共查询到20条相似文献,搜索用时 15 毫秒
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.
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
3.
We prove that all the non-negative Lyapunov exponents of difference Schrödinger equation
相似文献
4.
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
5.
G. D. Raikov 《Communications in Mathematical Physics》1989,124(4):665-685
We consider the discrete spectrum of the selfadjoint Schrödinger operatorA
h
=–h
2
+V defined inL
2(m) with potentialV which steadies at infinity, i.e.V(x)=g+|x|–
f(1+o(1)) as |x| for>0 and some homogeneous functionsg andf of order zero. Let
h
(),0, be the total multiplicity of the eigenvalues ofA
h
smaller thanM–, M being the minimum value ofg over the unit sphereS
m–1 (hence,M coincides with the lower bound of the essential spectrum ofA
h
). We study the asymptotic behaviour of
1() as0, or of
h
() ash0, the number0 being fixed. We find that these asymptotics depend essentially on the structure of the submanifold ofS
m–1, where the functiong takes the valueM, and generically are nonclassical, i.e. even as a first approximation (2)
m
h
() differs from the volume of the set {(x, )2m:h
2||2+V(x)<M–}.Partially supported by Contract No. 52 with the Ministry of Culture, Science and Education 相似文献
6.
George D. Raikov 《Letters in Mathematical Physics》1991,21(1):41-49
We consider the Schrödinger operator with electric potential V, which decays at infinity, and magnetic potential A. We study the asymptotic behaviour for large values of the electric field coupling constant of the eigenvalues situated under the essential-spectrum lower bound. We concentrate on the cases of rapidly decaying V (e.g. V L
m/2(
m
) for m 3) and arbitrary A, or slowly decaying V (i.e. V(x |x|–
, (0,2), as |x| ) and magnetic potentials A corresponding to constant magnetic fields B = curl A.Partially supported by the Ministry of Culture, Science and Education under Grant No. 52. 相似文献
7.
Ya. G. Sinai 《Journal of statistical physics》1987,46(5-6):861-909
The Schrödinger difference operator considered here has the form $$(H_\varepsilon (\alpha )\psi )(n) = - (\psi (n + 1) + \psi (n - 1)) + V(n\omega + \alpha )\psi (n)$$ whereV is aC 2-periodic Morse function taking each value at not more than two points. It is shown that for sufficiently small? the operatorH ?(α) has for a.e.α a pure point spectrum. The corresponding eigenfunctions decay exponentially outside a finite set. The integrated density of states is an incomplete devil's staircase with infinitely many flat pieces. 相似文献
8.
Inequalities on eigenvalues of the Schrödinger operator are re-examined in the case of spherically symmetric potentials. In particular, we obtain:
9.
O. V. Beloshapka 《Russian Journal of Mathematical Physics》2010,17(3):267-271
A Feynman formula for heat-type equations with respect to functions defined on the product of a real line and the space ℚp
n
is obtained. By a Feynman formula we mean a representation of a solution of the Cauchy problem for the differential evolution
equation as a limit of integrals over Cartesian powers of some space. The result thus obtained sharpens results of the paper
[1]. The role of the Laplace operator is played here by the Vladimirov operator. Equations of this type turned out to be useful
when describing the dynamics of proteins. 相似文献
10.
Yu. A. Semenov 《Letters in Mathematical Physics》1977,1(6):463-465
It is shown that the generalized eigenfunctions of the Schrödinger operator with singular potentials actins in L
2(3) are ordinary functions with determined asymptotic behaviour at infinity. 相似文献
11.
I. Ya. Goldsheid 《Communications in Mathematical Physics》1995,174(2):347-365
We consider the Schrödinger equation with a random potential
|