Space-dependent Potential Research Articles

Wavelet representation of hardcore bosons

We consider the one-dimensional Tonks–Girardeau gas with a space-dependent potential out of equilibrium. We derive the exact dynamics of the system when divided into n boxes and decomposed into energy eigenstates within each box. It is a representation of the wave function that is a mixture between real space and momentum space, with basis elements consisting of plane waves localized in a box, giving rise to the term ‘wavelet’. Using this representation, we derive the emergence of generalized hydrodynamics in appropriate limits without assuming local relaxation. We emphasize that a generalized hydrodynamic behaviour emerges in a high-momentum and short-time limit, in addition to the more common large-space and late-time limit, which is akin to a semi-classical expansion. In this limit, conserved charges do not require numerous particles to be described by generalized hydrodynamics. We also show that this wavelet representation provides an efficient numerical algorithm for a complete description of the out-of-equilibrium dynamics of hardcore bosons.

Lifespan estimates for semilinear wave equations with space dependent damping and potential

In this work, we investigate the influence of general damping and potential terms on the blow-up and lifespan estimates for energy solutions to power-type semilinear wave equations. The space-dependent damping and potential functions are assumed to be critical or short range, spherically symmetric perturbation. The blow up results and the upper bound of lifespan estimates are obtained by the so-called test function method. The key ingredient is to construct special positive solutions to the linear dual problem with the desired asymptotic behavior, which is reduced, in turn, to constructing solutions to certain elliptic “eigenvalue” problems.

A finite-volume scheme for modeling compressible magnetohydrodynamic flows at low Mach numbers in stellar interiors

Fully compressible magnetohydrodynamic (MHD) simulations are a fundamental tool for investigating the role of dynamo amplification in the generation of magnetic fields in deep convective layers of stars. The flows that arise in such environments are characterized by low (sonic) Mach numbers (ℳson ≲ 10−2). In these regimes, conventional MHD codes typically show excessive dissipation and tend to be inefficient as the Courant–Friedrichs–Lewy (CFL) constraint on the time step becomes too strict. In this work we present a new method for efficiently simulating MHD flows at low Mach numbers in a space-dependent gravitational potential while still retaining all effects of compressibility. The proposed scheme is implemented in the finite-volume SEVEN-LEAGUE HYDRO (SLH) code, and it makes use of a low-Mach version of the five-wave Harten–Lax–van Leer discontinuities (HLLD) solver to reduce numerical dissipation, an implicit–explicit time discretization technique based on Strang splitting to overcome the overly strict CFL constraint, and a well-balancing method that dramatically reduces the magnitude of spatial discretization errors in strongly stratified setups. The solenoidal constraint on the magnetic field is enforced by using a constrained transport method on a staggered grid. We carry out five verification tests, including the simulation of a small-scale dynamo in a star-like environment at ℳson ~ 10−3. We demonstrate that the proposed scheme can be used to accurately simulate compressible MHD flows in regimes of low Mach numbers and strongly stratified setups even with moderately coarse grids.

Thermal Hall response: Violation of gravitational analogs and Einstein relations

The response of solids to temperature gradients is often described in terms of a gravitational analogue: the effect of a space-dependent temperature is modeled using a space dependent metric. We investigate the validity of this approach in describing the bulk response of quantum Hall states and other gapped chiral topological states. To this end, we consider the prototypical Haldane model in two different cases of (i) a space-dependent electrostatic potential and gravitational potential and (ii) a space-dependent temperature and chemical potential imprinted by a weak coupling to non-interacting electron baths and phonons. We find that the thermal analogue is \textit{invalid}; while a space dependent gravitational potential induces transverse energy currents proportional to the third derivative of the gravitational potential, the response to an analogous temperature profile vanishes in limit of weak coupling to the thermal bath. Similarly, the Einstein relation, the analogy between the electrostatic potential and the internal chemical potential, is not valid in such a setup.

Out-of-equilibrium quantum dynamics of fermionic gases in the presence of localized particle loss

We address the effects of dissipative defects giving rise to a localized particle loss, in one-dimensional noninteracting lattice fermionic gases confined within a region of size $\ensuremath{\ell}$. We consider homogeneous systems within hard walls and inhomogeneous systems where the particles are trapped by space-dependent external potentials, such as harmonic traps. We model the dissipative particle-decay mechanism by Lindblad master equations governing the time evolution of the density matrix. The resulting quantum dynamics is analyzed in protocols starting from the ground state of the Hamiltonian for ${N}_{0}$ particles, then evolving under the effect of one dissipative particle-loss defect, for example, at the center of the system. We study the interplay between time, size $\ensuremath{\ell}$, and the number ${N}_{0}$ of initial particles, considering two different situations: (i) fixed number ${N}_{0}$ of initial particles; and (ii) fixed ratio ${N}_{0}/\ensuremath{\ell}$, corresponding to the thermodynamic limit of the initial equilibrium state. We show that the quantum evolutions of the particle number and density develop various intermediate and asymptotic dynamic regimes, and nontrivial large-time states when the dissipative mechanism acts at the center of the system.

The free path and generation rate of fast-moving electron interacting with dielectric media

In the framework of macroscopic continuous medium approach, we have studied interaction between a fast-moving charged particle and dielectric or semiconducting media with low energy electrically active excitations. The excitations contribute to frequency dispersion of the media dielectric permittivity. Two types of processes induced by a moving charged particle have been considered: electron-hole generation under interband transitions and excitation of polar optical phonons. For both processes, we calculated and analyzed the time- and space-dependent electric potential generated by the charged particle, polarization of the media, energy losses of the particle and other important constituents of the interaction patterns. Obtained results can contribute to deeper understanding of the charged particle beams interaction with a semiconducting medium, as well as may be useful for versatile applications of charged beams.

Lie symmetry properties of the two-component complex Ginzburg–Landau system with variable coefficients

We classify the Lie point symmetries of a system of nonlinearly coupled complex Ginzburg–Landau equations with time and space dependent potentials and nonlinearities. New families of systems with exact solutions and conservation laws are identified. When there is only space dependence, the system can be reduced to a set of ordinary differential equations for the amplitude and phase components, for which integrability and first integrals are discussed.

Finite codimensional controllability and optimal control problems with endpoint state constraints

In this paper, motivated by the study of optimal control problems for infinite dimensional systems with endpoint state constraints, we introduce the notion of finite codimensional (exact/approximate) controllability. Some equivalent criteria on the finite codimensional controllability are presented. In particular, the finite codimensional exact controllability is reduced to deriving a Gårding type inequality for the adjoint system, which is new for many evolution equations. This inequality can be verified for some concrete problems (and hence applied to the corresponding optimal control problems), say the wave equations with both time and space dependent potentials. Moreover, under some mild assumptions, we show that the finite codimensional exact controllability of this sort of wave equations is equivalent to the classical geometric control condition.

Pair production from the vacuum by a weakly inhomogeneous space-dependent electric potential

There exists a clear physical motivation for theoretical studies of the vacuum instability related to the production of electron-positron pairs from a vacuum due to strong external electric fields. Various nonperturbative (with respect to the external fields) calculation methods were developed. Some of these methods are based on possible exact solutions of the Dirac equation. Unfortunately, there are only few cases when such solutions are known. Recently, an approximate but still nonperturbative approach to treat the vacuum instability caused by slowly varying $t$-electric potential steps (time dependent external fields that vanish as $|t|\rightarrow\infty$), which does not depend on the existence of the corresponding exact solutions, was formulated in Ref. [S. P. Gavrilov, D. M. Gitman, Phys. Rev. D \textbf{95}, 076013 (2017)]. Here, we present an approximate calculation method to treat nonperturbatively the vacuum instability in arbitrary weakly inhomogeneous $x$-electric potential steps (time-independent electric fields of a constant direction that are concentrated in restricted space areas, which means that the fields vanish as $|x|\rightarrow\infty$) in the absence of the corresponding exact solutions. Defining the weakly inhomogeneous regime in general terms, we demonstrate the universal character of the vacuum instability. This universality is associated with a large density of states excited from the vacuum by the electric field. Such a density appears in our approach as a large parameter. We derive universal representations for the total number and current density of the created particles. Relations of these representations with a locally constant field approximation for Schwinger's effective action are found.

Hölder Stably Determining the Time-Dependent Electromagnetic Potential of the Schrödinger Equation

We consider the inverse problem of determining the time and space dependent electromagnetic potential of the Schrodinger equation in a bounded domain of R n , n 2, by boundary observation of the solution over the entire time span. Assuming that the divergence of the magnetic potential is fixed, we prove that the electric potential and the magnetic potential can be Holder stably retrieved from these data, whereas stability estimates for inverse time-dependent coefficients problems of evolution partial differential equations are usually of logarithmic type.

On the rate of convergence of Schwarz waveform relaxation methods for the time-dependent Schrödinger equation

This paper is dedicated to the analysis of the rate of convergence of the classical and quasi-optimal Schwarz waveform relaxation (SWR) method for solving the linear Schrödinger equation with space-dependent potential. The strategy is based on i) the rewriting of the SWR algorithm as a fixed point algorithm in frequency space, and ii) the explicit construction of contraction factors thanks to pseudo-differential calculus. Some numerical experiments illustrating the analysis are also provided.

Backstepping Control for the Schrödinger Equation with an Arbitrary Potential in a Confined Space

In this work the control design problem for the Schrödinger equation with an arbitrary potential is addressed. In particular a controller is designed which (i) for a space-dependent potential steers the state probability density function to a prescribed solution and (ii) for a space and state-dependent potential exponentially stabilizes the zero solution. The problem is addressed using a backstepping controller that steers to zero the deviation between the initial probability wave function and the target probability wave function. The exponential convergence property is rigorously established and the convergence behavior is illustrated using numerical simulations for the Morse and the Pöschl-Teller potentials as well as the semilinear Schrödinger equation with cubic potential.

Global Existence and Blow-up for Semilinear Wave Equations with Variable Coefficients

The authors consider the critical exponent problem for the variable coefficients wave equation with a space dependent potential and source term. For sufficiently small data with compact support, if the power of nonlinearity is larger than the expected exponent, it is proved that there exists a global solution. Furthermore, the precise decay estimates for the energy, L2 and Lp+1 norms of solutions are also established. In addition, the blow-up of the solutions is proved for arbitrary initial data with compact support when the power of nonlinearity is less than some constant.

Engineered swift equilibration for Brownian objects: from underdamped to overdamped dynamics

We propose an enlarged framework to study transformations that drive an underdamped Brownian particle in contact with a thermal bath from an equilibrium state to a new one in an arbitrarily short time. To this end, we make use of a time and space-dependent potential, that plays a dual role: confine the particle, and manipulate the system. In the special case of an isothermal compression or decompression of a harmonically trapped particle, we derive explicit protocols that perform this quick transformation, following an inverse engineering method. We focus on the properties of these protocols, which crucially depend on two key dimensionless numbers that characterize the relative values of the three timescales of the problem, associated with friction, oscillations in the confinement and duration of the protocol. In particular, we show that our protocols encompass the known overdamped version of this problem and extend it to any friction for decompression and to a large range of frictions for compression.

Circular Time Scale Yields a Recurrent Calculation of the Schrödinger Perturbation Energy

An essential simplification of approach to the Schrodinger perturbation series for energy does hold when the perturbation events are arranged along a circular scale of time. The aim of the present paper is to demonstrate how such a scale of time leads to the recurrence calculation process of the Schrodinger energy terms belonging to an arbitrary perturbation order N. This process seems to have never been represented before. Only a non-degenerate quantum state and its perturbation due to the space-dependent potential are considered in the paper.

Reconstruction of space-dependent potential and/or damping coefficients in the wave equation

In this paper, nonlinear reconstructions of the space-dependent potential and/or damping coefficients in the wave equation from Cauchy data boundary measurements of the displacement and the flux tension are investigated. This is a very interesting and challenging nonlinear inverse coefficient problem with important applications in wave propagation phenomena. The uniqueness and stability results that are revised and in some cases proved demonstrate an advancement in understanding the stability of the inverse coefficient problems. However, in practice, the inverse coefficient identification problems under investigation are still ill-posed since small random errors in the input data cause large errors in the output solution. In order to stabilize the solution we employ the nonlinear Tikhonov regularization method. Numerical reconstructions performed for the first time are presented and discussed to illustrate the accuracy and stability of the numerical solutions under finite difference mesh refinement and noise in the measured data.

An analysis of Schwarz waveform relaxation domain decomposition methods for the imaginary-time linear Schrödinger and Gross-Pitaevskii equations

The aim of this paper is to derive and numerically validate some asymptotic estimates of the convergence rate of Classical and quasi-Optimal Schwarz Waveform Relaxation (SWR) domain decomposition methods applied to the computation of the stationary states of the one-dimensional linear and nonlinear Schrodinger equations with a space-dependent potential. Although SWR methods are currently used for efficiently solving high dimensional partial differential equations, their convergence analysis and most particularly obtaining expressions of their convergence rate remains largely open even in one dimension, except in simple cases. In this work, we tackle this problem for linear and nonlinear one-dimensional Schrodinger equations by developing techniques which can be extended to higher dimensional problems and other types of PDEs. The approach combines the method developed in Gander and Halpern (SIAM J Num Anal 45(2):666–697, 2007) for the linear advection reaction diffusion equation and the theory of inhomogeneous pseudodifferential operators in conjunction with the associated symbolic asymptotic expansions. For computing the stationary states, we consider the imaginary-time formulation of the Schrodinger equation based on the Normalized Gradient Flow (NGF) method and use a semi-implicit Euler scheme for the discretization. Some numerical results in the one-dimensional case illustrate the analysis for both the linear Schrodinger and Gross-Pitaevskii equations.

Polarised black holes in AdS

We consider solutions in Einstein–Maxwell theory with a negative cosmological constant that asymptote to global AdS4 with conformal boundary . At the sphere at infinity we turn on a space-dependent electrostatic potential, which does not destroy the asymptotic AdS behaviour. For simplicity we focus on the case of a dipolar electrostatic potential. We find two new geometries: (i) an AdS soliton that includes the full backreaction of the electric field on the AdS geometry; (ii) a polarised neutral black hole that is deformed by the electric field, accumulating opposite charges in each hemisphere. For both geometries we study boundary data such as the charge density and the stress tensor. For the black hole we also study the horizon charge density and area, and further verify a Smarr formula. Then we consider this system at finite temperature and compute the Gibbs free energy for both AdS soliton and black hole phases. The corresponding phase diagram generalizes the Hawking–Page phase transition. The AdS soliton dominates the low temperature phase and the black hole the high temperature phase, with a critical temperature that decreases as the external electric field increases. Finally, we consider the simple case of a free charged scalar field on with conformal coupling. For a field in the SU(N ) adjoint representation we compare the phase diagram with the above gravitational system.

Dynamics of a classical particle in a quasi periodic potential

We study the dynamics of a one-dimensional classical particle in a space and time dependent potential with randomly chosen parameters. The focus of this work is a quasi-periodic potential, which only includes a finite number of Fourier components. The momentum is calculated analytically for short time within a self-consistent approximation, under certain conditions. We find that the dynamics can be described by a model of a random walk between the Chirikov resonances, which are resonances between the particle momentum and the Fourier components of the potential. We use numerical methods to test these results and to evaluate the important properties, such as the characteristic hopping time between the resonances. This work sheds light on the short time dynamics induced by potentials which are relevant for optics and atom optics.

Novel exact solutions of coupled nonlinear Schrödinger equations with time—space modulation

We construct various novel exact solutions of two coupled dynamical nonlinear Schrödinger equations. Based on the similarity transformation, we reduce the coupled nonlinear Schrödinger equations with time- and space-dependent potentials, nonlinearities, and gain or loss to the coupled dynamical nonlinear Schrödinger equations. Some special types of non-travelling wave solutions, such as periodic, resonant, and quasiperiodically oscillating solitons, are used to exhibit the wave propagations by choosing some arbitrary functions. Our results show that the number of the localized wave of one component is always twice that of the other one. In addition, the stability analysis of the solutions is discussed numerically.