Inverse scattering problem for internal waves with finite fluid depth

In this paper, we derive the Bäcklund transformation, an infinite set of conservation laws and the inverse scattering transform of nonlinear equation that describes the internal waves of fluids with finite depth, ø_t + 2øø_x + (C/S)ø_xx = 0, which has been shown before to have N-soliton solutions.     

Aspects of supersymmetric quantum mechanics

We review the properties of supersymmetric quantum mechanics for a class of models proposed by Witten. Using both Hamiltonian and path integral formulations, we give general conditions for which supersymmetry is broken (unbroken) by quantum fluctuations. The spectrum of states is discussed, and a virial theorem is derived for the energy. We also show that the euclidean path integral for supersymmetric quantum mechanics is equivalent to a classical stochastic process when the supersymmetry is unbroken (E₀ = 0). By solving a Fokker-Planck equation for the classical probability distribution, we find P_c(y) is identical to |Ψ₀(y)|² in the quantum theory.     

On the universality classes of the Henon map

Within an appropriate renormalization framework, we discuss a¹(b) and δ associated with the bifurcation road to chaos of a Hénon-like map generalized as follows: (x_t+1, y_t+1) = (1?a|x_t|^z + y_t, ?bx_t); (b?0, z?1). For fixed z, we obtained (i) only two universality classes, namely the conservative (b = 1) and non-conservative (b≠1) ones and (ii) $\text{a}{}^1\text{(}\text{1}\text{b}\text{) =}\text{a}{}^1\text{(b)}\text{b}{}^{\mathrm{z}}$. For b = 1, δ(z) presents a minimum, and diverges for z → 1 and z → ∞ (this contrasts with the b≠1 case).     

The derivative nonlinear Schrödinger equation on the half-line

We analyze the derivative nonlinear Schrödinger equation on the half-line using the Fokas method. Assuming that the solution q(x,t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter ζ. The jump matrix has explicit x,t dependence and is given in terms of the spectral functions a(ζ), b(ζ) (obtained from the initial data q₀(x)=q(x,0)) as well as A(ζ), B(ζ) (obtained from the boundary values g₀(t)=q(0,t) and g₁(t)=q_x(0,t)). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation. Given initial and boundary values {q₀(x),g₀(t),g₁(t)} such that there exist spectral functions satisfying the global relation, we show that the function q(x,t) defined by the above Riemann-Hilbert problem exists globally and solves the derivative nonlinear Schrödinger equation with the prescribed initial and boundary values.     

Parton densities beyond perturbation theory

I compute non-perturbative corrections to the kernel governing the evolution of non-singlet parton densities. The model used is QED in the limit of many charged particles. I find an infrared renormalon at b₀t = −1, where t is the Borel variable and b₀ is the first coefficient of the β-function. This term has a non-trivial dependence on the variable x = -q²/(2p · q) and its coefficient scales as x³/(1 − x)² (p is the momentum of the hadron and q is the momentum transfer). An extrapolation of my results to QCD implies a breakdown of the parton model near the elastic region.     

The parity of regge cuts

The exchange of two reggeons of naturalities η_a and η_b will in general produce a J plane cut in amplitudes of both naturalities ± η_aη_b. We show that the amplitudes of naturality ? η_aη_b are suppressed, and this suppression grows with the energy, regardless of the particular model used for the vertices coupling the reggeons to the external particles. In consequence, for cuts where the two reggeons are any of ?, A₂, f, ω, or the pomeron, the natural-parity cut (if it is present) will dominate over the unnatural parity. We also show that cut vertices which contribute to both naturalities satisfy the conspiracy condition at t = 0 by suppressing both contributions equally. Away from t = 0, the η = + η_aη_b contribution can recover from this suppression. This behaviour is of importance in, for example, the pion-pomeron cut.We compare these conclusions with experiment.     

Current Fluctuations in the One-Dimensional Symmetric Exclusion Process with Open Boundaries

We calculate the first four cumulants of the integrated current of the one-dimensional symmetric simple exclusion process of N sites with open boundary conditions. For large system size N, the generating function of the integrated current depends on the densities ρ _a and ρ _b of the two reservoirs and on the fugacity z, the parameter conjugated to the integrated current, through a single parameter. Based on our expressions for these first four cumulants, we make a conjecture which leads to a prediction for all the higher cumulants. In the case ρ _a =1 and ρ _b =0, our conjecture gives the same universal distribution as the one obtained by Lee, Levitov, and Yakovets for one-dimensional quantum conductors in the metallic regime.     

On a Quantum System with Memory

We consider the integro-differential equation for the classical trajectory of an oscillator coupled to another one. On the quantum level the elimination of the coordinate A of the “unvisible” oscillator leads to an effective path integral (X, Ξ, μ) for the associated imaginary time stochastic process t ?, (-∞,∞) → x(t). We prove reflection positivity of the measure dμ ≈ F · dξ, where dξ governes the free oscillator x and F is the counterpart of Feynman's influence functional. Finally, realizing the Hamiltonian semigroup exp(-tH), t ≧ 0, in the physical Hilbert space ?? = L²(X, Γ, μ), where Γ ? Ξ₊, we try to understand what is memory.     

Perturbation of the Laplacian supported by null sets,with applications to polymer measures and quantum fields

We discuss hamiltonians in L²(R^d, dx) of the form H = ?Δ + V, with V a potential supported by a zero measure set C. In particular if C is a path of a brownian motion b such that V(x) = ∫₀¹λ(x, ω)δ(x-b(s, ω)) ds, we show that H exists as a nontrivial, self-adjoint, lower bounded perturbation of ?Δ when d ?5. We must choose λ to be an infinitesimal, negative function for d = 4,5, but for d ? 3 any bounded real-valued function λ will do. The connection with Edward's model of polymers as well as with quantum fields of the ?_d⁴-type is also discussed. The proofs use methods of nonstandard analysis.     

Remark on the solution of the Schrödinger equation for anharmonic oscillators via the Feynman path integral

We give a simple proof of Feynman's formula for the Green's function of the n-dimensional harmonic oscillator valid for every time t with Im t ≤ 0. As a consequence the Schrödinger equation for the anharmonic oscillator is integrated and expressed by the Feynman path integral on Hilbert space.     

The isotonic oscillator

The motion of a particle in the potential V(x) = V₀ (a/x - x/a)² is exactly soluble both classically and quantum mechanically. We have studied the features of the dynamics and the energies of a coherent state of this nonlinear system.     

Quantum dynamics of a massless relativistic string (II)

The massless relativistic free string is studied in the gauge x₀ = τ. It is found that the classical solutions include transverse and longitudinal vibrations. The problem is treated both in the Lagrangian and Hamiltonian formalism. Different ways of quantizing the system are investigated. The path integral quantization leads to a Poincaré invariant quantum theory in any number of dimensions.     

Almost One Bit Violation for the Additivity of the Minimum Output Entropy

In a previous paper, we proved that, in the appropriate asymptotic regime, the limit of the collection of possible eigenvalues of output states of a random quantum channel is a deterministic, compact set K_k,t. We also showed that the set K_k,t is obtained, up to an intersection, as the unit ball of the dual of a free compression norm. In this paper, we identify the maximum of \({\ell^p}\) norms on the set K_k,t and prove that the maximum is attained on a vector of shape (a, b, . . . , b) where a > b. In particular, we compute the precise limit value of the minimum output entropy of a single random quantum channel. As a corollary, we show that for any \({\varepsilon > 0}\), it is possible to obtain a violation for the additivity of the minimum output entropy for an output dimension as low as 183, and that for appropriate choice of parameters, the violation can be as large as \({\log 2 -\varepsilon}\). Conversely, our result implies that, with probability one in the limit, one does not obtain a violation of additivity using conjugate random quantum channels and the Bell state, in dimension 182 and less.     

Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

We extend some recent results of Lubinsky, Levin, Simon, and Totik from measures with compact support to spectral measures of Schrödinger operators on the half-line. In particular, we define a reproducing kernel S _L for Schrödinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schrödinger operator with perturbed periodic potential. We show that if solutions u(ξ, x) are bounded in x by ${e^{\epsilon x}}We extend some recent results of Lubinsky, Levin, Simon, and Totik from measures with compact support to spectral measures of Schr?dinger operators on the half-line. In particular, we define a reproducing kernel S _L for Schr?dinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schr?dinger operator with perturbed periodic potential. We show that if solutions u(ξ, x) are bounded in x by e^ex{e^{\epsilon x}} uniformly for ξ near the spectrum in an average sense and the spectral measure is positive and absolutely continuous in a bounded interval I in the interior of the spectrum with x₀ ? I{\xi_0\in I}, then uniformly in I,

\fracSL(x0 + a/L, x0 + b/L)SL(x0, x0)? \fracsin(pr(x0)(a - b))pr(x0)(a - b),\frac{S_L(\xi_0 + a/L, \xi_0 + b/L)}{S_L(\xi_0, \xi_0)}\rightarrow \frac{\sin(\pi\rho(\xi_0)(a - b))}{\pi\rho(\xi_0)(a - b)},      17. Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions      Jean Zinn-Justin  Ulrich D. Jentschura 《Annals of Physics》2004,313(1):197-267       18. Relation between contiguous Mie coefficients for perfectly conducting spheres      《Physics letters. A》1986,115(5):191-192 It is shown that when the size parameter x = ka in the scattering of electromagnetic waves by a perfectly conducting sphere is equal to an integer n, the Mie coefficients an and an−1 are equal. It is also shown that the other Mie coefficient bn cannot be equal to bn−1 for any finite x. The latter conclusion also holds in the case of scattering of a scalar particle by a hard sphere in quantum mechanics.      19. Lithium-ion conductivity in the novel La1/3−xLi3xNbO3 solid solution with perovskite-related structure      《Solid State Ionics》1999,116(1-2):11-18 The stoichiometric range, crystal chemistry, ionic conductivity and electrochemical window of the La1/3−xLi3xNbO3 solid solution with a perovskite-related structure have been studied. The range of existence of the solid solution appears to be 0≤x≤0.06. These niobates have a basic diagonal unit cell a≈√2ap b√2ap c≈2ap. Ionic conductivity of the materials and its dependence with the composition and temperature have been examined. We have found that the highest conductivity value is 4.3×1O−5 S cm−1 at 300 K for x=0.04. The electrochemical window of the compounds has been investigated by potentiostatic discharge and charge. Electrochemical experiments show that the use of the materials as solid electrolytes in secondary batteries is limited down to 1.75 V using Li metal as anode.      20. The generalized exponential-integral $\text{V(x,y)=}\text{∫}\text{1}\text{∞}\text{exp(−xt)ln(t+y)}\text{dt}\text{t}$ and computer algorithms for y = 0, 1      D.G. Hummer 《Journal of Quantitative Spectroscopy & Radiative Transfer》1983,30(3):281-287 The generalized exponential-integral function V(x, y) defined here includes as special cases the function E(2)1(x) = V(x, 0) introduced by van de Hulst and functions M0(x) = V(x, 1) and N0(x) = V(x, -1) introduced by Kourganoff in connection with integrals of the form ∫ En)t)Em(t±x), which play an important role in the theory of monochromatic radiative transfer. Series and asymptotic expressions are derived and, for the most important special cases, y = 0 and y = 1, Chebyshev expansions and rational approximations are obtained that permit the function to be evaluated to at least 10 sf on 0<x<∞ using 16 sf arithmetic.

Relation between contiguous Mie coefficients for perfectly conducting spheres

It is shown that when the size parameter x = ka in the scattering of electromagnetic waves by a perfectly conducting sphere is equal to an integer n, the Mie coefficients a_n and a_n−1 are equal. It is also shown that the other Mie coefficient b_n cannot be equal to b_n−1 for any finite x. The latter conclusion also holds in the case of scattering of a scalar particle by a hard sphere in quantum mechanics.     

Lithium-ion conductivity in the novel La1/3−xLi3xNbO3 solid solution with perovskite-related structure

The stoichiometric range, crystal chemistry, ionic conductivity and electrochemical window of the La_1/3−xLi_3xNbO₃ solid solution with a perovskite-related structure have been studied. The range of existence of the solid solution appears to be 0≤x≤0.06. These niobates have a basic diagonal unit cell a≈√2a_p b√2a_p c≈2a_p. Ionic conductivity of the materials and its dependence with the composition and temperature have been examined. We have found that the highest conductivity value is 4.3×1O⁻⁵ S cm⁻¹ at 300 K for x=0.04. The electrochemical window of the compounds has been investigated by potentiostatic discharge and charge. Electrochemical experiments show that the use of the materials as solid electrolytes in secondary batteries is limited down to 1.75 V using Li metal as anode.     

The generalized exponential-integral V(x,y)=∫1∞ exp(−xt)ln(t+y)dtt and computer algorithms for y = 0, 1

The generalized exponential-integral function V(x, y) defined here includes as special cases the function E⁽²⁾₁(x) = V(x, 0) introduced by van de Hulst and functions M₀(x) = V(x, 1) and N₀(x) = V(x, -1) introduced by Kourganoff in connection with integrals of the form ∫ E_n)t)E_m(t±x), which play an important role in the theory of monochromatic radiative transfer. Series and asymptotic expressions are derived and, for the most important special cases, y = 0 and y = 1, Chebyshev expansions and rational approximations are obtained that permit the function to be evaluated to at least 10 sf on 0<x<∞ using 16 sf arithmetic.