共查询到20条相似文献,搜索用时 31 毫秒
1.
Thomas Chen 《Communications in Mathematical Physics》2006,267(2):355-392
We study the macroscopic scaling and weak coupling limit for a random Schrödinger equation on $\mathbb{Z}^3We study the macroscopic scaling and weak coupling limit for a random Schr?dinger equation on
. We prove that the Wigner transforms of a large class of “macroscopic” solutions converge in r
th mean to solutions of a linear Boltzmann equation, for any 1 ≤ r < ∞. This extends previous results where convergence in expectation was established. 相似文献
2.
We show that the knowledge of the set of the Cauchy data on the boundary of a bounded open set in ${\mathbb{R}^n}$ , ${n \geq 3}$ , for the magnetic Schrödinger operator with L ∞ magnetic and electric potentials, determines the magnetic field and electric potential inside the set uniquely. The proof is based on a Carleman estimate for the magnetic Schrödinger operator with a gain of two derivatives. 相似文献
3.
Heinz Langer Branko Najman Christiane Tretter 《Communications in Mathematical Physics》2006,267(1):159-180
In this paper we investigate an abstract Klein–Gordon equation by means of indefinite inner product methods. We show that, under certain assumptions on the potential which are more general than in previous works, the corresponding linear operator A is self-adjoint in the Pontryagin space induced by the so-called energy inner product. The operator A possesses a spectral function with critical points, the essential spectrum of A is real with a gap around 0, and the non-real spectrum consists of at most finitely many pairs of complex conjugate eigenvalues of finite algebraic multiplicity; the number of these pairs is related to the ‘size’ of the potential. Moreover, A generates a group of bounded unitary operators in the Pontryagin space . Finally, the conditions on the potential required in the paper are illustrated for the Klein–Gordon equation in ; they include potentials consisting of a Coulomb part and an L
p
-part with n ≤ p < ∞.Branko Najman: Deceased 相似文献
4.
Orlin Stoytchev 《Letters in Mathematical Physics》2007,79(3):235-249
Any -graded C
*-dynamical system with a self-adjoint graded-Kubo-Martin-Schwinger (KMS) functional on it can be represented (canonically)
as a -graded algebra of bounded operators on a -graded Hilbert space, so that the grading of the latter is compatible with the functional. The modular conjugation operator
plays a crucial role in this reconstruction. The results are generalized to the case of an unbounded graded-KMS functional
having as dense domain the union of a net of C
*-subalgebras. It is shown that the modulus of such an unbounded graded-KMS functional is KMS.
相似文献
5.
We establish the phenomenon of Anderson localisation for a quantum two-particle system on a lattice with short-range interaction and in presence of an IID external potential with sufficiently regular marginal distribution. 相似文献
6.
Edward Fan Zuhair Khandker Robert S. Strichartz 《Communications in Mathematical Physics》2009,287(1):351-382
We study the analog of the quantum-mechanical harmonic oscillator on infinite blowups of the Sierpinski Gasket, using the
standard Kigami Laplacian. Our main task is to find a class of potentials analogous to on the line. We describe a class of potentials u with the properties Δu = 1, u attains a minimum value zero, and u → ∞ at infinity. We show how to construct such potentials attaining the minimum value at any prescribed point, and we show
how to parameterize the class of potentials by a certain surface in . We obtain estimates for the growth rate of the eigenvalue counting function for −Δ + u. We obtain numerical approximations to the eigenfunctions, and in particular observe that the ground-state eigenfunction
resembles a Gaussian function.
Research supported by Chinese University Mathematics Alumni.
Research supported by the National Science Foundation through the Research Experiences for Undergraduates program at Cornell.
Research supported in part by the National Science Foundation, grant DMS 0652440. 相似文献
7.
Ran Wei 《Journal of statistical physics》2018,172(4):1057-1085
We study the Cauchy directed polymer model on \(\mathbb {Z}^{1+1}\), where the underlying random walk is in the domain of attraction to the 1-stable law. We show that, if the random walk satisfies certain regularity assumptions and its symmetrized version is recurrent, then the free energy is strictly negative at any inverse temperature \(\beta >0\). Moreover, under additional regularity assumptions on the random walk, we can identify the sharp asymptotics of the free energy in the high temperature limit, namely,
相似文献
$$\begin{aligned} \lim \limits _{\beta \rightarrow 0}\beta ^{2}\log (-p(\beta ))=-c. \end{aligned}$$
8.
Asao Arai 《Letters in Mathematical Physics》2007,80(3):211-221
Let H be a self-adjoint operator on a complex Hilbert space . A symmetric operator T on is called a time operator of H if, for all , (D(T) denotes the domain of T) and . In this paper, spectral properties of T are investigated. The following results are obtained: (i) If H is bounded below, then σ(T), the spectrum of T, is either (the set of complex numbers) or . (ii) If H is bounded above, then is either or . (iii) If H is bounded, then . The spectrum of time operators of free Hamiltonians for both nonrelativistic and relativistic particles is exactly identified.
Moreover spectral analysis is made on a generalized time operator.
This work is supported by the Grant-in-Aid No.17340032 for Scientific Research from the JSPS. 相似文献
9.
Let L = ?Δ? W be a Schrödinger operator with a potential $W\in L^{\frac{n+1}{2}}(\mathbb{R}^n)Let L = −Δ− W be a Schr?dinger operator with a potential
,
. We prove that there is no positive eigenvalue. The main tool is an
Carleman type estimate, which implies that eigenfunctions to positive eigenvalues must be compactly supported. The Carleman estimate builds on delicate dispersive estimates established in [7]. We also consider extensions of the result to variable coefficient operators with long range and short range potentials and gradient potentials.The first author was partially supported by DFG grant KO1307/1 and also by MSRI for Fall 2005The second author was partially supported by NSF grants DMS0354539 and DMS 0301122 and also by MSRI for Fall 2005 相似文献
10.
We consider the time-dependent Schrödinger-Hartree equation (1) $$iu_t + \Delta u = \left( {\frac{1}{r}*|u|^2 } \right)u + \lambda \frac{u}{r},(t, x) \in \mathbb{R} \times \mathbb{R}^3 ,$$ (2) $$u(0,x) = \phi (x) \in \Sigma ^{2,2} ,x \in \mathbb{R}^3 ,$$ where λ≧0 and \(\Sigma ^{2,2} = \{ g \in L^2 ;\parallel g\parallel _{\Sigma ^{2,2} }^2 = \sum\limits_{|a| \leqq 2} {\parallel D^a g\parallel _2^2 + \sum\limits_{|\beta | \leqq 2} {\parallel x^\beta g\parallel _2^2< \infty } } \} \) . We show that there exists a unique global solutionu of (1) and (2) such that $$u \in C(\mathbb{R};H^{1,2} ) \cap L^\infty (\mathbb{R};H^{2,2} ) \cap L_{loc}^\infty (\mathbb{R};\Sigma ^{2,2} )$$ with $$u \in L^\infty (\mathbb{R};L^2 ).$$ Furthermore, we show thatu has the following estimates: $$\parallel u(t)\parallel _{2,2} \leqq C,a.c. t \in \mathbb{R},$$ and $$\parallel u(t)\parallel _\infty \leqq C(1 + |t|)^{ - 1/2} ,a.e. t \in \mathbb{R}.$$ 相似文献
11.
Markus Heydenreich Remco van der Hofstad 《Communications in Mathematical Physics》2007,270(2):335-358
We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in sufficiently high dimensions, or when d > 6 for sufficiently spread-out percolation. We use a relatively simple coupling argument to show that this largest critical
cluster is, with high probability, bounded above by a large constant times V
2/3 and below by a small constant times , where V is the volume of the torus. We also give a simple criterion in terms of the subcritical percolation two-point function on
under which the lower bound can be improved to small constant times , i.e. we prove random graph asymptotics for the largest critical cluster on the high-dimensional torus. This establishes a conjecture by [1], apart from logarithmic
corrections. We discuss implications of these results on the dependence on boundary conditions for high-dimensional percolation.
Our method is crucially based on the results in [11, 12], where the scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on . We also strongly rely on mean-field results for percolation on proved in [17–20]. 相似文献
12.
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.
U.R.A. 760 C.N.R.S. 相似文献
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
13.
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
14.
15.
Recently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M, have been solved. Here we give further results, applicable to several problems:
Even more, accurate versions of this last result are obtained if the Cauchy hypersurface S were non-spacelike (including non-smooth, or achronal but non-acausal). 相似文献
(1) | Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, but a continuous extension (in fact, valid for any compact achronal subset K) is still possible. |
(2) | Given any spacelike Cauchy hypersurface S, a Cauchy temporal function (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy hypersurfaces as levels) with is constructed – thus, the spacetime splits orthogonally as in a canonical way. |
16.
We define the notion of a moment map and reduction in both generalized complex geometry and generalized Kähler geometry. As an application, we give very simple explicit constructions of bi-Hermitian structures on $\mathbb{C}\mathbb{P}^{N}We define the notion of a moment map and reduction in both generalized complex geometry and generalized K?hler geometry. As an application, we give very simple explicit constructions of bi-Hermitian structures on
, Hirzebruch surfaces, the blow up of
at arbitrarily many points, and other toric varieties, as well as complex Grassmannians. 相似文献
17.
We obtain exact results in α′ for open and closed A-model topological string amplitudes on a large class of toric Calabi-Yau threefolds by using their correspondence with five
dimensional gauge theories. The toric Calabi-Yaus that we analyze are obtained as minimal resolution of cones over Y
p,q
manifolds and give rise via M-theory compactification to SU(p) gauge theories on . As an application we present a detailed study of the local case and compute open and closed genus zero Gromov-Witten invariants of the orbifold. We also display the modular structure of the topological wave function and give predictions for higher genus amplitudes.
The mirror curve in this case is the spectral curve of the relativistic A
1 Toda chain. Our results also indicate the existence of a wider class of relativistic integrable systems associated to generic
Y
p,q
geometries. 相似文献
18.
We consider a random walk on the support of an ergodic simple point process on , d ≥ 2, furnished with independent energy marks. The jump rates of the random walk decay exponentially in the jump length and
depend on the energy marks via a Boltzmann–type factor. This is an effective model for the phonon–induced hopping of electrons
in disordered solids in the regime of strong Anderson localization. Under some technical assumption on the point process we
prove an upper bound for the diffusion matrix of the random walk in agreement with Mott law. A lower bound for d ≥ 2 in agreement with Mott law was proved in [8]. 相似文献
19.
R. B. Zhang 《Communications in Mathematical Physics》2008,280(2):545-562
A Schrödinger type equation on the superspace $\mathbb {R}^{D|2n}A Schr?dinger type equation on the superspace is studied, which involves a potential inversely proportional to the negative of the osp(D|2n) invariant “distance” away from the origin. An osp(2, D + 1|2n) dynamical supersymmetry for the system is explicitly constructed, and the bound states of the system are shown to form an
irreducible highest weight module for this superalgebra. A thorough understanding of the structure of the irreducible module
is obtained. This in particular enables the determination of the energy eigenvalues and the corresponding eigenspaces as well
as their respective dimensions. 相似文献
20.
We prove bounds on moments of the Smoluchowski coagulation equations with diffusion, in any dimension d ≥ 1. If the collision propensities α(n, m) of mass n and mass m particles grow more slowly than , and the diffusion rate is non-increasing and satisfies for some b
1 and b
2 satisfying 0 ≤ b
2 < b
1 < ∞, then any weak solution satisfies for every and T ∈(0, ∞), (provided that certain moments of the initial data are finite). As a consequence, we infer that these conditions
are sufficient to ensure uniqueness of a weak solution and its conservation of mass.
This work was performed while A.H. held a postdoctoral fellowship in the Department of Mathematics at U.B.C.
This work is supported in part by NSF grant DMS0307021. 相似文献