounded potential V-L : Lambda(L) -> R-d and Dirichlet, Neumann, or periodic boundary conditions. Our main result is of the type integral(Lambda L) |phi|(2) <= C-sfuc integral(W delta(L)) |phi|(2), where phi is an infinite complex linear combination of eigenfunctions of H-L with exponentially decaying coefficients, W-delta(L)is some union of equidistributed delta-balls in Lambda(L) and C-sfuc > 0 an L-independent constant. The exponential decay condition on phi can alternatively be formulated as an exponential decay condition of the map lambda (sic) ||x([lambda,infinity)) (H-L)phi||(2). The novelty is that at the same time we allow the function phi to be from an infinite dimensional spectral subspace and keep an explicit control over the constant C-sfuc in terms of the parameters. Moreover, we show that a similar result cannot hold under a polynomial decay condition.