Inverse Problem for the Discrete 1D Schrödinger Operator with Small Periodic Potentials

Consider the discrete 1D Schrödinger operator on ? with an odd 2k periodic potential q. For small potentials we show that the mapping: q→ heights of vertical slits on the quasi-momentum domain (similar to the Marchenko-Ostrovski maping for the Hill operator) is a local isomorphism and the isospectral set consists of 2 ^k distinct potentials. Finally, the asymptotics of the spectrum are determined as q→0.     

Asymptotics of Block Toeplitz Determinants and the Classical Dimer Model

We compute the asymptotics of a block Toeplitz determinant which arises in the classical dimer model for the triangular lattice when considering the monomer-monomer correlation function. The model depends on a parameter interpolating between the square lattice (t = 0) and the triangular lattice (t = 1), and we obtain the asymptotics for 0 < t ≤ 1. For 0 < t < 1 we apply the Szegö Limit Theorem for block Toeplitz determinants. The main difficulty is to evaluate the constant term in the asymptotics, which is generally given only in a rather abstract form.     

Dynamic correlations in a thermalized system described by the Burgers equation

For the field u(x, t) governed by the Burgers equation with a thermal noise, short-time asymptotics of multipoint correlators are obtained. Their exponential parts are independent of the correlator number. This means that they are determined by a single rare fluctuation and exhibit an intermittency phenomenon.     

Photoionization of the Bound Systems at High Energies

We consider photoionization of a system bound by the central potential V(r). We demonstrate that the high energy nonrelativistic asymptotics of the photoionization cross section can be obtained without solving the wave equation. The asymptotics can be expressed in terms of the Fourier transform of the potential by employing the Lippmann–Schwinger equation. We find the asymptotics for the screened Coulomb field. We demonstrate that the leading corrections to this asymptotics are described by the universal factor. The high energy nonrelativistic asymptotics is found to be determined by the analytic properties of the potential V(r). We show that the energy dependence of the asymptotics of photoionization cross sections of fullerenes is to large extent model-dependent. We demonstrate that if the fullerene field V(r) is approximated by the function with singularities in the complex plane, the power drop of the asymptotics is reached at the energies which are so high that the cross section becomes unobservably small. The preasymptotic behavior with a faster decrease of the cross sections becomes important in these cases.     

Asymptotics of the S-matrix for perturbed Hill operators

We consider the Schrödinger operator with a periodic potential p plus a compactly supported potential q on the real line. We assume that both p and q have m ? 0 derivatives. For generic p, the essential spectrum of the operator has an infinite sequence of open gaps. We determine the asymptotics of the S-matrix at high energy.     

Higher order corrections to the Lipatov asymptotics in the ϕ<Superscript>4</Superscript> theory

Higher orders in perturbation theory can be calculated by the Lipatov method [1]. For most field theories, the Lipatov asymptotics has the functional form ca ^N Γ(N+b (N is the perturbation theory order); relative corrections to this asymptotics have the form of a power series in 1/N. The coefficients of higher order terms of this series can be calculated using a procedure analogous to the Lipatov approach and are determined by the second instanton in the field theory in question. These coefficients are calculated quantitatively for the n-component ?⁴ theory under the assumption that the second instanton is (i) a combination of elementary instantons and (ii) a spherically asymmetric localized function. A technique of two-instanton computations, as well as the method for integrating over rotations of an asymmetric instanton in the coordinate state, is developed.     

On the measure of gaps and spectra for discrete 1D Schrödinger operators

We study the lebesgue measure of gaps and spectra, of ergodic Jacobi matrices. We show that: |σ/A|+|G|≥v, where: σ is the spectrum,G is the union of the gaps,A is the set of energies where the Lyaponov exponent vanishes andv is an appropriate seminorm of the potential. We also study in more detail periodic Jacobi matrices, and obtain a lower bound and large coupling asymptotics for the measure of the spectrum. We apply the results of the periodic case, to limit periodic Jacobi matrices, and obtain sufficient conditions for |G|≥v and for |σ|>0.     

Heavy-tailed targets and (ab)normal asymptotics in diffusive motion

We show that, under suitable confinement conditions, the ordinary Fokker-Planck equation may generate non-Gaussian heavy-tailed probability density functions (pdfs) (like, for example, Cauchy or more general Lévy stable distributions) in its long-time asymptotics. In fact, all heavy-tailed pdfs known in the literature can be obtained this way. For the underlying diffusion-type processes, our main focus is on their transient regimes and specifically the crossover features, when an initially infinite number of pdf moments decreases to a few or none at all. The time dependence of the variance (if in existence), ∼t^γ with 0<γ<2, may in principle be interpreted as a signature of subdiffusive, normal diffusive or superdiffusive behavior under confining conditions; the exponent γ is generically well defined in substantial periods of time. However, there is no indication of any universal time rate hierarchy, due to a proper choice of the driver and/or external potential.     

Determination of coupling constants from superconvergence sum rules in vector meson — Vector meson scattering

Dynamical relations between masses and coupling constants have been studied using the superconvergent sum rule technique in vector meson — vector meson scattering. Unessential complications due to the spin have been removed by defining a set of 25 KSF invariant amplitudes. Commonly accepted analyticity properties and asymptotics estimated arguing along the line of unitarity then lead to superconvergent sum rules for three amplitudes. Their saturation by one-intermediate-particle contributions in the processesωρ→ ωρ, ωB→ωB andωA ₁→ωA ₁ results in a system of nine coupled equations which have been approximately solved for coupling constants and aρ-ω-B- A ₁ mass sum rule.     

Asymptotics of unit Burger number rotating and stratified flows for small aspect ratio

Rotating and stably stratified Boussinesq flow is investigated for Burger number unity in domain aspect ratio (height/horizontal length) δ<1 and δ=1. To achieve Burger number unity, the non-dimensional rotation and stratification frequencies (Rossby and Froude numbers, respectively) are both set equal to a second small parameter ?<1. Non-dimensionalization of potential vorticity distinguishes contributions proportional to (?δ)⁻¹, δ⁻¹ and O(1). The (?δ)⁻¹ terms are the linear terms associated with the pseudo-potential vorticity of the quasi-geostrophic limit. For fixed δ=1/4 and a series of decreasing ?, numerical simulations are used to assess the importance of the δ⁻¹ contribution of potential vorticity to the potential enstrophy. The change in the energy spectral scalings is studied as ? is decreased. For intermediate values of ?, as the flow transitions to the (δ?)⁻¹ regime in potential vorticity, both the wave and vortical components of the energy spectrum undergo changes in their scaling behavior. For sufficiently small ?, the (δ?)⁻¹ contributions dominate the potential vorticity, and the vortical mode spectrum recovers k⁻³ quasi-geostrophic scaling. However, the wave mode spectrum shows scaling that is very different from the well-known k⁻¹ scaling observed for the same asymptotics at δ=1. Visualization of the wave component of the horizontal velocity at δ=1/4 reveals a tendency toward a layered structure while there is no evidence of layering in the δ=1 case. The investigation makes progress toward quantifying the effects of aspect ratio δ on the ?→0 asymptotics for the wave component of unit Burger number flows. At the lowest value of ?=0.002, it is shown that the horizontal kinetic energy spectral scalings are consistent with phenomenology that explains how linear potential vorticity constrains energy in the limit ?→0 for fixed δ.     

The asymptotic behaviour of a non-planar orientable dual two-loop amplitude

Both the s- and t-channel asymptotic behaviour of a non-planar orientable two-loop four-point amplitude are investigated. A factorized expression for the iterated pomeron singularity, contributing to the s-channel behaviour, has been obtained with an internal pomeron blob closely related to the off-shell “propagator” of Cremmer and Scherk. These results suggest an interpretation in terms of a renormalization scheme of the pomeron. For t-channel asymptotics Regge-cut behaviour arises. Remarks are also made for the corresponding six-point case. Direct expansions of all the relevant functions in terms of group parameters as well as a thorough study of Jacobi's transformation in the double parabolic limit are of crucial importance to prove these results.     

Asymptotics of Karhunen-Loeve Eigenvalues and Tight Constants for Probability Distributions of Passive Scalar Transport

In this paper we study the asymptotics of the probability distribution function for a certain model of freely decaying passive scalar transport. In particular we prove rigorous large n, or semiclassical, asymptotics for the eigenvalues of the covariance of a fractional Brownian motion. Using these asymptotics, along with some standard large deviations results, we are able to derive tight asymptotics for the rate of decay of the tails of the probability density for a generalization of the Majda model of scalar intermittency originally due to Vanden Eijnden. We are also able to derive asymptotically tight estimates for the closely related problem of small L₂ ball probabilities for a fractional Brownian motion.     

Summing divergent perturbative series in a strong coupling limit. The Gell-Mann-Low function of the ϕ4 theory

An algorithm is proposed for determining asymptotics of the sum of a perturbative series in the strong coupling limit using given values of the expansion coefficients. Application of the algorithm is illustrated, methods for estimating errors are developed, and an optimization procedure is described. Applied to the ?⁴ theory, the algorithm yields the Gell-Mann-Low function asymptotics of the type β(g)≈7.4g ^0.96 for large g. The fact that the exponent is close to unity can be interpreted as a manifestation of the logarithmic branching of the type β(g)～g (lng)^?γ (with γ≈0.14), which is confirmed by independent evidence. In any case, the ?⁴ theory is self-consistent. The procedure of summing perturbative series with arbitrary values of the expansion parameter is discussed.     

On perturbation theory for the gr2N anharmonic oscillator

Perturbation series (PS) in powers of the coupling constant g for the D-dimensional anharmonic oscillator with power anharmonicity gr^2N is considered. The high order PS coefficients ?_k for the ground state energy are calculated explicitly with the help of recurrence relations among intergers. The rate of approach ?_k to their asymptotics $\text{?}{}_{\mathrm{k}}$ as a function of space dimension D is discussed.     

Heat kernel asymptotics with mixed boundary conditions

We calculate the coefficient a₅ of the heat kernel asymptotics for an operator of Laplace type with mixed boundary conditions on a general compact manifold.     

Small x behaviour of the chirally-odd parton distribution h 1(x,Q 2)

The small x behaviour of the structure function h ₁ (x, Q ²) is studied within the leading logarithmic approximation of perturbative QCD. There are two contributions relevant at small x. The leading one behaves like (1/x)⁰ i.e. it is just a constant in this limit. The second contribution, suppressed by one power of x, includes the terms summed by the GLAP equation. Thus for h ₁ (x, Q ²) the GLAP asymptotics and Regge asymptotics are completely different, making h ₁(x, Q ²) quite an interesting quantity for the study of small x physics.     

The heat equation with singular coefficients

The small time asymptotics of the kernel ofe ^?tH is defined and derived for \(H = \frac{{d^2 }}{{dx^2 }} + \frac{\kappa }{{x^2 }}\) on ?¹. Lemmas on singular asymptotics in the sense of distributions are formulated and used. The results are applied to derive an index formula on ?¹.     

Autocorrelation functions in the generalized Boltzmann-Enskog model

The purpose of this paper is to study the asymptotic properties of time autocorrelation functions for the generalized nonlinear Boltzmann-Enskog model, which contains a long-range component of the interaction between the particles. On the basis of the analysis of non-linear features of the Boltzmann-Enskog kinetic equation, the role of nonlinear effects is directly revealed at the approach to an equilibrium state. It is shown that autocorrelation functions have power asymptotics t ^?3/2, and the effects that are related to the inclusion of the long-range component lead to a change in the coefficient at t ^?3/2. These results establish a closed expression for the determination of coefficients in the asymptotic expansion of the autocorrelation functions of rate and thermal diffusion.     

On the summation of divergent perturbation series in quantum mechanics and field theory

The possibility of recovering the Gell-Mann–Low function in the asymptotic strong-coupling regime by known first-order perturbation-theory (PT) terms β_n and their asymptotics as \(\tilde \beta _n \) as n → ∞ is investigated. Conditions are formulated that are necessary for recovering the required function at the physical level of rigor: (1) a large number of PT coefficients are known whose asymptotics has already been established, and (2) there is no intermediate asymptotics. Higher orders of PT, their asymptotic behavior, and power corrections are calculated in quantum mechanical problems that involve divergent PT series (including series for a funnel potential, the ? ₍₀₎ ⁴ model, and the Stark effect in a strong field). The scalar field theory ? ₍₄₎ ⁴ is considered in the \(\overline {MS} \) and MOM regularization schemes. It is shown that one cannot make any definite conclusion about the asymptotics of the Gell-Mann–Low function as g → ∞ on the basis of information available for the above theory.