Localization at large disorder and at extreme energies: An elementary derivations (original) (raw)

Abstract

The work presents a short proof of localization under the conditions of either strong disorder (λ > λ0) or extreme energies for a wide class of self adjoint operators with random matrix elements, acting in_l_ 2 spaces. A prototypical example is the discrete Schrödinger operator_H_=−Δ+U 0(x)+λ_V_ x on_Z_ d,d_≧1, with_U 0(x) a specified background potential and {V x } generated as random variables. The general results apply to operators with −Δ replaced by a non-local self adjoint operator_T_ whose matrix elements satisfy: ∑ y |T x,y |S_≦Const., uniformly in_x, for some_s_<1. Localization means here that within a specified energy range the spectrum of_H_ is of the pure-point type, or equivalently — the wave functions do not spread indefinitely under the unitary time evolution generated by_H_. The effect is produced by strong disorder in either the potential or in the off-diagonal matrix elements_T_ x, y_ . Under rapid decay of_T_ x, y , the corresponding eigenfunctions are also proven to decay exponentially. The method is based on resolvent techniques. The central technical ideas include the use of low moments of the resolvent kernel, i.e. <_|G_ _E_ _(x, y)_|_s_> with_s small enough (<1) to avoid the divergence caused by the distribution's Cauchy tails, and an effective use of the simple form of the dependence of_G_ E (x, y) on the individual matrix elements of_H_ in elucidating the implications of the fundamental equation (H_−_E)G E (x,x 0)=δ x,x0 . This approach simplifies previous derivations of localization results, avoiding the small denominator difficulties which have been hitherto encountered in the subject. It also yields some new results which include localization under the following sets of conditions: i) potentials with an inhomogeneous non-random part_U_ 0 (x), ii) the Bethe lattice, iii) operators with very slow decay in the off-diagonal terms (T _x,y_≈1/|x_−_y|(d+ε)), and iv) localization produced by disordered boundary conditions.

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Note added in proof. The method presented here can be extended also to localization at weak disorder and moderate energies. Such supplementary results, and further developments, are found in

• [A2] Aizenman, M.: Localization at weak disorder: some elementary bounds. To appear in Rev. Math. Phys.

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Author notes

1. Michael Aizenman (Visiting as Fairchild Scholar at Caltech)
   Present address: , 91125, Pasadena, CA, USA
2. Stanislav Molchanov
   Present address: Department of Mathematics, TRB 314, Univ. South. Cal., 90089, Los Angeles, CA, USA

Authors and Affiliations

1. Departments of Physics and Mathematics, Princeton University, Jadwin Hall, P.O.Box 708, 08544, Princeton, New Jersey, USA
   Michael Aizenman (Visiting as Fairchild Scholar at Caltech)
2. Department of Probability Theory, Faculty of Mathematics and Mechanics, Moscow State University, 117234, Moscow, Russia
   Stanislav Molchanov

Authors

1. Michael Aizenman
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2. Stanislav Molchanov
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Additional information

Communicated by T. Spencer

Work supported in part by NSF Grants PHY-8912067 and PHY-9214654 (MA), and ONR Grant N00014-91-J-1526 (SM)

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Aizenman, M., Molchanov, S. Localization at large disorder and at extreme energies: An elementary derivations.Commun.Math. Phys. 157, 245–278 (1993). https://doi.org/10.1007/BF02099760

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• Received: 29 September 1992
• Issue Date: October 1993
• DOI: https://doi.org/10.1007/BF02099760

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