Hardy Type Inequalities, Mourre Estimate and A-priori Decay for Eigenfunctions

Abstract

We address the problem of finding upper bounds on the decay at infinity for the eigenfunctions of a large class of self-adjoint operators, covering many interesting quantum Hamiltonians and valid even for eigenfunctions associated to embedded eigen-values. Inspired by [AHS89], [AMG87], [FH82] and [FHHO82], we put into evidence a strategy [MP00a] that starts from a local Mourre inequality with respect to a conjugate operator (that is not supposed to be self-adjoint) and produces a local weighted estimation; from this, a cut-off procedure (both on the test function and on the weights [Ag82], [AMG87]) leads to a Hardy type estimation and a simple argument [MP00a] gives then an a-priori bound (see Proposition 1.1 below) for the decay of eigenfunctions associated to eigenvalues belonging to the interval on which the initial Mourre inequality is valid. We apply this method to a large class of perturbations of convolution operators in $\mathbb{R}^n$ and among them a class of perturbed periodic Schrödinger Hamiltonians.