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A. Ya. Reznikova

Limit theorems for the spectra of ID random Schrödinger operator

Let

be an one-dimensional Schrödinger operator on L ² ([0, L]) with classical boundary conditions (say, ψ (0) = ψ (L) = 0 at t = 0, t = L.) Assume that the random potential q (·) has a Markov structure: q (t, ω) = F (x_t), where x_t , t ≥ 0 is a Brownian motion on the Euclidean ?-dimensional torus T^? and F : T^? → R is a smooth Morse type function on T^? . We can assume also that min F (·) = 0. Operator H_L has the discrete spectrum

Put

(the last limit exists P-a.s.). The goal of the paper is the studying of the fluctuations

We prove that the distributions of the process (where λ plays now a role of the time parameter) converge for L → ∞ weakly in C (0, ∞) to the distribution of the (continuous and Gaussian) process N ^∗ (λ). The limiting process N ^∗ (λ) has the non-degenerated densities of the finite dimensional distributions and its increments are locally "almost independent".

Random Operators and Stochastic Equations, Walter de Gruyter

Print ISSN: 0926-6364
Volume: 12, 09/2004
Pages: 235 - 254

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