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1.
We investigate the spectrum of Schrödinger operatorsH
of the type:H
=–+q
i
()f(x–x
i
+
i
())(q
i
() and
i
() independent identically distributed random variables,i
d
). We establish a strong connection between the spectrum ofH
and the spectra of deterministic periodic Schrödinger operators. From this we derive a condition for the existence of forbidden zones in the spectrum ofH
. For random one- and three-dimensional Kronig-Penney potentials the spectrum is given explicitly. 相似文献
2.
Frédéric Klopp 《Communications in Mathematical Physics》1995,167(3):553-569
We study the spectrum of random Schrödinger operators acting onL
2(R
d
) of the following type
. The
are i.i.d. random variables. Under weak assumptions onV, we prove exponential localization forH at the lower edge of its spectrum. In order to do this, we give a new proof of the Wegner estimate that works without sign assumptions onV.
U.R.A. 760 C.N.R.S. 相似文献
Résumé Dans ce travail, nous étudions le spectre d'opérateurs de Schrödinger aléatoires agissant surL 2(R d ) du type suivant . Les sont des variables aléatoires i.i.d. Sous de faibles hypothèses surV, nous démontrons que le bord inférieur du spectre deH n'est composé que de spectre purement ponctuel et, que les fonctions propres associées sont exponentiellement décroissantes. Pour ce faire nous donnons une nouvelle preuve de l'estimée de Wegner valable sans hypothèses de signe surV.
U.R.A. 760 C.N.R.S. 相似文献
3.
We give new examples of discrete Schrödinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hullX of the potential is strictly ergodic, then the existence of just one potentialx inX for which the operator has no eigenvalues implies that there is a generic set inX for which the operator has purely singular continuous spectrum. A sufficient condition for the existence of such anx is that there is azX that contains arbitrarily long palindromes. Thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inX. The class includes well-known substitutions like Fibonacci, Thue-Morse, Period Doubling, binary non-Pisot and ternary non-Pisot. We also show that the operator has no absolutely continuous spectrum for allxX ifX derives from a primitive substitution. For potentials defined by circle maps,x
n
=1
J
(0+n), we show that the operator has purely singular continuous spectrum for a generic subset inX for all irrational and every half-open intervalJ.Work partially supported by NSERC.This material is based upon work supported by the National Science Foundation under Grant No. DMS-91-1715. The Government has certain rights in this material. 相似文献
4.
Robert S. Maier 《Journal of statistical physics》1987,48(3-4):425-447
Bounds are obtained on the unintegrated density of states ρ(E) of random Schrödinger operatorsH=?Δ + V acting onL 2(? d ) orl 2(? d ). In both cases the random potential is $$V: = \sum\limits_{y \in \mathbb{Z}^d } {V_y \chi (\Lambda (y))}$$ in which the \(\left\{ {V_y } \right\}_{y \in \mathbb{Z}^d }\) areIID random variables with densityf. The χ denotes indicator function, and in the continuum case the \(\left\{ {\Lambda (y)} \right\}_{y \in \mathbb{Z}^d }\) are cells of unit dimensions centered ony∈? d . In the finite-difference case Λ(y) denotes the sitey∈? d itself. Under the assumptionf ∈ L 0 1+? (?) it is proven that in the finitedifference casep ∈ L ∞(?), and that in thed= 1 continuum casep ∈ L loc ∞ (?). 相似文献
5.
We prove localization at high disorder or low energy for lattice Schrödinger operators with random potentials whose values at different lattice sites are correlated over large distances. The class of admissible random potentials for our multiscale analysis includes potentials with a stationary Gaussian distribution whose covariance functionC(x,y) decays as |x–y|–, where >0 can be arbitrarily small, and potentials whose probability distribution is a completely analytical Gibbs measure. The result for Gaussian potentials depends on a multivariable form of Nelson's best possible hypercontractive estimate.Partially supported by the NSF under grant PHY8515288Partially supported by the NSF under grant DMS8905627 相似文献
6.
We investigate the spectrum of the following random Schrödinger operators:
相似文献
7.
Y. A. Gordon V. Jakšić S. Molčanov B. Simon 《Communications in Mathematical Physics》1993,157(1):23-50
We investigate spectral properties of random Schrödinger operators H = - + n()(1 + |n|) acting onl
2(Z
d), where n are independent random variables uniformly distributed on [0, 1].Research partially supported by a Sloan Doctoral Dissertation Fellowship and NSERC under grant OGP-0007901Research partially supported by NSF grant DMS-9101716 相似文献
8.
We consider the energy dependent super Schrödinger linear problem
which is a direct generalization of the purely even, energy dependent Schrödinger equation discussed in [1]. We show that the isospectral flows of that problem possess (N+1) compatible Hamiltonian structures. We also extend a generalised factorisation approach of [2] to this case and derive a sequence ofN modifications for the 2N component systems. Then
th such modification possesses (N–n+1) compatible Hamiltonian structures.On leave of absence from Institute of Theoretical Physics, Warsaw University, Hoza 69, PL-00-681 Warsaw, Poland (present address) 相似文献
9.
We consider a random Schrödinger operator onL
2(v) of the form
, {C
i} being a covering of
v
with unit cubes around the sites of
v
and {q
i} i.i.d. random variables with values in [0, 1]. We assume that theq
i's are continuously distributed with bounded densityf(q) and that 0<P(q
0<1/2)=<1. Then we show that an ergodic mean of the quantity dx|x|2|(exp(itH
))(x)|2t
–1 vanishes provided =g
E(H
), where is well-localized around the origin andg
E is a positiveC
-function with support in (0,E),EE*(, |f|). Our estimate ofE*(, |f|) is such that the set {x
v
|V
(x) E*(, |f|)} may contain with probability one an infinite cluster of cubes {C
i} which are nearest neighbours. The proof is based on the technique introduced by Fröhlich and Spencer for the analysis of the Anderson model.Work supported in part by C.N.R. (Italy) and NAVF (Norway)On leave of absence from Instituto di Fisica Università di Roma, Italia 相似文献
10.
We prove that one-dimensional Schrödinger operators with even almost periodic potential have no point spectrum for a denseG
in the hull. This implies purely singular continuous spectrum for the almost Mathieu equation for coupling larger than 2 and a denseG
in even if the frequency is an irrational with good Diophantine properties.This material is based upon work supported by the National Science Foundation under Grant No. DMS-9101715. The Government has certain rights in this material. 相似文献
11.
Schrödinger operator with a nonlocal potential whose absolutely continuous and point spectra coexist
We consider the Schrödinger-like operatorH in which the role of a potential is played by the lattice sum of rank 1 operators \(|\left. {v_n } \right\rangle \left\langle {v_n |} \right.\) multiplied by g tan π[(α,n)+ω],g>0, α∈? d ,n∈? d , ω∈[0, 1]. We show that if the vector α satisfies the Diophantine condition and the Fourier transform support of the functionsv n (x)=v(x-n),x∈? d ,n∈? d , is small then the spectrum ofH consists of a dense point component coinciding with? and an absolutely continuous component coinciding with [?, ∞), where ? is the radius of the mentioned support. Besides, we find the integrated density of statesN(λ) (it has a jump at λ=?) and zero temperature a.c. conductivityσ λ (v), that also has a jump at λ=? and vanishes faster than any power of the external field frequency ν as ν→0 and λ≠?. 相似文献
12.
A. Ja. Režnikova 《Journal of statistical physics》1981,25(2):291-308
Let HL = –d2/dt2+q(t,) be an one-dimensional random Schrödinger operator in 2(–L, L) with the classical boundary conditions. The random potential q(t,) has a form q(t, )=F(xt), where xt is a Brownian motion on the Euclidean v-dimensional torus, FSv R1 is a smooth function with the nondegenerated critical points, mins
v F = 0. Let
are the eigenvalues of HL) be a spectral distribution function in the volume [– L,L] and N() = limL(1/2L)NL() be a corresponding limit distribution function.Theorem 1. If L then the normalized difference N
L
*
()=[NL() -2L·N()]2L tends (in the sense of Levi-Prokhorov) to the limit Gaussian process N*(); N*()0, 0, and N*() has nondegenerated finitedimensional distributions on the spectrum (i.e., > 0). Theorem 2. The limit process N*() is a continuous process with the locally independent increments. 相似文献
13.
Shu Nakamura 《Communications in Mathematical Physics》1994,161(1):63-76
Low energy behavior of Schrödinger operators with potentials which decay slowly at infinity is studied. It is shown that if the potential is positive then the zero energy is very regular and the resolvent is smooth near 0. This implies rapid local decay for the solutions of the Schrödinger equation. On the other hand, if the potential is negative then the resolvent has discontinuity at zero energy. Thus one cannot expect local decay faster than ordert
–1 ast. 相似文献
14.
We study the influence of disorder on propagation of waves in one-dimensional structures. Transmission properties of the process governed by the Schrödinger equation with the white noise potential can be expressed through the Lyapunov exponent γ which we determine explicitly as a function of the noise intensity σ and the frequency ω. We find uniform two-parameter asymptotic expressions for γ which allow us to evaluate γ for different relations between σ and ω. The value of the Lyapunov exponent is also obtained in the case of a short-range correlated noise, which is shown to be less than its white noise counterpart. 相似文献
15.
F. Delyon 《Journal of statistical physics》1985,40(5-6):621-630
We consider a discrete Schrödinger operator on l2() with a random potential decaying at infinity as ¦n¦–1/2. We prove that its spectrum is purely singular. Together with previous results, this provides simple examples of random Schrödinger operators having a singular continuous component in its spectrum. 相似文献
16.
We consider the Schrödinger operatorH = – +V(|x|) onR
3. Letn
denote the number of bound states with angular momentum (not counting the 2 + 1 degeneracy). We prove the following bounds onn
. LetV 0 and d/dr r
1-2p
(-V)1 –p
0 for somep [1/2, 1) then
|