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1.
Walter Craig 《Communications in Mathematical Physics》1989,126(2):379-407
This paper discusses certain aspects of the spectral and inverse spectral problems for the Schrödinger operator
, for q(x)C(), the space of bounded continuous functions. The trace formula of the title is the relation
相似文献
2.
A. Hof 《Journal of statistical physics》1995,81(3-4):851-855
We prove that for a large class of Schrödinger operators on aperiodic tilings the spectrum and the integrated density of states are the same for all tilings in the local isomorphism class, i.e., for all tilings in the orbit closure of one of the tilings. This generalizes the argument in earlier work from discrete strictly ergodic operators onl
2(
d
) to operators on thel
2-spaces of sets of vertices of strictly ergodic tilings. 相似文献
3.
LetS ?=??Δ+V, withV smooth. If 0<E 2
4.
A. Hof 《Journal of statistical physics》1993,72(5-6):1353-1374
We consider Schrödinger operators onl
2(
) with deterministic aperiodic potential and Schrödinger operators on the l2-space of the set of vertices of Penrose tilings and other aperiodic self-similar tilings. The operators onl
2(
) fit into the formalism of ergodic random Schrödinger operators. Hence, their Lyapunov exponent, integrated density of states, and spectrum are almost-surely constant. We show that they are actually constant: the Lyapunov exponent for one-dimensional Schrödinger operators with potential defined by a primitive substitution, the integrated density of states, and the spectrum in arbitrary dimension if the system is strictly ergodic. We give examples of strictly ergodic Schrödinger operators that include several kinds of almost-periodic operators that have been studied in the literature. For Schrödinger operators on Penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular Penrose tiling under consideration. 相似文献
5.
We studyH=–d
2/dx
2+V(x) withV(x) limit periodic, e.g.V(x)=a
n
cos(x/2
n
) with a
n
<. We prove that for a genericV (and for generica
n
in the explicit example), (H) is a Cantor ( nowhere dense, perfect) set. For a dense set, the spectrum is both Cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous.Research partially supported by NSF Grant MCS78-01885On leave from Department of Physics, Princeton UniversityOn leave from Departments of Mathematics and Physics, Princeton University; during 1980–81 Sherman Fairchild Visiting Scholar at Caltech 相似文献
6.
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
7.
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.
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 8.
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
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