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1.
On the Schrödinger equation and the eigenvalue problem 总被引:1,自引:0,他引:1
If
k
is thek
th eigenvalue for the Dirichlet boundary problem on a bounded domain in
n
, H. Weyl's asymptotic formula asserts that
, hence
. We prove that for any domain and for all
. A simple proof for the upper bound of the number of eigenvalues less than or equal to - for the operator –V(x) defined on
n
(n3) in terms of
is also provided.Research partially supported by a Sloan Fellowship and NSF Grant No. 81-07911-A1 相似文献
2.
H. C. Hsieh 《International Journal of Infrared and Millimeter Waves》1981,2(1):131-147
The expression for free carrier Faraday rotation and for ellipticity , as the function of the applied parallel static electric field
and static magnetic field
for a given value of wave angular frequency and electron concentration N0, are obtained and theoretically analyzed with the aid of one-dimensional linearized wave theory and Kane's non-parabolic isotropic dispersion law. It is shown that the maximum Faraday rotation occurs near the cyclotron resonance condition, which can be expressed as
, where
,
, and
. Here m* and e denote the effective mass and charge of electron, respectively. g is the forbidden bandgap of semiconductor. v0 is the carrier drift velocity, which is a non-linear function of E0 in high field condition. A possibility of a simple way of determining the non-linear v0 vs E0 characteristics of semiconductors by the measurement of Faraday rotation is also discussed. 相似文献
3.
We prove that Gibbs states for the Hamiltonian
, with thes
x varying on theN-dimensional unit sphere, obtained with nonrandom boundary conditions (in a suitable sense), are almost surely rotationally invariant if
withJ
xy i.i.d. bounded random variables with zero average, 1 in one dimension, and 2 in two dimensions. 相似文献
4.
For Lax-pair isospectral deformations whose associated spectrum, for given initial data, consists of the disjoint union of a finitely denumerable discrete spectrum (solitons) and a continuous spectrum (continuum), the matrix Riemann–Hilbert problem approach is used to derive the leading-order asymptotics as
of solutions
to the Cauchy problem for the defocusing nonlinear Schrödinger equation (
NLSE),
, with finite-density initial data
.The
NLSE dark soliton position shifts in the presence of the continuum are also obtained. 相似文献
5.
Claudio Albanese 《Communications in Mathematical Physics》1990,134(2):237-272
This is the continuation of a series of articles concerning a class of quantum spin systems with Hamiltonian operators of the form
相似文献
6.
An approximation procedure for the solution of stochastic nonlinear equations, which was derived from a variational principle in a previous paper, is applied to the problem of a particle that diffuses in a symmetric bistable potential starting from the point of unstable equilibrium. The second moment
and variance
for the particle's position
are calculated as functions of the timet. Good agreement is found with results recently obtained by Baibuzet al. from an approximate evaluation of a path integral expression for the probability density. 相似文献
7.
Hugo Parr 《Zeitschrift für Physik B Condensed Matter》1976,25(4):359-361
We have calculated analytically the superheating fieldH
sh
for bulk superconductors, correct to second order in. We find
, which agrees well with numerical computations for<0.5. The surface order parameter is
, and the penetration depth is
. 相似文献
8.
James Glimm 《Communications in Mathematical Physics》1967,5(5):343-386
A renormalization procedure is proposed. It gives rigorous mathematical meaning to the infinite cancellations in this model. A space cutoff is introduced in the interaction termV and soV has the form
, but there are no momentum cutoffs inV. There is an infinite constant and an infinite boson mass renormalization in this model. The main result is that the renormalized Hamiltonian is rigorously defined as a bilinear form in the Fock Hilbert space.This work was supported in part by the National Science Foundation, GP-6165. 相似文献
9.
We consider the Dirichlet Laplacian for astrip in
with one straight boundary and a width
, where $f$ is a smooth function of acompact support with a length 2b. We show that in the criticalcase,
, the operator has nobound statesfor small
.On the otherhand, a weakly bound state existsprovided
. In thatcase, there are positive c
1,c
2 suchthat the corresponding eigenvalue satisfies
for all
sufficiently small. 相似文献
10.
LetR be an expanding rational function with a real bounded Julia set, and let
be a Ruelle operator acting in a space of functions analytic in a neighbourhood of the Julia set. We obtain explicit expressions for the resolvent function
and, in particular, for the Fredholm determinantD()=det(I-L). It gives us an equation for calculating the escape rate. We relate our results to orthogonal polynomials with respect to the balanced measure ofR. Two examples are considered.The first named author was sponsored in part by the Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany) 相似文献
11.
We consider the large time asymptotic behavior of solutions to the Cauchy problem for the modified Korteweg–de Vries equation
, with initial data
. We assume that the coefficient
is real, bounded and slowly varying function, such that
, where
. We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space
. In comparison with the previous paper (Internat. Res. Notices
8 (1999), 395–418), here we exclude the condition that the integral of the initial data u
0 is zero. We prove the time decay estimates
and
for all
, where
. We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution. 相似文献
12.
We consider the Zakharov equation in space dimension two
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