Localization for one-dimensional random potentials with large local fluctuations

Abstract

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 $\int\D{x} \smean{V(x)V(0)}=\infty$. 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, ${\ell }_{\mathrm{loc}}\propto E/\ln E$, ${\ell }_{\mathrm{loc}}\propto E^{\mu /2}$ with $0<\mu <2$ or ${\ell }_{\mathrm{loc}}\propto {\ln }^{\mu -1}E$ for $\mu >1$, or superlocalization (decay of the wave functions faster than a simple exponential).

9 pages, LaTeX, 2 eps figures ; v2: Refs. added, small paragraph added p.4, additional table in conclusion

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• Supersymmetric quantum mechanics with Levy disorder in one dimension
• Fluctuations of the product of random matrices and generalized Lyapunov exponent
• Dyson's disordered linear chain from a random matrix theory viewpoint
• The generalized Lyapunov exponent for the one-dimensional Schrödinger equation with Cauchy disorder: some exact results
• Disordered harmonic chain with random masses and springs: a combinatorial approach