共查询到20条相似文献,搜索用时 12 毫秒
1.
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
2.
Lieven Smits 《Communications in Mathematical Physics》1989,123(3):527-528
We prove that, under mild regularity conditions, the magnetic Schrödinger semigroup generated by
has its range inside the bounded continuous functions. We also give a counterexample for the general case.Sponsored by the Belgian National Science Foundation NFWO 相似文献
3.
H. Tamura 《Letters in Mathematical Physics》1989,17(1):31-36
We consider a N-body Schrödinger operator H=H
0+V. The interaction V is given by a sum of pair potentials V
jk(y)(=V
jk
s
+V
jk
l
), y R3. We assume that: V
jk
s
=O(|y|-(1+p)), p>0, as |y| for the short-range part V
jk
s
;
for the long-range part V
jk
l
. Under this assumption, we prove the principle of limiting absorption for H. The obtained result is essentially as good as those obtained in the two-body case. The proof is done by a slight modification of the remarkable commutator method due to Mourre. 相似文献
4.
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
5.
LetS ?=??Δ+V, withV smooth. If 0<E 2V(x), the spectrum ofS ? nearE 2 consists (for ? small) of finitely-many eigenvalues,λ j (?). We study the asymptotic distribution of these eigenvalues aboutE 2 as ?→0; we obtain semi-classical asymptotics for $$\sum\limits_j {f\left( {\frac{{\sqrt {\lambda _j (\hbar )} - E}}{\hbar }} \right)} $$ with \(\hat f \in C_0^\infty \) , in terms of the periodic classical trajectories on the energy surface \(B_E = \left\{ {\left| \xi \right|^2 + V(x) = E^2 } \right\}\) . This in turn gives Weyl-type estimates for the counting function \(\# \left\{ {j;\left| {\sqrt {\lambda _j (\hbar )} - E} \right| \leqq c\hbar } \right\}\) . We make a detailed analysis of the case when the flow onB E is periodic. 相似文献
6.
7.
E. Balslev 《Communications in Mathematical Physics》1977,52(2):127-146
The complex-dilated many-body Schrödinger operatorH(z) is decomposed on invariant subspaces associated with the cuts {+z
–2
R
+}, where is any threshold, and isolated spectral points. The interactions are dilation-analytic multiplicative two-body potentials, decaying asr
–1+ atr=0 and asr
–1+ atr=. 相似文献
8.
We discuss stochastic Schrödinger operators and Jacobi matrices with wave functions, taking values in
l
so there are 2l Lyaponov exponents 1...l0
l+1...2l
=–1. Our results include the fact that if 1=0 on a set positive measure, thenV is deterministic and one that says that {E|exactly 2j 's are zero} is the essential support of the a.c. spectrum of multiplicity 2j.Research partially supported by USNSF under grant DMS-8416049 相似文献
9.
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
10.
Yu. A. Semenov 《Communications in Mathematical Physics》1977,53(3):277-284
We discuss the question of when the closure of the Schrödinger operator, –+V, acting inL
p(R
l,d
lx), generates a strongly continuous contraction semigroup. We prove a series of theorems proving the stability for –:L
pL
p of the property of having am-accretive closure under perturbations by functions inL
loc
q
(1<pq). The connection with form sums and the Trotter product formula are considered. These results generalize earlier results of Kato, Kalf-Walter, Semenov and Beliy-Semenov in that we allow more general local singularities, including arbitrary singularities at one point, and arbitrary growth at infinity. We exploit bilinear form methods, Kato's inequality and certain properties of infinitesimal generators of contractions. 相似文献
11.
Jingbo Xia 《Communications in Mathematical Physics》1994,159(3):619-645
We give two formulas for the lowest point in the spectrum of the Schrödinger operatorL=–(d/dt)p(d/dt)+q, where the coefficientsp andq are real-valued, bounded, uniformly continuous functions on the real line. We determine whether or not is an eigenvalue forL in terms of a set of probability measures on the maximal ideal space of theC
*-algebra generated by the translations ofp andq.Research supported in part by the National Science Foundation under Grant DMS-910496 相似文献
12.
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
13.
《Physica D: Nonlinear Phenomena》2001,148(3-4):221-226
The critical speed for the nucleation of quantized vortices in the nonlinear Schrödinger equation (NLS) for a flow around a disk in two spatial dimensions is discussed in this paper. This problem is closely related to a compressible flow around a disk. The flow is computed via a Janzen–Rayleigh expansion for low Mach number. The calculation leads to an estimate for the critical Mach number Mc=0.36969(7)… 相似文献
14.
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
15.
We propose a technical reformulation of the measurement problem of quantum mechanics, which is based on the postulate that the final state of a measurement is classical; this accords with experimental practice as well as with Bohr’s views. Unlike the usual formulation (in which the post-measurement state is a unit vector in Hilbert space), our version actually opens the possibility of admitting a purely technical solution within the confines of conventional quantum theory (as opposed to solutions that either modify this theory, or introduce unusual and controversial interpretative rules and/or ontologies). To that effect, we recall a remarkable phenomenon in the theory of Schrödinger operators (discovered in 1981 by Jona-Lasinio, Martinelli, and Scoppola), according to which the ground state of a symmetric double-well Hamiltonian (which is paradigmatically of Schrödinger’s Cat type) becomes exponentially sensitive to tiny perturbations of the potential as ?→0. We show that this instability emerges also from the textbook wkb approximation, extend it to time-dependent perturbations, and study the dynamical transition from the ground state of the double well to the perturbed ground state (in which the cat is typically either dead or alive, depending on the details of the perturbation). Numerical simulations show that adiabatically arising perturbations may (quite literally) cause the collapse of the wave-function in the classical limit. Thus, at least in the context of a simple mathematical model, we combine the technical and conceptual virtues of decoherence (which fails to solve the measurement problem but launches the key idea that perturbations may come from the environment) with those of dynamical collapse models à la grw (which do solve the measurement problem but are ad hoc), without sharing their drawbacks: single measurement outcomes are obtained (instead of merely diagonal reduced density matrices), and no modification of quantum mechanics is needed. 相似文献
16.
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 17.
We investigate the spectrum of the following random Schrödinger operators:
|