共查询到20条相似文献,搜索用时 609 毫秒
1.
D. Ya. Petrina 《Ukrainian Mathematical Journal》2008,60(9):1448-1476
We consider the BCS Hamiltonian with sources, as proposed by Bogolyubov and Bogolyubov, Jr. We prove that the eigenvectors
and eigenvalues of the BCS Hamiltonian with sources can be exactly determined in the thermodynamic limit. Earlier, Bogolyubov
proved that the energies per volume of the BCS Hamiltonian with sources and the approximating Hamiltonian coincide in the
thermodynamic limit.
Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1243–1269, September, 2008. 相似文献
2.
We investigate the spectrum of a model Hamiltonian with BCS and mean-field interaction in a finite domain under periodic boundary conditions. The model Hamiltonian is considered on the states of pairs and waves of density charges and their excitations. It is represented as the sum of three operators that describe noninteracting pairs, the interaction between pairs, and the interaction between pairs and waves of density charges. The last two operators tend to zero in the thermodynamic limit, and the spectrum of the model Hamiltonian coincides with the spectrum of noninteracting pairs with chemical potential shifted by mean-field interaction. It is shown that the model and approximating Hamiltonians coincide in the thermodynamic limit on their ground and excited states and both have two branches of eigenvalues and eigenvectors. 相似文献
3.
We establish that the averages per volume of the BCS and approximating Hamiltonians over all excited states coincide in the thermodynamic limit. Earlier, this was established only for the ground state. 相似文献
4.
D. Ya. Petrina 《Ukrainian Mathematical Journal》2005,57(11):1763-1791
The BCS Hamiltonian of superconductivity has the second branch of eigenvalues and eigenvectors. It consists of wave functions
of pairs of electrons in ground and excited states. The continuous spectrum of excited pairs is separated by a nonzero gap
from the point of the discrete spectrum that corresponds to the pair in the ground state. The corresponding grand partition
function and free energy are exactly calculated. This implies that, for low temperatures, the system is in the condensate
of pairs in the ground state. The sequence of correlation functions is exactly calculated in the thermodynamic limit, and
it coincides with the corresponding sequence of the system with approximating Hamiltonian. The gap in the spectrum of excitations
depends continuously on temperature and is different from zero above the critical temperature corresponding to the first branch
of the spectrum. In our opinion, this fact explains the phenomenon of “pseudogap.”
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1508–1533, November, 2005. 相似文献
5.
V. Zh. Sakbaev 《P-Adic Numbers, Ultrametric Analysis, and Applications》2012,4(2):115-129
We study the dynamics of quantum system with degenerated Hamiltonian. To this end we consider the approximating sequence of
regularized Hamiltonians and corresponding sequence of dynamical semigroups acting in the space of quantum states. The limit
points set of the sequence of regularized semigroups is obtained as the result of averaging by finitely additive measure on
the set of regularizing parameters. We establish that the family of averaging dynamical maps does not possess the semigroup
property and the injectivity property. We define the functionals on the space of maps of the time interval into the quantum
states space such that the maximum points of this functionals coincide with the trajectories of the family of averaging dynamical
maps. 相似文献
6.
D. Ya. Petrina 《Ukrainian Mathematical Journal》2000,52(5):765-788
The BCS Hamiltonian in a finite cube with periodic boundary condition is considered. The special subspace of pairs of particles
with opposite momenta and spin is introduced. It is proved that, in this subspace, the spectrum of the BCS Hamiltonian is
defined exactly for one pair, and for n pairs the spectrum is defined through the eigenvalues of one pair and a term that
tends to zero as the volume of the cube tends to infinity. On the subspace of pairs, the BCS Hamiltonian can be represented
as a sum of two operators. One of them describes the spectra of noninteracting pairs and the other one describes the interaction
between pairs that tends to zero as the volume of the cube tends to infinity. It is proved that, on the subspace of pairs,
as the volume of the cube tends to infinity, the BCS Hamiltonian tends to the approximating Hamiltonian, which is a quadratic
form with respect to the operators of creation and annihilation. 相似文献
7.
Yuan Shan 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(14):4809-4819
This paper consider the multiple solutions for even Hamiltonian systems satisfying Sturm-Liouville boundary conditions. The gradient of Hamiltonian function is generalized asymptotically linear. The solutions obtained are shown to coincide with the critical points of a dual functional. Thanks to the index theory for linear Hamiltonian systems by Dong (2010) [1], we find critical points of this dual functional by verifying the assumptions of a lemma about multiple critical points given by Chang (1993) [2]. 相似文献
8.
A rigorous mathematical theory of approximations is developed for abstract nonrelativistic quantum scattering systems within the two-Hilbert-space framework. An approximate space of asymptotic states and an approximate asymptotic Hamiltonian must be specified initially. An approximate N-particle Hamiltonian is then constructed and proved to be self-adjoint. Approximate wave operators are shown to exist and, in certain interesting cases, to be asymptotically complete. Certain sequences of the approximate wave operators are proved to converge to the exact wave operators in an appropriate limit. Thus approximate scattering operators are unitary and converge to the exact scattering operator. 相似文献
9.
V. V. Losyakov 《Theoretical and Mathematical Physics》2011,166(1):94-103
We obtain an exact scheme describing the dynamics of the N-particle state in the BCS model that is similar to the Gel’fand-Yaglom
procedure for finding the transition amplitude of the harmonic oscillator with a time-dependent frequency. We find the many-particle
spectrum and the eigenfunctions of the BCS Hamiltonian in the limit as the system volume tends to infinity. 相似文献
10.
We consider the model and approximating Hamiltonians directly in the case of infinite volume. We show that each of these Hamiltonians has two branches of the spectrum and two systems of eigenvectors, which represent excitations of the ground states of the model and approximating Hamiltonians as well as the ground states themselves. On both systems of eigenvectors, the model and approximating Hamiltonians coincide with one another. In both branches of the spectrum, there is a gap between the eigenvalues of the ground and excited states. 相似文献
11.
I. G. Bostrem A. S. Ovchinnikov V. E. Sinitsyn 《Theoretical and Mathematical Physics》2006,149(2):1527-1544
We propose a double-pass method of exact diagonalization of a finite cluster on the base of functions that have a definite
total spin and transform by a definite irreducible representation of the point symmetry group of the lattice. We also propose
the method for approximating the energy spectrum in the thermodynamic limit using the spectrum of the surrounding states,
which increases the calculation accuracy leaving the cluster size invariant. The algorithm details are extensively illustrated
with an example of clusters of spin 1/2 on a simple square lattice.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 2, pp. 262–280, November, 2006. 相似文献
12.
We study the mean-field approximation of quantum electrodynamics (QED) by means of a thermodynamic limit. The QED Hamiltonian is written in Coulomb gauge and does not contain any normal ordering or choice of bare electron/positron subspaces. Neglecting photons, we properly define this Hamiltonian in a finite box [−L/2; L/2)3, with periodic boundary conditions and an ultraviolet cutoff λ. We then study the limit of the ground state (i.e., the vacuum) energy and of the minimizers as L goes to infinity, in the Hartree-Fock approximation. In the case with no external field, we prove that the energy per volume converges and obtain in the limit a translation-invariant projector describing the free Hartree-Fock vacuum. We also define the energy per unit volume of translation-invariant states and prove that the free vacuum is the unique minimizer of this energy. In the presence of an external field, we prove that the difference between the minimum energy and the energy of the free vacuum converges as L goes to infinity. We obtain in the limit the so-called Bogoliubov-Dirac-Fock functional. The Hartree-Fock (polarized) vacuum is a Hilbert-Schmidt perturbation of the free vacuum and it minimizes the Bogoliubov-Dirac-Fock energy. © 2006 Wiley Periodicals, Inc. 相似文献
13.
We prove that a single-jump unitary quantum stochastic evolution is unitarily equivalent to the Dirac boundary-value problem on the half-line in an extended space. It is shown that this solvable model can be derived from the Schrödinger boundary-value problem for a positive relativistic Hamiltonian on the half-line as the inductive ultrarelativistic limit corresponding to the input flow of Dirac particles with asymptotically infinite momenta. Thus the problem of stochastic approximation can be reduced to a quantum mechanical boundary-value problem in the extended space. The problem of microscopic time reversibility is also discussed in the paper. 相似文献
14.
Andrea Carati Luigi Galgani Alberto Maiocchi Fabrizio Gangemi Roberto Gangemi 《Regular and Chaotic Dynamics》2016,21(6):660-664
A review is given of the studies aimed at extending to the thermodynamic limit stability results of Nekhoroshev type for nearly integrable Hamiltonian systems. The physical relevance of such an extension, i. e., of proving the persistence of regular (or ordered) motions in that limit, is also discussed. This is made in connection both with the old Fermi–Pasta–Ulam problem, which gave origin to such discussions, and with the optical spectral lines, the existence of which was recently proven to be possible in classical models, just in virtue of such a persistence. 相似文献
15.
L. D. Kudryavtsev 《Journal of Mathematical Sciences》2004,124(4):5154-5162
The concept of almost-normed spaces is introduced. It is proved that the space of sufficiently smooth functions asymptotically approximating to polynomials (of degrees no higher than a given one) as their argument tends to infinity is an almost-normed space. It is demonstrated that this space is a complete metric space with respect to the metrics generated by the almost-norm introduced. The space of functions strongly asymptotically approximating to polynomials is defined, and its embedding into the space of functions asymptotically approximating to polynomials is proved. The results obtained give a new approach to studying boundary-value problems with asymptotic initial value data at singular points of ordinary differential equations. 相似文献
16.
Rong Cheng 《Acta Appl Math》2010,110(1):209-214
In many fields of applications, especially in applications from mechanics, many equations of motion can be written as Hamiltonian
systems. In this paper, we study a class of asymptotically linear Hamiltonian systems. We construct a symplectic transformation
which reduces the linear systems of the Hamiltonian systems. This reduction method can be applied to study the existence of
periodic solutions for a class of asymptotically linear Hamiltonian systems under weaker conditions on the linear systems
of the Hamiltonian systems. 相似文献
17.
I. N. Sergeev 《Differential Equations》2017,53(4):479-484
We show that, for any linear Hamiltonian system, there exists an arbitrarily close (in the uniform metric on the half-line) linear Hamiltonian system whose upper and lower Lyapunov exponents coincide with the upper and lower upper-limit central Vinograd–Millionshchikov exponents, respectively, of the original system and whose upper and lower Perron exponents coincide with the respective lower-limit exponents of the original system. 相似文献
18.
Banach空间中几乎渐近非扩张型映象的不动点的迭代逼近 总被引:6,自引:0,他引:6
在Banach空间中引入了一类新的几乎渐近非扩张型映象,概括了Banach空间中若干熟知的非线性的Lipschitz映象类与非Lipschitz映象类成特例;例如,熟知的非扩张映象类,渐近非扩张映象类与渐近非扩张型映象类.考虑了用于逼近几乎渐近非扩张型映象不动点的带误差的修改了的Ishikawa迭代序列的收敛性问题.关于Banach空间范数的S.S.Chang的不等式与H.K.Xu的不等式皆被用于做精确不动点与近似不动点间的误差估计.而且,张石生教授用于做带误差的修改了的Ishikawa迭代序列收敛性分析的方法(应用数学和力学,2001,22(1):23-31)被推广到几乎渐近非扩张型映象的情况.给出了用于求一致凸Banach空间中几乎渐近非扩张型映象不动点的带误差的修改了的Ishikawa迭代序列的新的收敛判据.并且,由该判据,立即得到了此类映象的带误差的修改了的Mann迭代序列的新的收敛判据.上述结果统一、改进与推广了张石生教授关于用带误差的修改了的Ishikawa与Mann迭代序列来逼近渐近非扩张型映象不动点方面的结果. 相似文献
19.
A closed, entirely quantum mechanical Josephson oscillator model is considered,
consisting of two superconductors in tunneling contact, which weakly interact
with the photon field. For each superconductor we use, for notational simplicity,
the BCS model in the strong coupling approximation and restrict ourselves
to Andersons quasi-spin formulation. The thermodynamic limit of the global (non-equilibrium)
dynamics is formulated for a large variety of states. It arises a generalization
of previously developed cocycle equations, connecting the collective behaviour
of the two superconductors with the photon field dynamics. In the physically
most interesting situations, where the two superconductors are in thermal
equilibrium (below the critical temperature) at voltage difference V, we show, that
for arbitrary initial states the outgoing multi-photon states are quantum optically
all-order coherent and constitute an almost monochromatic radiation of frequency
$ 2eV/\hbar $. Furthermore, we deduce in detail, how the collective behaviour of the superconductors
influences the collective (that are the optical) features of the emitted
microwave photons.
Communicated by Joel Feldman
submitted 14/02/03, accepted: 04/03/03 相似文献
20.
We study non-adiabatic transitions in scattering theory for the timedependent molecular Schr?dinger equation in the Born-Oppenheimer
limit. We assume the electron Hamiltonian has finitely many levels and consider the propagation of coherent states with high
enough total energy.
When two of the electronic levels are isolated from the rest of the electron Hamiltonian’s spectrum and display an avoided
crossing, we compute the component of the nuclear wave function associated with the non-adiabatic transition that is generated
by propagation through the avoided crossing. This component is shown to be exponentially small in the square of the Born-Oppenheimer
parameter, due to the Landau-Zener mechanism. It propagates asymptotically as a free Gaussian in the nuclear variables, and
its momentum is shifted. The total transition probability for this transition and the momentum shift are both larger than
what one would expect from a naive approximation and energy conservation.
Communicated by Yosi Avron
submitted 14/10/04, accepted 18/01/05
An erratum to this article is available at . 相似文献