Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Theoretical studies of the motion of an electron in a disordered system have led to a number of remarkable results especially those concerned with localization and with the importance of dimensionality. The theoretical approaches previously used run into some severe technical difficulties. This article sets out the bases of the recently introduced supersymmetry method which has already, as the article shows, provided unified derivations of the major results in this subject, free of the technical problems. The method reduces the systems it treats to field-theoretical models involving both bosons and fermions.

Supersymmetry and theory of disordered metals

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 inl
2 spaces. A prototypical example is the discrete Schrodinger operatorH=−Δ+U
0(x)+λV

 x
 onZ

 d
 ,d≧1, withU
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 operatorT whose matrix elements satisfy: ∑
 y
 |T

 x,y
 |
 S
 ≦Const., uniformly inx, for somes<1. Localization means here that within a specified energy range the spectrum ofH is of the pure-point type, or equivalently — the wave functions do not spread indefinitely under the unitary time evolution generated byH. The effect is produced by strong disorder in either the potential or in the off-diagonal matrix elementsT

 x, y
 . Under rapid decay ofT

 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
 > withs 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 ofG

 E
 
(x, y) on the individual matrix elements ofH 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 partU
0
(x), ii) the Bethe lattice, iii) operators with very slow decay in the off-diagonal terms (T
x,y≈1/|x−y|(d+e)), and iv) localization produced by disordered boundary conditions.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1104253939

Localization at large disorder and at extreme energies: an elementary derivation

Simple models of quantum chaos: Spectrum and eigenfunctions

Random matrix theory is a wide and growing field with a variety of concepts, results, and techniques and a vast range of applications in mathematics and the related sciences. The book, written by well-known experts, offers beginners a fairly balanced collection of basic facts and methods (Part 1 on classical ensembles) and presents experts with an exposition of recent advances in the subject (Parts 2 and 3 on invariant ensembles and ensembles with independent entries). The text includes many of the authors' results and methods on several main aspects of the theory, thus allowing them to present a unique and personal perspective on the subject and to cover many topics using a unified approach essentially based on the Stieltjes transform and orthogonal polynomials. The exposition is supplemented by numerous comments, remarks, and problems. This results in a book that presents a detailed and self-contained treatment of the basic random matrix ensembles and asymptotic regimes. This book will be an important reference for researchers in a variety of areas of mathematics and mathematical physics. Various chapters of the book can be used for graduate courses; the main prerequisite is a basic knowledge of calculus, linear algebra, and probability theory.

Eigenvalue Distribution of Large Random Matrices

Let 
$$H_V = - \frac{{d^{\text{2}} }}{{dt^{\text{2}} }} + q(t,\omega )$$

 be an one-dimensional random Schrodinger operator in ℒ2(−V,V) with the classical boundary conditions. The random potentialq(t, ω) has a formq(t, ω)=F(x

 t
 ), wherex

 t
 is a Brownian motion on the compact Riemannian manifoldK andF:K→R
1 is a smooth Morse function,
 
$$\mathop {\min }\limits_K F = 0$$

 . Let 
$$N_V (\Delta ) = \sum\limits_{Ei(V) \in \Delta } 1 $$

 , where Δ∈(0, ∞),E

 i
 (V) are the eigenvalues ofH

 V
 . The main result (Theorem 1) of this paper is the following. IfV→∞,E
0>0,k∈Z
+ anda>0 (a is a fixed constant) then
 
$$P\left\{ {N_V \left( {E_0 - \frac{a}{{2V}},E_0 + \frac{a}{{2V}}} \right) = k} \right\}\xrightarrow[{V \to \infty }]{}e^{ - an(E_0 )} (an(E_0 ))^k |k!,$$

 wheren(E
0) is a limit state density ofH

 V
 ,V→∞. This theorem mean that there is no repulsion between energy levels of the operatorH

 V
 ,V→∞. The second result (Theorem 2) describes the phenomen of the repulsion of the corresponding wave functions.

/pdf/the-local-structure-of-the-spectrum-of-the-one-dimensional-3tmb1wz4fz.pdf

The local structure of the spectrum of the one-dimensional Schrödinger operator

Eigenvalue asymptotics of the Neumann Laplacian of regions and manifolds with cusps

The purpose of this paper is to study a limit probability distribution of the set of the first κ eigenvalues λ1(ℒ)<λ2(ℒ)<...<λκ(ℒ) (with a fixed κ and ℒ→∞) of the boundary problem on the interval [0, ℒ]

wherea(t, ω),q(t, ω) are the random stationary processes. Particularly the question of the repulsion between the first eigenvalues (small energetic levels) is studied. It has been proved that in the “divergent” case (q(t, ω)=0,a(t, ω)≠0) levels repulsion exists. As for the “potential” case (a(t, ω)≡1,q(t, ω)≠0) there is not any repulsion at all. This is one of the main differences between these two cases.

/pdf/on-the-basic-states-of-one-dimensional-disordered-structures-2an9vlh4by.pdf

On the Basic States of One-Dimensional Disordered Structures

We investigate spectral properties of random Schrodinger operators Hω = - Δ + ξn(ω)(1 + |n|α) acting onl
2(Z
d), where ξn are independent random variables uniformly distributed on [0, 1].

/pdf/spectral-properties-of-random-schrodinger-operators-with-sjlhtexja7.pdf

S. A. Molčanov

Papers

The local structure of the spectrum of the one-dimensional Schrödinger operator

Eigenvalue asymptotics of the Neumann Laplacian of regions and manifolds with cusps

On the Basic States of One-Dimensional Disordered Structures

Spectral properties of random Schrödinger operators with unbounded potentials