Exponential and super-exponential localizations for one-dimensional Schrödinger operators with Lévy noise potentials
The aim of this paper is to extend the main facts of the spectral theory of the one-dimensional Schrödinger operator with random potential [see e.g. \textit{R. Carmona}, Lect. Notes Math. 1180, 1-124 (1986; Zbl 0607.35089), and the reviewer, J. Sov. Math. 46, No.4, 1979-2021 (1989); translation from Itogi Nauki Tekh., Ser. Teor. Veroyatn., Mat. Stat., Teor. Kibern. 25, 3-67 (1987; Zbl 0661.60079)]] to the case, when the random potential is the ``derivative'' of the homogeneous Lévy process \(Q_{\omega}(t)\), i.e., intuitively speaking, is a continuous-parameter family of i.i.d. random variables, which can be called ``Lévy noise''. This random process can be represented as a sum of the Wiener process and the Poisson process defined by the intensity measure \(\nu\) (dx). The author starts from the definition of the respective Schrödinger operator \(H_{\omega}\) by the corresponding bilinear form and proves the nonrandomness of the spectrum and its location (Propositions 1 and 2). Furthermore he establishes the existence of the Lyapunov exponent \(\gamma\) (\(\lambda)\) and a subspace of \({\mathbb{R}}^ 2\) which is stable under the action of the transfer matrix, i.e. analogues of the Fürstenberg and Oseledic theorems, \(\gamma\) (\(\lambda)\) is finite if \(\int_{| x| >1}\ell n | x| \nu (dx)\) is finite and \(\gamma (\lambda)=\infty\) otherwise (Theorems 1 and 2). In the latter case if \(\int_{| x| <1}| x| \nu (dx)<\infty\) the Oseledic theorem is still valid (Theorem 3). Furthermore, if \(\int_{| x| >1}\ell n | x| \nu (dx)<\infty\) and one of the three conditions holds: a) \(Q_{\omega}(x)\) has a Gaussian part; b) the Gaussian part is absent, \(\nu ((- \infty,0)=0\), \(\nu ((0,\infty))<\infty\); c) \(\nu\) (dx) has an absolutely continuous component; then the spectrum of \(H_{\omega}\) is pure point and the eigenfunctions are exponentially decreasing with rate \(\gamma\) (\(\lambda)\) (exponential localization) (Theorem 5). If \(\int_{| x| >1}\ell n | x| \nu dx=\infty\) then under the same conditions the rate is infinite (Theorem 6). If one assumes some additional regularity conditions for the measure \(\nu\) (dx) then a rather precise asymptotic behaviour (faster than exp\(\{\)-const\(\cdot t\})\) can be established (Theorem 7). This is the so- called superexponential localization.
Schrödinger operator with random potential
scientific article; zbMATH DE number 4137128