Finite Space Interval Research Articles

Exact time-evolving scattering states in open quantum-dot systems with an interaction: discovery of time-evolving resonant states

We study exact time-evolving many-electron states of an open double quantum-dot system with an interdot Coulomb interaction. A systematic construction of the time-evolving states for arbitrary initial conditions is proposed. For any initial states of one- and two-electron plane waves on the electrical leads, we obtain exact solutions of the time-evolving scattering states, which converge to known stationary scattering eigenstates in the long-time limit. For any initial states of localized electrons on the quantum dots, we find exact time-evolving states of a new type, which we refer to as time-evolving resonant states. In contrast to stationary resonant states, whose wave functions spatially diverge and not normalizable, the time-evolving resonant states are normalizable since their wave functions are restricted to a finite space interval due to causality. The exact time-evolving resonant states enable us to calculate the time-dependence of the survival probability of electrons on the quantum dots for the system with the linearized dispersions. It decays exponentially in time on one side of an exponential point of resonance energies while, on the other side, it oscillates during the decay as a result of the interference of the two resonance energies.

A fundamental sequences method with time-reduction for one-dimensional lateral Cauchy problems

A fundamental sequences method is derived for the numerical solution of an ill-posed one-dimensional lateral Cauchy problem for a hyperbolic damped wave equation, including as a special case the parabolic heat equation. Either the Laguerre transform or the Houbolt finite difference scheme is applied to reduce the time-dependent lateral Cauchy problem to a sequence of second-order ordinary differential equations (ODEs) with function values and derivatives specified at the right endpoint of a finite space interval. A set of fundamental solutions is constructed, termed a fundamental sequence, to the differential equations. The solution of the obtained ODEs is approximated by a linear combination of elements in the fundamental sequence. Source points are placed outside of the solution interval in space, and by collocating at the endpoints of this interval a sequence of linear equations is obtained for finding the unknown coeficients. Tikhonov regularization is used to render a stable solution to the obtained systems of linear equations. Numerical results both for the parabolic and hyperbolic case confirm the efficiency of the proposed method including noisy data. The presented results complement the higher-dimensional case initiated in our previous researches.

Quantum Field Theory Beyond the S-Matrix Formalism

Particle transitions passing over finite space and time intervals are self-consistently described within a novel quantum field-theoretical approach based on Feynman diagrams in configuration space. The transition to momentum space is performed following the rules modified so as to consider the geometry of neutral-meson and neutrino oscillation experiments. Using the processes of decay of an unstable scalar particle at a macroscopic distance from the source and of neutral kaon and neutrino oscillations as examples, we demonstrate that their known characteristics are adequately reproduced within the proposed approach.

Quantum Field Theory Description of Processes Passing at Finite Space and Time Intervals

We consider a new approach to the quantum field theory description of processes passing at finite space-time intervals. The formalism is based on the Feynman diagram technique in the coordinate representation supplemented with modified rules for the transition to the momentum representation reflecting specific experimental situations. This effectively implies that only the particle propagators in the momentum representation are modified, while the standard Feynman rules in the momentum representation remain unchanged. No wave packets are used in the approach, i.e., the initial and final states of particles are described by plane waves, which significantly simplifies the calculations. Three processes—neutrino oscillations, unstable particle decay, and neutral kaon oscillations—are used as examples to show that the proposed approach correctly reproduces the well-known results.

Description of processes passing at finite space and time intervals in the framework of QFT

We consider a new approach to the description in the framework of QFT of processes passing at finite space and time intervals. The formalism is based on the Feynman diagram technique in the coordinate representation, in which the rules of passing to the momentum representation are modified in accordance with the experimental setup of neutrino oscillation experiments. In effect, only the propagators of particles in the momentum representation are modified, while all the other standard Feynman rules in the momentum representation remain the same. Since the initial and final particle states are described by plane waves, the approach does not need the use of wave packets, which greatly simplifies the calculations of amplitudes. Taking as examples the processes of displaced pion decay, neutral kaon and neutrino oscillations we show that the approach under consideration correctly reproduces the known standard results.

Large-N \u2102\u2119N\u2009\u2212\u20091 sigma model on a finite interval: general Dirichlet boundary conditions

This is the third of the series of articles on the large-N two-dimensional ℂℙN − 1 sigma model, defined on a finite space interval L with Dirichlet boundary conditions. Here the cases of the general Dirichlet boundary conditions are studied, where the relative ℂℙN − 1 orientations at the two boundaries are generic, and numerical solutions are presented. Distinctive features of the ℂℙN − 1 sigma model, as compared e.g., to an O(N) model, which were not entirely evident in the basic properties studied in the first two articles in the large N limit, manifest themselves here. It is found that the total energy is minimized when the fields are aligned in the same direction at the two boundaries.

Large-N mathbb{C}{mathrm{mathbb{P}}}^{mathrm{N}-1} sigma model on a finite interval and the renormalized string energy

We continue the analysis started in a recent paper of the large-N two-dimensional mathbb{C}{mathrm{mathbb{P}}}^{N-1} sigma model, defined on a finite space interval L with Dirichlet (or Neumann) boundary conditions. Here we focus our attention on the problem of the renormalized energy density mathrm{mathcal{E}} (x, Λ, L) which is found to be a sum of two terms, a constant term coming from the sum over modes, and a term proportional to the mass gap. The approach to mathrm{mathcal{E}}left(x,varLambda, Lright)to frac{N}{4pi }{varLambda}^2 at large LΛ is shown, both analytically and numerically, to be exponential: no power corrections are present and in particular no Lüscher term appears. This is consistent with the earlier result which states that the system has a unique massive phase, which interpolates smoothly between the classical weakly-coupled limit for LΛ → 0 and the “confined” phase of the standard mathbb{C}{mathrm{mathbb{P}}}^{N-1} model in two dimensions for LΛ → ∞.

Large-N ℂP N − 1 sigma model on a finite interval

We analyze the two-dimensional ℂP N − 1 sigma model defined on a finite space interval L, with various boundary conditions, in the large N limit. With the Dirichlet boundary condition at the both ends, we show that the system has a unique phase, which smoothly approaches in the large L limit the standard 2D ℂP N − 1 sigma model in confinement phase, with a constant mass generated for the n i fields. We study the full functional saddle-point equations for finite L, and solve them numerically. The latter reduces to the well-known gap equation in the large L limit. It is found that the solution satisfies actually both the Dirichlet and Neumann conditions.

Models of optimum radio-systems on the vector combinatorial configurations

Method for construction of optimum radio systems, based on a new conceptual model of the systems - Ideal Ring Vector structures (clusters of the IRV) is proposed. IRV clusters are cyclic ordered sequences of t- integer sub-sequences of the sequence which form perfect relationships of t-dimensional partitions over a virtual t-dimensional lattice covered surface of a finite space interval. The sums of connected sub-sequences of an IRV enumerate the set of tcoordinates specified with respect to cyclic frame reference exactly R-times. This property makes IRVs useful in applications which need to partition multidimensional objects with the smallest possible number of intersections. This sort of models can be used for finding optimal solution for wide classes of technological problems based on the idea of “perfect” vector combinatorial constructions, and expanding the applicability of two-, three- and multidimensional IRV as multidimensional cyclic relationships for fundamental and applied research in systems engineering, for improving such quality indices as vector data coding and signal reconstruction, resolving ability and low side lobe antenna design. There are regarded basic properties these models and made comparative analysis of the models with difference sets. It is shown that the IRVs to be in exceed of difference sets multiply, and set of the classical difference sets is subset of the IRVs. Short review of the methods relating to constructing of the optimum models of non-uniform antenna arrays with respect to low side lobe is given. The problem statement involves development the regular method for construction of the optimum radio-systems using two- and three-dimensional IRVs, and some examples are regarded for illustration its technical merits including algorithm of synthesis of two-dimensional symmetric antenna with 12 elements which provides sufficiently low side lobe radiation levels and shows a flat gain over the frequency range is presented. Method for design of two- or multidimensional vector signals coded based on the optimum binary monolithic code is presented.

Existence and uniqueness of solutions of a class of third order dissipative problems with various boundary conditions describing the Josephson effect

We prove existence and uniqueness of solutions of a large class of initial–boundary-value problems characterized by a quasi-linear third order equation (the third order term being dissipative) on a finite space interval with Dirichlet, Neumann or pseudoperiodic boundary conditions. The class includes equations arising in superconductor theory, in the form of various modified sine–Gordon equation describing the Josephson effect, and in the theory of viscoelastic materials.

Kink localization under asymmetric double-well potentials

We study diffuse phase interfaces under asymmetric double-well potential energies with degenerate minima and demonstrate that the limiting sharp profile, for small interface energy cost, on a finite space interval is in general not symmetric and its position depends exclusively on the second derivatives of the potential energy at the two minima (phases). We discuss an application of the general result to porous media in the regime of solid-fluid segregation under an applied pressure and describe the interface between a fluid-rich and a fluid-poor phase.

Existence and Uniqueness for Some 3rd Order Dissipative Problems with Various Boundary Conditions

We briefly present new results on the existence and uniqueness of solutions u of a large class of initial-boundary-value problems characterized by a quasi-linear third order equation on a finite space interval with Dirichlet, Neumann or pseudoperiodic boundary conditions. The class includes equations arising in Superconductor Theory and in the Theory of Viscoelastic Materials.

Well-posedness and long-time behavior of Lipschitz solutions to generalized extremal surface equations

We show that in one space dimension, Lipschitz solutions of generalized extremal surface equations are equivalent to entropy solutions in \documentclass[12pt]{minimal}\begin{document}$L^\infty ({\bb R})$\end{document}L∞(R) of a non-strictly hyperbolic system of conservation laws. We obtain an explicit representation formula and the uniqueness of the entropy solutions to the Cauchy problem of the system. By using this formula, we also obtain the convergence and convergence rates as t → +∞ of the entropy solutions to explicit traveling waves in the \documentclass[12pt]{minimal}\begin{document}$L^1({\bb R})$\end{document}L1(R) norm. Moreover, when initial data are constants outside of a finite space interval, the entropy solutions become the explicit traveling waves after a finite time. Finally, we prove L1 stabilities of the entropy solutions.

Lyapunov exponential stability of linear hyperbolic systems of balance laws

AbstractExplicit boundary dissipative conditions are given for the exponential stability in L2-norm of one-dimensional linear hyperbolic sytems of balance laws ∂tξ+Λ∂xξ-Mξ=0 over a finite space interval, when the matrix M is marginally diagonally stable. The result is illustrated with an application to boundary feedback stabilisation of open channels represented by linearised Saint-Venant-Exner equations.

Spatial neuron model with two-parameter Ornstein–Uhlenbeck input current

We consider a new and extended spatial neuron model in which the neuronal electrical depolarization from resting level satisfies a cable partial differential equation. The synaptic input current is also a function of space and time and satisfies a first order linear partial differential equation driven by a two-parameter random process. A natural choice for these random input processes is to make them two-parameter Poisson processes for both excitation and inhibition. For such inputs the mean subthreshold voltage is found in the case of an infinite cable and of finite cables. Assuming uniform amplitudes and rates exact expressions are obtained in the case of particular boundary conditions. We then consider a diffusion approximation and show that the membrane current is a two-parameter Ornstein–Uhlenbeck process, whose statistical properties are derived. Using representations for the voltage in terms of stochastic integrals in the plane we find, in the case of finite space intervals the mean, variance and covariance of the subthreshold voltage. For large times the voltage process is shown to be covariance stationary in time and the corresponding spectral density is found and compared with the result for a purely (two-parameter) white noise driven cable. The limiting white noise case is obtained from the extended model as the decay parameter becomes infinite. Finally, we develop useful simulation methods for the solution of the extended spatial model using properties of stochastic integrals involving eigenfunctions to obtain one-dimensional representations which are easily implemented.

An exact solution for the linearized multifrequency radiation diffusion equation

An exact solution, based on Fourier and Laplace (FL) transforms, is developed for a linearization of the system modeling the multifrequency radiation diffusion and matter energy balance equations. The model uses an ideal gas equation of state. Opacities are proportional to the inverse of the cube of the frequency, thereby simulating free–free transitions. The solution is obtained in terms of integrals over the FL coefficients of the initial conditions and explicit sources. Results are presented for two special cases. (1) No sources, initially cold radiation field, and a localized matter energy profile. (2) Initially cold matter and radiation fields and a source of matter energy extending over finite space and time intervals.

Fine structure of the Vavilov-Cherenkov radiation.

The aim of this paper is to study the fine structure of the Cherenkov rings. We analyze the experiments performed by one of authors (Zrelov) in which no special focusing devices were used. The broad Cherenkov ring was observed in the plane perpendicular to the motion axis. Using the exact and approximate formulas we investigate how a charge moving uniformly in a medium radiates in a finite space interval (the Tamm problem). The formulas obtained describe the radiation intensity in the whole space, inside and outside the Cherenkov ring. In the plane perpendicular to the motion axis, the radiation fills mainly the finite ring. Its width (proportional to the motion interval) and the energy released in this ring do not depend on the position of the observation plane. Outside the Cherenkov ring, the radiation intensity suddenly drops. Inside it, the radiation intensity exhibits small oscillations which are due to the interference of the Vavilov-Cherenkov radiation and bremsstrahlung. The bursts of the radiation intensity at the ends of the Cherenkov ring are associated with the shock waves arising at the instant when the charge velocity coincides with the light velocity in a medium. For the chosen motion interval, the well-known Tamm formula does not describe the radiation intensity inside the Cherenkov ring for any position of the observation plane. Outside the Cherenkov ring, the Tamm formula is valid only at very large distances. Theoretical calculations are in satisfactory agreement with experimental data. Thus, the combined experimental and theoretical study of the unfocused Cherenkov rings allows one to obtain information on the physical processes accompanying the Cherenkov radiation in the finite spatial interval (bremsstrahlung, transition of the light velocity barrier, etc.).

Massive scalar field in a one-dimensional oscillating region

We study the classical scalar massive field satisfying the Klein–Gordon equation in a finite one-dimensional space interval of periodically varying length with Dirichlet boundary conditions. For a sufficiently small mass, the energy can exponentially grow with time under the same conditions as for the massless case. The proofs are based on estimates of exactly given mass-induced corrections to the massless case.

Cherenkov radiation versus bremsstrahlung in the Tamm problem

The charge motion in medium on a finite space interval is considered. We analyse recent alternative attempts to interpret the radiation described by the Tamm formula as an interference of two instantaneous accelerations arising at the beginning and termination of motion. Exact solution of the Tamm problem in the time representation shows that in some time interval, only the bremsstrahlungshock wave associated with the beginning of motion and the Cherenkovshock wave exist, and there is no bremsstrahlung shock wave associated with the end of motion. This proves that in the time representation the Cherenkov radiation is not necessarily related to the interference of initial and final bremsstrahlung shock waves.In the spectral representation, we consider the motion consistingof accelerated, decelerated, and uniform parts. Analytic formulae are obtained that describe radiation intensities for this motion. Approximating the instantaneous acceleration in the original Tamm problem by the acceleration on a finitepath and then tending its length to zero, we prove that the radiation intensityproduced on the accelerated part of the charge trajectory also tends to zero (despite the infinite value of acceleration in this limit). This means that in the original Tamm problem the instantaneous accelerationand deceleration do not contribute to the radiation intensity (as it is usually believed). It seems that only the combined consideration of the Tamm problem in the time and spectral representations shows that the above-mentioned alternative interpretation of the Cherenkov radiation fails.

On the Smoothed Tamm Problem

To remove obstacles against an instantaneous acceleration and deceleration of charge in the Tamm problem, we consider charge motion on a finite space interval with finite acceleration and deceleration. When the charge velocity coincides with the velocity of light in the medium, a new shock wave (in addition to the Cherenkov shock wave) arises which propagates with the light velocity in the medium. The angular-frequency distribution of the emitted radiation crucially depends on the radius of the sphere on which the observations are made.