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where {Q'Jf); ―co<?< + oo} is a temporally homogeneous Levy process and Q'JS) is the "derivative" of its sample function. Intuitively speaking, {Q'm(t)}teR is a continuous parameter family of i.i.d. random variables,which we will call "Levy noise", so that the above HM can be viewed as an idealization of the Schrodinger operator with random potential, and it may be of some interest to analyze in detail such an idealized model of disordered system. On the other hand, it is well known that almost every sample function of a Levy process is not differentiable(except the case of a trivial Levy process Qa(t)=ct, with a real constant c). Hence the expression (0-1) has only a symbolical meaning. The precise definition of Hw was given by the present author ([25]), and it was shown that H^ can actually be realized as a random self-adjoint operator in L2(R;dt). Moreover, the exact location of the spectrum of Hw was determined. The purpose of this paper is to study the properties of spectrum and eigenfunctions of Hw in more detail than in [25]. It will be shown that under some condition on {Qm(t)}, almost every realizationof H^ has pure point spectrum with exponentially decaying eigenfunctions (exponential localization―see Theorem 5). A remarkable fact is that in some other cases, the eigenfunctions decay faster than exponentially (Theorem 6). We would like to refer to this phenomenon as "super-exponential localization". Moreover, it will be shown that under some conditions on the Levy measure of {QJf)}, the eigenfunctions, in a rough sense, behave like exp[― \t\a~\ with a>l, or even like exp[―exp[exp[--exp[U|a] ・・・]] with a>0, as |*|->oo (Theorem 7).

Exponential and super-exponential localizations for one-dimentional Schroedinger operators with Lévy noise potentials