2008/07/31 by Tom Bienaimé, Tom Bienaime, Christophe Texier
Mathematics · Physics and Astronomy · #Quantum and electron transport phenomena #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #cond-mat.dis-nn
paper · pdf · doi:10.1088/1751-8113/41/47/475001
published as J. Phys. A: Math. Theor. 41 (2008) 475001 · 9 pages, LaTeX, 2 eps figures ; v2: Refs. added, small paragraph added p.4, additional table in conclusion
openalex publication_date 2008/10/20 · arxiv created 2008/10/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We study the localization of wave functions for one-dimensional Schrödinger Hamiltonians with random potentials V (x) with short range correlations and large local fluctuations such that ∫ dx 〈V (x)V (0) 〉 = ∞. A random supersymmetric Hamiltonian is also considered. Depending on how large the fluctuations of V (x) are, we find either new energy dependences of the localization length, ℓloc ∝ E / lnE, ℓloc ∝ E µ/2 with 0 < µ < 2 or ℓloc ∝ ln µ−1 E for µ> 1, or superlocalization (decay of the wave functions faster than a simple exponential). PACS numbers: 72.15.Rn; 73.20.Fz; 02.50.-r. Introduction. – The phenomenon of Anderson localization [1] in one dimension has been widely studied since the pioneering work of Mott & Twose arguing that all states are localized in onedimension (1d) [2]. This statement was rigorously proven in Refs. [3, 4]. A general method to study the spectral and localization properties of 1d random Hamiltonian was proposed in Refs. [5, 6] : let us consider the one-dimensional Schrödinger Hamiltonian H = − d2