Unstable and Stable Galaxy Models

To determine the stability and instability of a given steady galaxy configuration is one of the fundamental problems in the Vlasov theory for galaxy dynamics. In this article, we study the stability of isotropic spherical symmetric galaxy models f ₀(E), for which the distribution function f ₀ depends on the particle energy E only. In the first part of the article, we derive the first sufficient criterion for linear instability of f ₀(E) : f ₀(E) is linearly unstable if the second-order operator

Quantum Computer Algorithm for Parity Determination Based on Quantum Counting

A new quantum computer algorithm is proposed for determining the parity of function f(x) by using quantum counting algorithm. The parity of function f(x) can be determined by counting exactly the number of satisfying f(x)=−1, which is equivalent to determine the number of solutions, M, to an N item search problem. The algorithm can be accomplished in time of order .     

A sharp decrease of resistivity in La0.7Ca0.3Mn0.96Cu0.04O3: Evidence for Cu-assisted coherent tunneling of spin Polarons

Nearly a 50% decrease of the resistivity ρ(T, x) is observed upon just 4% Cu doping at the Mn site of La_2/3Ca_1/3Mn_1−x Cu_xO₃. When the observed phenomenon is attributed to a decrease of the spin-polaron energy E _σ(x) below T _C (x), all of the data are found to be well fitted by the nonthermal coherent tunneling expression , assuming that the magnetization in the ferromagnetic state is given by the expression . The best fits through all the data points suggest and E _σ(x)≃E _σ(0)(1−x)⁴ for the explicit x dependence of the Cu-induced modifications of the Mn-spin-dominated zero-temperature spontaneous magnetization, residual paramagnetic contribution, and spin-polaron tunneling energy, respectively, with E _σ(0)=0.12 eV. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 11, 812–815 (10 June 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

A Variational Principle in the Dual Pair of Reproducing Kernel Hilbert Spaces and an Application

Given a positive definite, bounded linear operator A on the Hilbert space ₀≔l ²(E), we consider a reproducing kernel Hilbert space ₊ with a reproducing kernel A(x,y). Here E is any countable set and A(x,y), x,y∊ E, is the representation of A w.r.t. the usual basis of ₀. Imposing further conditions on the operator A, we also consider another reproducing kernel Hilbert space ₋ with a kernel function B(x,y), which is the representation of the inverse of A in a sense, so that ₋⊃ ₀⊃ ₊ becomes a rigged Hilbert space. We investigate the ratios of determinants of some partial matrices of A and B. We also get a variational principle on the limit ratios of these values. We apply this relation to show the Gibbsianness of the determinantal point process (or fermion point process) defined by the operator A(I+A)⁻¹ on the set E. 2000 Mathematics Subject Classification: Primary: 46E22 Secondary: 60K35     

Dual parametrization of GPDs versus the double distribution Ansatz

We establish a link between the dual parametrization of GPDs and a popular parametrization based on the double distribution Ansatz, which is in prevalent use in phenomenological applications. We compute several first forward-like functions that express the double distribution Ansatz for GPDs in the framework of the dual parametrization and show that these forward-like functions make the dominant contribution into the GPD quintessence function. We also argue that the forward-like functions with 1 contribute to the leading singular small-x_Bj behavior of the imaginary part of DVCS amplitude. This makes the small-x_Bj behavior of independent of the asymptotic behavior of PDFs. Assuming analyticity of Mellin moments of GPDs in the Mellin space we are able to fix the value of the D -form factor in terms of the GPD quintessence function N(x, t) and the forward-like function Q ₀(x, t) .     

Analyticity of Smooth Eigenfunctions and Spectral Analysis of the Gauss Map

We provide a sufficient condition of analyticity of infinitely differentiable eigenfunctions of operators of the form Uf(x)=a(x,y)f(b(x,y))(dy) acting on functions (evolution operators of one-dimensional dynamical systems and Markov processes have this form). We estimate from below the region of analyticity of the eigenfunctions and apply these results for studying the spectral properties of the Frobenius–Perron operator of the continuous fraction Gauss map. We prove that any infinitely differentiable eigenfunction f of this Frobenius–Perron operator, corresponding to a non-zero eigenvalue admits a (unique) analytic extension to the set . Analyzing the spectrum of the Frobenius–Perron operator in spaces of smooth functions, we extend significantly the domain of validity of the Mayer and Röpstorff asymptotic formula for the decay of correlations of the Gauss map.     

Schwinger mechanism for gluon-pair production in the presence of arbitrary time-dependent chromo-electric field

We study the Schwinger mechanism for gluon-pair production in the presence of an arbitrary time-dependent chromo-electric background field E ^a (t) with arbitrary color index a=1,2,…,8 in SU(3) by directly evaluating the path integral. We obtain an exact expression for the probability of non-perturbative gluon-pair production per unit time per unit volume and per unit transverse momentum, , from arbitrary E ^a (t). We show that the tadpole (or single-gluon) effective action does not contribute to the non-perturbative gluon-pair production rate, . We find that the exact result for non-perturbative gluon-pair production is independent of all the time derivatives , where n=1,2,…,∞, and that it has the same functional dependence on the two Casimir invariants, [E ^a (t)E ^a (t)] and [d _abc E ^a (t)E ^b (t)E ^c (t)]², as the constant chromo-electric field E ^a result with the replacement: E ^a →E ^a (t). This result relies crucially on the validity of the shift conjecture, which has not yet been established. This result may be relevant to the study of the production of a non-perturbative quark–gluon plasma at RHIC and LHC.     

Semiclassical analysis of edge state energies in the integer quantum Hall effect

Analysis of edge-state energies in the integer quantum Hall effect is carried out within the semiclassical approximation. When the system is wide so that each edge can be considered separately, this problem is equivalent to that of a one dimensional harmonic oscillator centered at x = x_c and an infinite wall at x = 0, and appears in numerous physical contexts. The eigenvalues E_n(x_c) for a given quantum number n are solutions of the equation S(E,x_c)=π[n+ γ(E,x_c)] where S is the WKB action and 0 < γ < 1 encodes all the information on the connection procedure at the turning points. A careful implication of the WKB connection formulae results in an excellent approximation to the exact energy eigenvalues. The dependence of γ[E_n(x_c),x_c] ≡γ_n(x_c) on x_c is analyzed between its two extreme values as x_c ↦-∞ far inside the sample and as x_c ↦∞ far outside the sample. The edge-state energiesE_n(x_c) obey an almost exact scaling law of the form and the scaling function f(y) is explicitly elucidated.     

Existence of Solutions of a Kinetic Equation Modeling Cometary Flows

A global existence theorem is presented for a kinetic problem of the form _t f+v· _x f=Q(f), f(t=0)=f ₀, where Q(f) is a simplified model wave–particle collision operator extracted from quasilinear plasma physics. Evaluation of Q(f) requires the computation of the mean velocity of the distribution f. Therefore, the assumptions on the data are such that vacuum regions, where the mean velocity is not well defined, are excluded. Also the initial data are assumed to have bounded total energy. As additional results conservation laws for mass, momentum, and energy are derived, as well as an entropy dissipation law and the propagation of higher order moments.     

The Asymptotic Limits of Zero Modes of Massless Dirac Operators

Asymptotic behaviors of zero modes of the massless Dirac operator H = α · D + Q(x) are discussed, where α = (α₁, α₂, α₃) is the triple of 4 × 4 Dirac matrices, , and Q(x) = (q _jk (x)) is a 4 × 4 Hermitian matrix-valued function with | q _jk (x) | ≤ C 〈x〉^−ρ, ρ > 1. We shall show that for every zero mode f, the asymptotic limit of |x|² f (x) as |x| → + ∞ exists. The limit is expressed in terms of the Dirac matrices and an integral of Q(x) f (x).      

No Phase Transition for Gaussian Fields with Bounded Spins

Let a<b, and H be the (formal) Hamiltonian defined on Ω by

The Fixed Point of a Generalization of the Renormalization Group Maps for Self-Avoiding Paths on Gaskets

Let W(x,y) = ax ³+ bx ⁴+ f ₅ x ⁵+ f ₆ x ⁶+ (3 ax ²)² y+ g ₅ x ⁵ y + h ₃ x ³ y ² + h ₄ x ⁴ y ² + n ₃ x ³ y ³+a ₂₄ x ² y ⁴+a ₀₅ y ⁵+a ₁₅ xy ⁵+a ₀₆ y ⁶, and X = , , where the coefficients are non-negative constants, with a > 0, such that X ²(x,x ²)−Y(x,x ²) is a polynomial of x with non-negative coefficients. Examples of the 2 dimensional map Φ: (x,y)↦ (X(x,y),Y(x,y)) satisfying the conditions are the renormalization group (RG) maps (modulo change of variables) for the restricted self-avoiding paths on the 3 and 4 dimensional pre-gaskets. We prove that there exists a unique fixed point (x _f ,y _f ) of Φ in the invariant set . 2000 Mathematics Subject Classification Numbers: 82B28; 60G99; 81T17; 82C41.     

Future cosmological evolution in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) gravity using two equations of state parameters

We investigate the issues of future oscillations around the phantom divide (FOPD) for f(R) gravity. For this purpose, we introduce two types of energy density and pressure arisen from the f(R)-higher order curvature terms. One has the conventional energy density and pressure even in the beginning of the Jordan frame, whose continuity equation defines the native equation of state w _DE. On the other hand, the other has the different energy density and pressure which do not obviously satisfy the continuity equation. This needs to introduce the effective equation of state w _eff to describe the f(R)-fluid, in addition to the native equation of state [(w)\tilde]_DE\tilde{w}_{\mathrm{DE}}. We show that the FOPD occur in f(R) gravities by introducing two types of equation of state. Finally, we point out that the singularity appears ar x=x _c because the stability condition of f(R) gravity violates.     

A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation

We consider the problem of identifying sharp criteria under which radial H ¹ (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) i? _t u + Δu + |u|² u = 0 scatter, i.e., approach the solution to a linear Schrödinger equation as t → ±∞. The criteria is expressed in terms of the scale-invariant quantities ${\|u_0\|_{L^2}\|\nabla u_0\|_{L^2}}We consider the problem of identifying sharp criteria under which radial H ¹ (finite energy) solutions to the focusing 3d cubic nonlinear Schr?dinger equation (NLS) i∂ _t u + Δu + |u|² u = 0 scatter, i.e., approach the solution to a linear Schr?dinger equation as t → ±∞. The criteria is expressed in terms of the scale-invariant quantities and M[u]E[u], where u ₀ denotes the initial data, and M[u] and E[u] denote the (conserved in time) mass and energy of the corresponding solution u(t). The focusing NLS possesses a soliton solution e ^it Q(x), where Q is the ground-state solution to a nonlinear elliptic equation, and we prove that if M[u]E[u] < M[Q]E[Q] and , then the solution u(t) is globally well-posed and scatters. This condition is sharp in the sense that the soliton solution e ^it Q(x), for which equality in these conditions is obtained, is global but does not scatter. We further show that if M[u]E[u] < M[Q]E[Q] and , then the solution blows-up in finite time. The technique employed is parallel to that employed by Kenig-Merle [17] in their study of the energy-critical NLS.     

Convergence in Energy-Lowering (Disordered) Stochastic Spin Systems

We consider stochastic processes, with finite, in which spin flips (i.e., changes of S ^t _x ) do not raise the energy. We extend earlier results of Nanda–Newman–Stein that each site x has almost surely only finitely many flips that strictly lower the energy and thus that in models without zero-energy flips there is convergence to an absorbing state. In particular, the assumption of finite mean energy density can be eliminated by constructing a percolation-theoretic Lyapunov function density as a substitute for the mean energy density. Our results apply to random energy functions with a translation-invariant distribution and to quite general (not necessarily Markovian) dynamics.     

Baxterization of Solutions to Reflection Equation with Hecke R-matrix

Let R be a Hecke solution to the Yang–Baxter equation and K be a reflection equation matrix with coefficients in an associative algebra . Let R(x) be the baxterization of R and suppose that K satisfies a polynomial equation with coefficients in the center of . We construct solutions to the reflection equation with spectral parameter relative to R(x), in the form of polynomials in K.     

Bethe Algebra of Homogeneous <Emphasis Type="Italic">XXX</Emphasis> Heisenberg Model has Simple Spectrum

We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dimensional -modules. As a byproduct we show that there exist exactly two-dimensional vector subspaces with a basis such that deg f = l, deg g = n − l + 1 and f (u)g(u − 1) − f (u − 1)g(u) = (u + 1) ⁿ . Supported in part by NSF grant DMS-0601005. Supported in part by RFFI grant 08-01-00638. Supported in part by NSF grant DMS-0555327.     

Non-hermitian quantum mechanics in non-commutative space

A recent investigation of the possibility of having a -symmetric periodic potential in an optical lattice stimulated the urge to generalize non-hermitian quantum mechanics beyond the case of commutative space. We thus study non-hermitian quantum systems in non-commutative space as well as a -symmetric deformation of this space. Specifically, a -symmetric harmonic oscillator together with an iC(x ₁+x ₂) interaction are discussed in this space, and solutions are obtained. We show that in the deformed non-commutative space the Hamiltonian may or may not possess real eigenvalues, depending on the choice of the non-commutative parameters. However, it is shown that in standard non-commutative space, the iC(x ₁+x ₂) interaction generates only real eigenvalues despite the fact that the Hamiltonian is not -symmetric. A complex interacting anisotropic oscillator system also is discussed.     

Wave Decay on Convex Co-Compact Hyperbolic Manifolds

For convex co-compact hyperbolic quotients , we analyze the long-time asymptotic of the solution of the wave equation u(t) with smooth compactly supported initial data f = (f ₀, f ₁). We show that, if the Hausdorff dimension δ of the limit set is less than n/2, then where and . We explain, in terms of conformal theory of the conformal infinity of X, the special cases , where the leading asymptotic term vanishes. In a second part, we show for all the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip . As a byproduct we obtain a lower bound on the remainder R(t) for generic initial data f.     

(1+1)-Dirac Particle with Position-Dependent Mass in Complexified Lorentz Scalar Interactions: Effectively $\mathcal{PT}$ -Symmetric

The effect of the built-in supersymmetric quantum mechanical language on the spectrum of the (1+1)-Dirac equation, with position-dependent mass (PDM) and complexified Lorentz scalar interactions, is re-emphasized. The signature of the “quasi-parity” on the Dirac particles’ spectra is also studied. A Dirac particle with PDM and complexified scalar interactions of the form S(z)=S(x−ib) (an inversely linear plus linear, leading to a symmetric oscillator model), and S(x)=S _r (x)+iS _i (x) (a -symmetric Scarf II model) are considered. Moreover, a first-order intertwining differential operator and an η-weak-pseudo-Hermiticity generator are presented and a complexified -symmetric periodic-type model is used as an illustrative example.