Description: A random unitary channel is one that is given by a convex combination of unitary channels. It is shown that the conjectures on the additivity of the minimum output entropy and the multiplicativity of the maximum output p-norm can be equivalently restated in terms of random unitary channels. This is done by constructing a random unitary approximation to a general quantum channel. This approximation can be constructed efficiently, and so it is also applied to ...

By: H. D. Cornean, P. Duclos, G. Nenciu, and R. Purice

Description: Consider a three dimensional system which looks like a cross connected pipe system, i.e., a small sample coupled to a finite number of leads. We investigate the current running through this system, in the linear response regime, when we adiabatically turn on an electrical bias between leads. The main technical tool is the use of a finite volume regularization, which allows us to define the current coming out of a lead as the time derivative of its charge. We...

Description: We consider the system = a(z−a1x3−a2x2−bx), = −z, = −b1x+y+b2z, where a and b are parameters and b1 = 7/10, b2 = 6/25, a1 = 44/3, and a2 = 41/2. We analyze the existence of local and global analytic first integrals.

Description: The renormalization group (RG) method for differential equations is one of the perturbation methods for obtaining approximate solutions. This article shows that the RG method is effectual for obtaining an approximate center manifold and an approximate flow on it when applied to equations having a center manifold.

Description: We derive an asymptotic bound for the error of state estimation when we are allowed to use the quantum correlation in the measuring apparatus. It is also proven that this bound can be achieved in any statistical model in the qubit system. Moreover, we show that this bound cannot be attained by any quantum measurement with no quantum correlation in the measuring apparatus except for several specific statistical models. That is, in such a statistical model, th...

By: Gonca L. Aki, Peter A. Markowich, and Christof Sparber

Description: We consider the three-dimensional semirelativistic Hartree model for fast quantum mechanical particles moving in a self-consistent field. Under appropriate assumptions on the initial density matrix as a (fully) mixed quantum state we prove by using Wigner transformation techniques that its classical limit yields the well known relativistic Vlasov–Poisson system. The result holds for the case of attractive and repulsive mean-field interactions, with an additi...

Description: We develop a general technique for proving convergence of repeated quantum interactions to the solution of a quantum stochastic differential equation. The wide applicability of the method is illustrated in a variety of examples. Our main theorem, which is based on the Trotter–Kato theorem, is not restricted to a specific noise model and does not require boundedness of the limit coefficients.

By: Metin Gürses, Ismagil Habibullin, and Kostyantyn Zheltukhin

Description: The concept of integrable boundary conditions is applied to hydrodynamic type systems. Examples of such boundary conditions for dispersionless Toda systems are obtained. The close relation of integrable boundary conditions with integrable reductions in multifield systems is observed. The problem of consistency of boundary conditions with the Hamiltonian formulation is discussed. Examples of Hamiltonian integrable hydrodynamic type systems on a segment and a semiline are presented.

Description: A model for the first order phase transitions as ice-water and liquid-vapor is proposed using the Ginzburg–Landau equation for the order parameter φ. In this model the density ρ is composed of two quantities ρ0 and ρ1 such that 1/ρ = 1/ρ0+1/ρ1, where ρ1 is strictly connected to the order parameter φ. By means of this decomposition, we are able to represent the Andrew diagram without the use of the heuristic van der Waals equation.

By: Johannes Giannoulis, Michael Herrmann, and Alexander Mielke

Description: Studying spatially extended Hamiltonian systems with coherent microstructure, it is an important and challenging problem to identify reduced Hamiltonian models that describe the effective dynamics on large spatial and temporal scales. Such models require macroscopically varying, deterministic initial data which can possess a well-organized microstructure. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can be...

By: Nicolae Angelescu, Robert A. Minlos, Jean Ruiz, and Valentin A. Zagrebnov

Description: We study the structure of the spectrum of a two-level quantum system weakly coupled to a boson field (spin-boson model). Our analysis allows to avoid the cutoff in the number of bosons, if their spectrum is bounded below by a positive constant. We show that, for small coupling constant, the lower part of the spectrum of the spin-boson Hamiltonian contains (one or two) isolated eigenvalues and (one or two) manifolds of atom +1-boson states indexed by the boso...

Description: The goal of this paper is to consider the asymptotic behavior of solutions of nonautonomous classical reaction-diffusion equations in unbounded domains with nonlinearity having a polynomial growth of arbitrary order. The existence and structure of a uniform attractor are obtained in the spaces L2(n) and Lp(n), respectively.

Description: A mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics is proposed. For this a Hilbert space H of functions of four variables x,t furnished with an additional indefinite inner product invariant under Poincaré transformations is introduced. For a class of functions in H that are well localized in the time variable, the usual formalism of nonrelativistic quantum mechanics is derived. In particular, the interfe...

By: Nakao Hayashi, Pavel I. Naumkin, and Ratno Bagus Edy Wibowo

Description: We consider the initial value problem for systems of nonlinear Klein–Gordon equations with quadratic nonlinearities. We prove the existence of scattering states, namely, the asymptotic stability of small solutions in the neighborhood of the free solutions for small initial data in the weighted Sobolev space H4,3(R3)×H3,3(R3). If nonlinearities satisfy the strong null condition, then the same result is true in two space dimensions for small initial data in H5...

By: J. M. Velázquez-Arcos, C. A. Vargas, J. L. Fernández-Chapou, and A. L. Salas-Brito

Description: A method for computing the trace of the kernel of the homogeneous Fredholm’s equation for resonant states arising from nonlocal potentials is proposed. We show that this integral formulation is convergent.

By: Ismagil Habibullin, Natalya Zheltukhina, and Aslý Pekcan

Description: We study a differential-difference equation of the form tx(n+1) = f(t(n),t(n+1),tx(n)) with unknown t = t(n,x) depending on x and n. The equation is called a Darboux integrable if there exist functions F (called an x-integral) and I (called an n-integral), both of a finite number of variables x,t(n),t(n±1),t(n±2),…,tx(n),txx(n),…, such that DxF = 0 and DI = I, where Dx is the operator of total differentiation with respect to x and D is the shift operator: Dp...

By: Richard Durran, Andrew Neate, Aubrey Truman, and Feng-Yu Wang

Description: The correspondence limit of the atomic elliptic state in three dimensions is discussed in terms of Nelson’s stochastic mechanics. In previous work we have shown that this approach leads to a limiting Nelson diffusion, and here we discuss in detail the invariant measure for this process and show that it is concentrated on the Kepler ellipse in the plane z = 0. We then show that the limiting Nelson diffusion generator has a spectral gap; thereby proving that i...

Description: We revisit Weyl geometry in the context of recent higher-dimensional theories of space-time. After introducing the Weyl theory in a modern geometrical language we present some results that represent extensions of Riemannian theorems. We consider the theory of local embeddings and submanifolds in the context of Weyl geometries and show how a Riemannian space-time may be locally and isometrically embedded in a Weyl bulk. We discuss the problem of classical con...

Description: We prove that potential conservation laws have characteristics depending only on local variables if and only if they are induced by local conservation laws. Therefore, characteristics of pure potential conservation laws have to essentially depend on potential variables. This statement provides a significant generalization of results of the recent paper by Bluman et al. [J. Math. Phys. 47, 113505 (2006)] . Moreover, we present extensions to gauged potential s...