Resonances for ”large” ergodic systems in one dimension: a review

The present note reviews recent results on resonances for one-dimensional quantum ergodic systems constrained to a large box. We restrict ourselves to one dimensional models in the discrete case. We consider two type of ergodic potentials on the half-axis, periodic potentials and random potentials. For both models, we describe the behavior of the resonances near the real axis for a large typical sample of the potential. In both cases, the linear density of their real parts is given by the density of states of the full ergodic system. While in the periodic case, the resonances distribute on a nice analytic curve (once their imaginary parts are suitably renormalized), in the random case, the resonances (again after suitable renormalization of both the real and imaginary parts) form a two dimensional Poisson cloud. 0. Introduction On l2(N), consider V a bounded potential and the operatorH = −∆+V satisfying the Dirichlet boundary condition at 0. The potentials V we will consider with are of two types: • V periodic; • V = Vω random e.g. a collection of i.i.d. random variables. The spectral theory of such models has been studied extensively (see e.g. [10]) and it is well known that, when considered on l2(Z), the spectrum of H is purely absolutely continuous when V is periodic ([24]) while it is pure point when V = Vω is the Anderson potential ([2, 20]). On l2(N), the picture is the same except for possible discrete eigenvalues outside the essential spectrum which coincides and is of the same nature as the essential spectrum of the operator on l2(Z). Let L > 0. The object of our study is the following operator on l2(N) (0.1) HL = −∆+ V 1J0,LK when L becomes large; here −∆ is the free Lapalce operator defined by −(∆u)(n) = u(n+1)+u(n− 1) for n ≥ 0 where u = (u(n))n≥0 ∈ l 2(N) and u(−1) = 0 (Dirichlet boundary condition at 0). Clearly, the essential spectrum of HL is that of the discrete Laplace operator, that is, [−2, 2], and it is absolutely continuous. Moreover, outside this absolutely continuous spectrum, HL has only discrete eigenvalues associated to exponentially decaying eigenfunctions. We are interested in the resonances of the operator HL. These can be defined as the poles of the meromorphic continuation of the resolvent of HL through the continuous spectrum of HL (see e.g. [25]). One proves that