Asymptotic and numerical study of resonant tunneling in two-dimensional quantum waveguides of variable cross section

A waveguide is considered that coincides with a strip having two narrows of width ?. The electron wave function satisfies the Helmholtz equation with Dirichlet boundary conditions. The part of the waveguide between the narrows plays the role of a resonator, and there arise conditions for electron resonant tunneling. This phenomenon means that, for an electron of energy E, the probability T(E) of passing from one part of the waveguide to the other through the resonator has a sharp peak at E = E _res, where E _res is a “resonant” energy. To analyze the operation of electronic devices based on resonant tunneling, it is important to know E _res and the behavior of T(E) for E close to E _res. Asymptotic formulas for the resonance energy and the transition and reflection coefficients as ? → 0 are derived. These formulas depend on the limit shape of the narrows. The limit waveguide near each narrow is assumed to coincide with a pair of vertical angles. The asymptotic results are compared with numerical ones obtained by approximately computing the waveguide scattering matrix. Based on this comparison, the range of ? is found in which the asymptotic approach agrees with the numerical results. The methods proposed are applicable to much more complicated models than that under consideration. Specifically, the same approach is suitable for an asymptotic and numerical analysis of tunneling in three-dimensional quantum waveguides of variable cross section.     

Adiabatic asymptotics of reflection coefficients of a quantum electron moving in a two-dimensional waveguide

We study the adiabatic asymptotics of reflection coefficients of a quantum electron moving in a two-dimensional waveguide. The direction of the waveguide axis can slowly change, whereas the cut-section is a periodic function slowly varying along the waveguide axis. The motion of an electron is described by the free Helmoltz equation. We study turning points in a neighborhood of which considerable reflection of an electron is observed. We describe the asymptotic behavior of reflection coefficients and uniform asymptotic formulas for the wave function of electron; moreover, these formulas remain valid even if the turning points approach each other. An example of a waveguide with four turning points (with the so-called resonance tunneling) is considered. The quantization condition characterizing the asymptotic behavior of resonances is described. Bibliography: 26 titles. Translated from Problemy Matematicheskogo Analiza, No. 38, December 2008, pp. 93–119.     

High‐energy and smoothness asymptotic expansion of the scattering amplitude for the Dirac equation and application

We obtain an explicit formula for the diagonal singularities of the scattering amplitude for the Dirac equation with short‐range electromagnetic potentials. Using this expansion we uniquely reconstruct an electric potential and magnetic field from the high‐energy limit of the scattering amplitude. Moreover, supposing that the electric potential and magnetic field are asymptotic sums of homogeneous terms we give the unique reconstruction procedure for these asymptotics from the scattering amplitude, known for some energy E. Furthermore, we prove that the set of the averaged scattering solutions to the Dirac equation is dense in the set of all solutions to the Dirac equation that are in L²(Ω), where Ω is any connected bounded open set in with smooth boundary, and we show that if we know an electric potential and a magnetic field for , then the scattering amplitude, given for some energy E, uniquely determines these electric potential and magnetic field everywhere in . Combining this uniqueness result with the reconstruction procedure for the asymptotics of the electric potential and the magnetic field we show that the scattering amplitude, known for some E, uniquely determines a electric potential and a magnetic field, that are asymptotic sums of homogeneous terms, which converges to the electric potential and the magnetic field respectively. Moreover, we discuss the symmetries of the kernel of the scattering matrix, which follow from the parity, charge‐conjugation and time‐reversal transformations for the Dirac operator. Copyright © 2014 John Wiley & Sons, Ltd.     

Vector Lattice Powers: f-Algebras and Functional Calculus

Let s ∈ {2.3,…} and E be an Archimedean vector lattice. We prove that there exists a unique pair (E ^? ,?), where E ^? is an Archimedean vector lattice and ?:E× ··· ×E (s times) → E ^? is a symmetric lattice s-morphism, such that for every Archimedean vector lattice F and every symmetric lattice s-morphism T:E × ··· × E (s times) → F, there exists a unique lattice homomorphism T ^? :E ^?  → F such that T = T ^? ○?. We give two approaches to construct (E ^? ,?) based on f-algebras and functional calculus, respectively, provided that E is also uniformly complete.     

Constants of Derivation and Differential Ideals

ABSTRACT

Let R be a differential domain finitely generated over a differential field F of characteristic 0. Let C be the subfield of differential constants of F. This paper investigates conditions on differential ideals of R that are necessary or sufficient to guarantee that C is also the set of constants of differentiation of the quotient field, E, of R. In particular, when C is algebraically closed and R has a finite number of height one differential prime ideals, there are no new constants in E. An example where F is infinitely generated over C shows the converse is false. If F is finitely generated over C and R is a polynomial ring over F, sufficient conditions on F are given so that no new constants in E does imply only finitely many height one prime differential ideals in R. In particular, F can be Q¯(T) where T is a finite transcendence set.

     

Exact small ball asymptotics in weighted <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>-norm for some Gaussian processes

We find the exact small ball asymptotics under weighted L ₂-norm for a wide class of Gaussian processes which generate boundary-value problems for ordinary differential equations. Sharp constants in the asymptotics are derived for a number of processes connected with special functions. Bibliography: 23 titles.     

Multiphase double‐porosity homogenization for perimeter functionals

In this paper, we study a homogenization problem for perimeter energies in highly contrasted media; the analysis of the previous paper is carried out by removing the hypothesis that the perforated medium Rⁿ ? E is composed of disjoint compact components. Assuming E to be the union of a finite number N of connected components E¹, … ,E^N, the Γ‐limit F is a multiphase energy with a ‘decoupled’ surface part, obtained by homogenization from the surface tensions in each E ^j, a trivial bulk term obtained as a weak limit, and a further interacting term between the phases, involving an asymptotic formula for a family minimum problems on invading an asymptotic formula for a family of minimum problems on invading domains with prescribed boundary conditions. Copyright © 2012 John Wiley & Sons, Ltd.     

Uniform Estimates of Remainders in Asymptotic Expansions of Solutions to the Problem on Eigenoscillations of a Piezoelectric Plate

A method for obtaining estimates of asymptotic remainders is presented. The constants in estimates are independent of the number of the eigenvalue, as well as of the small parameter h, the thickness of the plate. Owing to an information about connections between frequencies of eigenoscillations of the three-dimensional plates and its two-dimensional model obtained under various restrictions to h, it is possible to divide the asymptotics in collective and individual ones. Only in the case of the individual asymptotics, i.e., under rigid restrictions on h, it is possible to construct asymptotic expansions for the corresponding eigenvectors. We consider arbitrarily anizotropic composed cylindrical plates in whcih piezoeffects can dominate along longitudinal directions, as well as along transverse directions. The connectedness of elastic and electric fields Implies the appearance of a nontrivial dissipative components of the operator of the problem under consideration, but its spectrum remains real and positive. Bibliography: 43 titles.     

Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses

A spectral problem for the Laplace operator in a thick cascade junction with concentrated masses is considered. This cascade junction consists of the junction's body and a great number of ?‐alternating thin rods belonging to two classes. One class consists of rods of finite length, and the second one consists of rods of small length of order . The density of the junction is of order on the rods from the second class and outside of them. The asymptotic behavior of eigenvalues and eigenfunctions of this problem is studied as ? → 0. There exist five qualitatively different cases in the asymptotic behavior of eigenmagnitudes as ? → 0, namely the case of ‘light’ concentrated (α ∈ (0,1)), ‘middle’ concentrated (α = 1), and ‘heavy’ concentrated masses (α ∈ (1, + ∞ )) that we divide into ‘slightly heavy’ concentrated (α ∈ (1,2)), ‘intermediate heavy’ concentrated (α = 2), and ‘very heavy’ concentrated masses (α > 2). In the paper, we study in detail the influence of the concentrated masses on the asymptotic behavior if α ∈ (1,2). We construct the leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions and prove the corresponding asymptotic estimates. Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotics of rank order statistics for ARCH residual empirical processes

This paper gives the asymptotic theory of a class of rank order statistics {T_N} for two-sample problem pertaining to empirical processes based on the squared residuals from two classes of ARCH models. An important aspect is that, unlike the residuals of ARMA models, the asymptotics of {T_N} depend on those of ARCH volatility estimators. Such asymptotics provide a useful guide to the reliability of confidence intervals, asymptotic relative efficiency and ARCH affection. We consider these aspects of {T_N} for some ARCH residual distributions via numerical illustrations. Moreover, a measure of robustness for {T_N} is introduced. These studies help to highlight some important features of ARCH residuals in comparison with the i.i.d. or ARMA settings.     

Asymptotic Expansion for the Null Distribution of the <Emphasis Type="Italic">F</Emphasis>-statistic in One-way ANOVA under Non-normality

In this paper we derive the asymptotic expansion of the null distribution of the F-statistic in one-way ANOVA under non-normality. The asymptotic framework is when the number of treatments is moderate but sample size per treatment (replication size) is small. This kind of asymptotics will be relevant, for example, to agricultural screening trials where large number of cultivars are compared with few replications per cultivar. There is also a huge potential for the application of this kind of asymptotics in microarray experiments. Based on the asymptotic expansion we will devise a transformation that speeds up the convergence to the limiting distribution. The results indicate that the approximation based on limiting distribution are unsatisfactory unless number of treatments is very large. Our numerical investigations reveal that our asymptotic expansion performs better than other methods in the literature when there is skewness in the data or even when the data comes from a symmetric distribution with heavy tails.     

The integrated density of states in strong magnetic fields

We consider three-dimensional Schrödinger operators with constant magnetic fields and ergodic electric potentials. We study the strong magnetic field asymptotic behaviour of the integrated density of states, distinguishing between the asymptotics far from the Landau levels, and the asymptotics near a given Landau level.     

Liouville–Green (WKB) asymptotics for almost-diagonal linear second-order matrix difference equations

A Liouville–Green (or WKB) asymptotic approximation theory is developed for a class of almost-diagonal (‘asymptotically diagonal’) linear second-order matrix difference equations. Rigorous and explicitly computable bounds for the error terms are obtained, the asymptotics being made with respect to both, the index and some parameter affecting the equation. The case of the associated inhomogeneous equations is also considered in detail. Some examples and a number of applications are presented for the purpose of illustration.     

Three classes of smooth Banach submanifolds in B(E,F)

Let E, F be two Banach spaces, and B(E, F), Φ(E, F), SΦ(E, F) and R(E,F) be the bounded linear, Fredholm, semi-Frdholm and finite rank operators from E into F, respectively. In this paper, using the continuity characteristics of generalized inverses of operators under small perturbations, we prove the following result Let ∑ be any one of the following sets {T ∈ Φ(E, F) IndexT =const, and dim N(T) = const.}, {T ∈ SΦ(E, F) either dim N(T) = const. ＜ ∞ or codim R(T) = const.＜ ∞} and {T ∈ R(E, F) RankT =const.＜∞}. Then ∑ is a smooth submanifold of B(E, F) with the tangent space TA∑ = {B ∈ B(E,F) BN(A) (∪) R(A)} for any A ∈ ∑. The result is available for the further application to Thom's famous results on the transversility and the study of the infinite dimensional geometry.     

A Note on <Emphasis Type="Italic">b</Emphasis>-Weakly Compact Operators

We consider a continuous operator T: E → X where E is a Banach lattice and X is a Banach space. We characterize the b-weak compactness of T in terms of its mapping properties.     

On identification of the threshold diffusion processes

We consider the problems of parameter estimation for several models of threshold ergodic diffusion processes in the asymptotics of large samples. These models are the direct continuous time analogues of the well known in time series analysis threshold autoregressive models. In such models, the trend is switching when the observed process attaints some (unknown) values and the problem is to estimate it or to test some hypotheses concerning these values. The related statistical problems correspond to the singular estimation or testing, for example, the rate of convergence of estimators is T and not ?T{\sqrt{T}} as in regular estimation problems. We study the asymptotic behavior of the maximum likelihood and Bayesian estimators and discuss the possibility of the construction of the goodness-of-fit test for such models of observation.     

Asymptotic efficiency and local optimality of tests based on two-sample <Emphasis Type="Italic">U</Emphasis>-and <Emphasis Type="Italic">V</Emphasis>-statistics

Many well-known homogeneity tests are based on two-sample U-and V-statistics. Using their large deviation asymptotics, we calculate for them the local Bahadur efficiency for common alternatives and find general conditions of their local asymptotic optimality. Bibliography: 16 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 219–231.     

Sign change of the function <Emphasis Type="Italic">S</Emphasis>(<Emphasis Type="Italic">t</Emphasis>) on short intervals

A theorem for the sign variation of the argument of the Riemann zeta function S(t) in the interval (t − A, t + A) with A = 4.39 ln ln ln ln T for each t, T ≤ t ≤ T + H excluding values from the set E with the measure mes(E) = O(H(ln ln T)⁻¹(ln ln ln T)^−0,5) is proved.     

Asymptotics for M/G/1 low-priority waiting-time tail probabilities

We consider the classical M/G/1 queue with two priority classes and the nonpreemptive and preemptive-resume disciplines. We show that the low-priority steady-state waiting-time can be expressed as a geometric random sum of i.i.d. random variables, just like the M/G/1 FIFO waiting-time distribution. We exploit this structures to determine the asymptotic behavior of the tail probabilities. Unlike the FIFO case, there is routinely a region of the parameters such that the tail probabilities have non-exponential asymptotics. This phenomenon even occurs when both service-time distributions are exponential. When non-exponential asymptotics holds, the asymptotic form tends to be determined by the non-exponential asymptotics for the high-priority busy-period distribution. We obtain asymptotic expansions for the low-priority waiting-time distribution by obtaining an asymptotic expansion for the busy-period transform from Kendall's functional equation. We identify the boundary between the exponential and non-exponential asymptotic regions. For the special cases of an exponential high-priority service-time distribution and of common general service-time distributions, we obtain convenient explicit forms for the low-priority waiting-time transform. We also establish asymptotic results for cases with long-tail service-time distributions. As with FIFO, the exponential asymptotics tend to provide excellent approximations, while the non-exponential asymptotics do not, but the asymptotic relations indicate the general form. In all cases, exact results can be obtained by numerically inverting the waiting-time transform. This revised version was published online in June 2006 with corrections to the Cover Date.