共查询到20条相似文献,搜索用时 580 毫秒
1.
We study spectral properties of
the discrete Laplacian H on the half-space
with random boundary condition ;
the V(n) are independent random variables on a
probability
space and λ is the coupling constant.
It is known that if the
V(n) have densities, then on the interval
[-2(d+1), 2(d+1)] (=σ(H
0),
the spectrum of the
Dirichlet Laplacian) the spectrum of H is P-a.s.
absolutely continuous for all λ [JL1].
Here we show that if the random potential P
satisfies the
assumption of Aizenman–Molchanov [AM], then there
are constants λ
d
and
Λ
d
such that for
|λ|<lambda;
d
and |λ|>
Λ
d
the spectrum of
H outside σ(H
0)
is P-a.s. pure point with exponentially decaying
eigenfunctions.
Received: 3 December 1998 / Accepted: 27 May 1999 相似文献
2.
Remco van der Hofstad Frank den Hollander Gordon Slade 《Communications in Mathematical Physics》2002,231(3):435-461
We construct the incipient infinite cluster measure (IIC) for sufficiently spread-out oriented percolation on ℤ
d
× ℤ+, for d +1 > 4+1. We consider two different constructions. For the first construction, we define ℙ
n
(E) by taking the probability of the intersection of an event E with the event that the origin is connected to (x,n) ℤ
d
× ℤ+, summing this probability over x ℤ
d
, and normalising the sum to get a probability measure. We let n → ∞ and prove existence of a limiting measure ℙ∞, the IIC. For the second construction, we condition the connected cluster of the origin in critical oriented percolation
to survive to time n, and let n → ∞. Under the assumption that the critical survival probability is asymptotic to a multiple of n
−1, we prove existence of a limiting measure ℚ∞, with ℚ∞ = ℙ∞. In addition, we study the asymptotic behaviour of the size of the level set of the cluster of the origin, and the dimension
of the cluster of the origin, under ℙ∞. Our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented
percolation to super-Brownian motion, for d+1 > 4+1.
Received: 13 December 2001 / Accepted: 11 July 2002 Published online: 29 October 2002
RID="*"
ID="*" Present address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513,
5600 MB Eindhoven, The Netherlands. E-mail: rhofstad@win.tue.nl 相似文献
3.
We eliminate by KAM methods the time dependence in a class of linear differential equations in ℓ2 subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator
H
0+εP(ωt) for ε small. Here H
0 is the one-dimensional Schr?dinger operator p
2+V, V(x)∼|x|α, α <2 for |x|→∞, the time quasi-periodic perturbation P may grow as |x|β, β <(α−2)/2, and the frequency vector ω is non resonant. The proof extends to infinite dimensional spaces the result valid
for quasiperiodically forced linear differential equations and is based on Kuksin's estimate of solutions of homological equations
with non-constant coefficients.
Received: 3 October 2000 / Accepted: 20 December 2000 相似文献
4.
We consider the time evolution of a disk under the action of a constant force and interacting with a free gas in the mean-field
approximation. Letting V0>0 be the initial velocity of the disk and V∞>0 its equilibrium velocity, namely the one for which the external field is balanced by the friction force exerted by the
background, we show that, if V∞−V0 is positive and sufficiently small, then the disk reaches V∞ with the power law t−(d+2), d=1,2,3 being the dimension of the physical space. The reason for this behavior is the long tail memory due to recollisions.
Any Markovian approximation (or simply neglecting the recollisions) yields an exponential approach to equilibrium. 相似文献
5.
J. Bourgain 《Communications in Mathematical Physics》1999,204(1):207-247
In this paper, we consider the following problem. Let iu
t
+Δu+V(x,t)u= 0 be a linear Schr?dinger equation ( periodic boundary conditions) where V is a real, bounded, real analytic potential which is periodic in x and quasi periodic in t with diophantine frequency vector λ. Denote S(t) the corresponding flow map. Thus S(t) preserves the L
2-norm and our aim is to study its behaviour on H
s
(T
D
), s> 0. Our main result is the growth in time is at most logarithmic; thus if φ∈H
s
, then
More precisely, (*) is proven in 1D and 2D when V is small. We also exhibit examples showing that a growth of higher Sobolev norms may occur in this context and (*) is thus
essentially best possible.
Received: 16 October 1997 / Accepted: 28 January 1999 相似文献
6.
We obtain bounds for the spectrum and for the total width of the spectral gaps for Jacobi matrices on ℓ2(ℤ) of the form (Hψ)
n
=a
n−1
ψ
n−1
+b
nψ
n
+a
nψ
n+1
, where a
n=a
n+q
and b
n=b
n+q
are periodic sequences of real numbers. The results are based on a study of the quasimomentum k(z) corresponding to H. We consider k(z) as a conformal mapping in the complex plane. We obtain the trace identities which connect integrals of the Lyapunov exponent
over the gaps with the normalised traces of powers of H.
Received: 17 April 2002 / Accepted: 1 October 2002 Published online: 13 January 2003
Communicated by B. Simon 相似文献
7.
Let H=−Δ+V be a two dimensional Schr?dinger operator with a real potential V(x) satisfying the decay condition , δ > 6. Let H
0=−Δ. We show that the wave operators are bounded in L
p
(R
2) under the condition that H has no zero resonances or bound states. In this paper the condition , imposed in a previous paper (K. Yajima, Commun. Math. Phys. 208, 125–152 (1999)), is removed.
Received: 13 September 2001 / Accepted: 15 October 2001 相似文献
8.
Pablo A. Ferrari Beat M. Niederhauser Eugene A. Pechersky 《Journal of statistical physics》2007,128(5):1159-1176
We consider the Harmonic crystal, a measure on
with Hamiltonian H(x)=∑
i,j
J
i,j
(x(i)−x(j))2+h∑
i
(x(i)−d(i))2, where x, d are configurations, x(i), d(i)∈ℝ, i,j∈ℤ
d
. The configuration d is given and considered as observations. The ‘couplings’ J
i,j
are finite range. We use a version of the harness process to explicitly construct the unique infinite volume measure at finite
temperature and to find the unique ground state configuration m corresponding to the Hamiltonian. 相似文献
9.
Hiroshi Isozaki 《Communications in Mathematical Physics》2001,222(2):371-413
We construct a generalized Fourier transformation ℱ(λ) associated with the 3-body Schr?dinger operator H=−Δ+Σ
a
V
a
(x
a
) and characterize all solutions of (H−λ)u= 0 in the Agmon–H?rmander space ℬ* as the image of ℱ(λ)*. These stationary solutions admit asymptotic expansions in ℬ* in terms of spherical waves associated with scattering channels.
Received: 20 September 2000 / Accepted: 20 May 2001 相似文献
10.
Photoacoustic spectroscopy of iodine molecule has been studied in gas phase using nitrogen laser-pumped tunable dye laser.
The experiment yielded the vibrational spectrum corresponding toX
1Σ+(0
g
+
)→B
3Π(0
g
+
) transition up to the convergence limit. The photo-acoustic spectrum in the region 17580–18850 cm−1 is presented along with the vibrational analysis. Five of the vibrational bands reported earlier by Venkateswarlu, Kumar
and McGlynn have been partially resolved and the structure of one of them has been analyzed and shown to be due to an overlap
of (14, 2) and (12, 1) bands. The analysis was based on a comparison with the highly resolved spectrum of Gerstenkorn and
Luc. The structure observed in the region 20200–20750 cm−1 which is beyond the convergence limit of the transitionX
1Σ+(0
g
+
)→B
3Π(0
u
+
) has been analyzed as due to two-photon absorption. Most of the bands could be assigned to two transitions both originating
in the ground state and terminating in two different electronic states 1
g
andE(0
g
+
), atT
e=40821 cm−1 (orT
0=41355 cm−1) andT
e=41411 cm−1 (orT
0=41355 cm−1) respectively. 相似文献
11.
On the Energy Growth of Some Periodically Driven Quantum Systems with Shrinking Gaps in the Spectrum
We consider quantum Hamiltonians of the form H(t)=H+V(t) where the spectrum of H is semibounded and discrete, and the eigenvalues behave as E
n
∼n
α
, with 0<α<1. In particular, the gaps between successive eigenvalues decay as n
α−1. V(t) is supposed to be periodic, bounded, continuously differentiable in the strong sense and such that the matrix entries with
respect to the spectral decomposition of H obey the estimate ‖V(t)
m,n
‖≤ε|m−n|−p
max {m,n}−2γ
for m≠n, where ε>0, p≥1 and γ=(1−α)/2. We show that the energy diffusion exponent can be arbitrarily small provided p is sufficiently large and ε is small enough. More precisely, for any initial condition Ψ∈Dom(H
1/2), the diffusion of energy is bounded from above as 〈H〉
Ψ
(t)=O(t
σ
), where
. As an application we consider the Hamiltonian H(t)=|p|
α
+ε
v(θ,t) on L
2(S
1,dθ) which was discussed earlier in the literature by Howland. 相似文献
12.
In this paper we show that for a.e. x∈[ 0,2 π) the operators defined on as
and with Dirichlet condition ψ− 1= 0, have pure point spectrum in with exponentially decaying eigenfunctions where δ > 0 and are small. As it is a simple consequence of known techniques that for small λ one has [− 2 +δ, 2−δ]⊂ spectrum (H(x)) for a.e.x∈[ 0, 2 π), we thus established Anderson localization on the spectrum up to the edges and the center. More general potentials
than cosine can be treated, but only those energies with nonzero spectral density are allowed. Finally, we prove the same
result for operators on the whole line ℤ with potential , where A:?2→?2 is a hyperbolic toral automorphism, F∈C
1(?2), ∫F= 0, and λ small. The basis for our analysis is an asymptotic formula for the Lyapunov exponent for λ→ 0 by Figotin–Pastur,
and generalized by Chulaevski–Spencer. We combine this asymptotic expansion with certain martingale large deviation estimates
in order to apply the methods developed by Bourgain and Goldstein in the quasi-periodic case.
Received: 28 January 2000 / Accepted: 14 June 2000 相似文献
13.
14.
Hellmut Baumgärtel 《International Journal of Theoretical Physics》2011,50(7):2002-2008
To asymptotic complete scattering systems {M
++V,M
+} on H+:=L2(R+,K{\mathcal{H}}_{+}:=L^{2}(\mathbf{R}_{+},{\mathcal{K}}, d
λ), where M
+ is the multiplication operator on H+{\mathcal{H}}_{+} and V is a trace class operator with analyticity conditions, a decay semigroup is associated such that the spectrum of the generator
of this semigroup coincides with the set of all resonances (poles of the analytic continuation of the scattering matrix into
the lower half plane across the positive half line), i.e. the decay semigroup yields a “time-dependent” characterization of
the resonances. As a counterpart a “spectral characterization” is mentioned which is due to the “eigenvalue-like” properties
of resonances. 相似文献
15.
S. Albeverio S. N. Lakaev K.A. Makarov Z.I. Muminov 《Communications in Mathematical Physics》2006,262(1):91-115
For a wide class of two-body energy operators h(k) on the d-dimensional lattice ℤd, d≥3, k being the two-particle quasi-momentum, we prove that if the following two assumptions (i) and (ii) are satisfied, then for
all nontrivial values k, k≠0, the discrete spectrum of h(k) below its threshold is non-empty. The assumptions are: (i) the two-particle Hamiltonian h(0) corresponding to the zero value of the quasi-momentum has either an eigenvalue or a virtual level at the bottom of its
essential spectrum and (ii) the one-particle free Hamiltonians in the coordinate representation generate positivity preserving
semi-groups. 相似文献
16.
17.
O.L. Safronov 《Communications in Mathematical Physics》1998,193(1):233-243
Given two selfadjoint operators A and V=V
+ -V
-, we study the motion of the eigenvalues of the operator A(t)=A-tV as t increases. Let α>0 and let λ be a regular point for A. We consider the quantities N
+(λ,α), N
-(λ,α), N
0(λ,α) defined as the number of the eigenvalues of the operator A(t) that pass point λ from the right to the left, from the left to the right or change the direction of their motion exactly
at point λ, respectively, as t increases from 0 to α>0. An abstract theorem on the asymptotics for these quantities is presented. Applications to Schr?dinger
operators and its generalizations are given.
Received: 9 April 1997 / Accepted: 26 August 1997 相似文献
18.
We state and prove a generalized adiabatic theorem for Markov chains and provide examples and applications related to Glauber
dynamics of the Ising model over ℤ
d
/nℤ
d
. The theorems derived in this paper describe a type of adiabatic dynamics for
l1(\mathbbRn+)\ell^{1}(\mathbb{R}^{n}_{+}) norm preserving, time inhomogeneous Markov transformations, while quantum adiabatic theorems deal with ℓ
2(ℂ
n
) norm preserving ones, i.e. gradually changing unitary dynamics in ℂ
n
. 相似文献
19.
T. Sato T. Majima Kenro Hashimoto Kouhei Hashimoto Y. Zama J. Matsumoto H. Shiromaru K. Okuno H. Tanuma T. Azuma 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2011,63(2):189-194
Microhydrated methylene blue cations,
MB+(H2O)
n
, are produced in an
electrospray ion source and their size-distributions are measured as a function of the
source temperature. A series of
MB+(H2O)
n
ions is observed up to
n ≃ 60. A striking feature observed in the mass spectra is that the
series of hydrated ions starts at n = 4; intensities of
n = 1–3 are extremely suppressed. The absence of
n = 1–3 ions is well explained by the energetics concerning evaporation
processes of water molecules, based on stable structures and the binding energies of
MB+(H2O)
n
ions calculated by DFT
calculations up to n = 5.
MB+(H2O)
n
ions for
n > 4 evaporate a single water molecule
sequentially, while MB+(H2O)4 tends to fragment into
MB+ and (H2O)4 rather than
MB+(H2O)3 and an H2O molecule. We have
observed a clear magic peak at n = 24, which strongly suggests that the
MB+(H2O)24 ion is formed by attaching a neutral
(H2O)20 cage onto an MB+(H2O)4
ion. 相似文献
20.
Frank Hövermann Herbert Spohn Stefan Teufel 《Communications in Mathematical Physics》2001,215(3):609-629
We consider the dynamics generated by the Schr?dinger operator H=−?Δ+V(x)+W(ɛx), where V is a lattice periodic potential and W an external potential which varies slowly on the scale set by the lattice spacing. We prove that in the limit ɛ→ 0 the time
dependent position operator and, more generally, semiclassical observables converge strongly to a limit which is determined
by the semiclassical dynamics.
Received: 7 February 2000 / Accepted: 7 July 2000 相似文献