Space-dependent Potential Research Articles

Absorbing boundary conditions for the two-dimensional Schrödinger equation with an exterior potential

Four families of ABCs where built in Antoine et al. (Math Models Methods Appl Sci, 22(10), 2012) for the two-dimensional linear Schrodinger equation with time and space dependent potentials and for general smooth convex fictitious surfaces. The aim of this paper is to propose some suitable discretization schemes of these ABCs and to prove some semi-discrete stability results. Furthermore, the full numerical discretization of the corresponding initial boundary value problems is considered and simulations are provided to compare the accuracy of the different ABCs.

Renormalization-group flow and asymptotic behaviors at the Berezinskii-Kosterlitz-Thouless transitions

We investigate the general features of the renormalization-group flow at the Berezinskii-Kosterlitz-Thouless (BKT) transition, providing a thorough quantitative description of the asymptotc critical behavior, including the multiplicative and subleading logarithmic corrections. For this purpose, we consider the RG flow of the sine-Gordon model around the renormalizable point which describes the BKT transition. We reduce the corresponding beta-functions to a universal canonical form, valid to all perturbative orders. Then, we determine the asymptotic solutions of the RG equations in various critical regimes: the infinite-volume critical behavior in the disordered phase, the finite-size scaling limit for homogeneous systems of finite size, and the trap-size scaling limit occurring in 2D bosonic particle systems trapped by an external space-dependent potential.

Resonant Tunneling Controlled by Laser and Constant Electric Fields

The aim of this paper is to present a numerically stable algorithm for investigations of nonrelativistic quantum processes occurring in arbitrary space-dependent scalar potential and a timeand space-dependent vector potential. Vector potential is periodic in time and describes a laser eld. Such conditions are met for example in semiconductor nanostructures [1 8] (like quantum wires or wells), photoemission from a metal tip [9], carbon nanotubes [10, 11] or in surface physics [12 16]. To make our presentation as clear as possible we shall restrict ourselves to the one-space-dimensional case, although extension of this algorithm to two and three-space dimensional systems, also with magnetic eld accounted for, is possible (see, e.g. [17]). We shall apply our method to investigation of the tunneling process and its dependence on relative phases of multichromatic laser pulses (multicolor processes have been considered for instance in [16, 18]). Multiple barrier, eld-assisted resonant tunneling is an interesting problem because it provides an insight into the physics of nanostructure quantum systems and because it is a fundamental e ect to use in a wide variety of technological applications. As concerns the latter, it is enough to mention all sorts of detectors and generators of microwave radiation based on double barrier structures with external electric eld added; for more examples, see [19 24]. Here we analyze resonant tunneling through semiconductor structures in the presence of both oscillating and constant in time external electric elds. The elds are supposed to be nonperturbative. We assume that single-

Entanglement and particle correlations of Fermi gases in harmonic traps

We investigate quantum correlations in the ground state of noninteracting Fermi gases of N particles trapped by an external space-dependent harmonic potential, in any dimension. For this purpose, we compute one-particle correlations, particle fluctuations and bipartite entanglement entropies of extended space regions, and study their large-N scaling behaviors. The half-space von Neumann entanglement entropy is computed for any dimension, obtaining S_HS = c_l N^(d-1)/d ln N, analogously to homogenous systems, with c_l=1/6, 1/(6\sqrt{2}), 1/(6\sqrt{6}) in one, two and three dimensions respectively. We show that the asymptotic large-N relation S_A\approx \pi^2 V_A/3, between the von Neumann entanglement entropy S_A and particle variance V_A of an extended space region A, holds for any subsystem A and in any dimension, analogously to homogeneous noninteracting Fermi gases.

Analytical Formulation for ϕ 4 Field Potential Dynamics

An analytical model for adding a space-dependent potential to the ϕ 4 field equation of motion is presented, by constructing a collective coordinate system for the solitary solutions of this model. The interaction of ϕ 4 solitons with a delta function potential barrier and also delta function potential well is investigated. Most of the characters of interaction are derived analytically while they are calculated by other models numerically. We will find that the behaviour of a solitary solution is like a point particle which is moved under the influence of a complicated effective potential. The effective potential is a function of the field initial conditions and also of parameters of the added potential.

Exact analytical solutions of three-dimensional Gross—Pitaevskii equation with time—space modulation

By the generalized sub-equation expansion method and symbolic computation, this paper investigates the (3 + 1)-dimensional Gross—Pitaevskii equation with time- and space-dependent potential, time-dependent nonlinearity, and gain or loss. As a result, rich exact analytical solutions are obtained, which include bright and dark solitons, Jacobi elliptic function solutions and Weierstrass elliptic function solutions. With computer simulation, the main evolution features of some of these solutions are shown by some figures. Nonlinear dynamics of a soliton pulse is also investigated under the different regimes of soliton management.

Quasi-long-range order in trapped systems

We investigate the effects of a trapping space-dependent potential on the low-temperature quasi-long-range order phase of two-dimensional particle systems with a relevant U(1) symmetry, such as quantum atomic gases. We characterize the universal features of the trap-size dependence using scaling arguments. The resulting scenario is supported by numerical Monte Carlo simulations of a classical two-dimensional XY model with a space-dependent hopping parameter whose inhomogeneity is analogous to that arising from the trapping potential in experiments of atomic gases.

Decay Properties for the Damped Wave Equation with Space Dependent Potential and Absorbed Semilinear Term

We consider the Cauchy problem for the damped wave equation with space dependent potential V(x)u t and absorbed semilinear term |u|ρ−1 u in R N . Our assumption on V(x) ∼ (1 + |x|2)−α/2 (0 ≤ α <1) still implies the diffusion phenomena and the decay rates of solutions are expected to be the same as the corresponding parabolic problem. In this paper we obtain two kinds of decay rates of the solution effective for ρ > ρ c (N, α): = 1 + 2/(N − α) and for ρ < ρ c (N, α). We believe that in the “supercritical” exponent the decay rates obtained are almost the same as those for the linear parabolic problem, while, in the “subcritical” exponent the solution decays faster than that of linear equation, thanks to the absorbed semilinear term. So we believe that ρ c (N, α) is a critical exponent. Note that ρ c (N, α) with α = 0 coincides to the Fujita exponent .

Dynamics of multi-kinks in the presence of wells and barriers

Sine-Gordon kinks are a much studied integrable system that possesses multi-soliton solutions. Recent studies on sine-Gordon kinks with space-dependent square-well-type potentials have revealed interesting dynamics of a single kink interacting with wells and barriers. In this paper, we study a class of smooth space-dependent potentials and discuss the dynamics of one kink in the presence of different wells. We also present values for the critical velocity for different types of barriers. Furthermore, we study two kinks interacting with various wells and describe interesting trajectories such as double-trapping, kink knock-out and double-escape.

Analytical three-dimensional bright solitons and soliton pairs in Bose-Einstein condensates with time-space modulation

We provide analytical three-dimensional bright multisoliton solutions to the $(3+1)$-dimensional Gross-Pitaevskii equation with time- and space-dependent potential, time-dependent nonlinearity, and gain or loss. The zigzag propagation trace and the breathing behavior of solitons are observed. Different shapes of bright solitons and fascinating interactions between two solitons can be achieved with different parameters. The obtained results may raise the possibility of relative experiments and potential applications.

Decay estimates for wave equations with variable coefficients

We establish weighted L 2 − L^2- estimates for dissipative wave equations with variable coefficients that exhibit a dissipative term with a space dependent potential. These results yield decay estimates for the energy and the L 2 − L^2- norm of solutions. The proof is based on the multiplier method where multipliers are specially engineered from asymptotic profiles of related parabolic equations.

Maximal regularity for non-autonomous Schrödinger type equations

In this paper we study the maximal regularity property for non-autonomous evolution equations ∂ t u ( t ) + A ( t ) u ( t ) = f ( t ) , u ( 0 ) = 0 . If the equation is considered on a Hilbert space H and the operators A ( t ) are defined by sesquilinear forms a ( t , ⋅ , ⋅ ) we prove the maximal regularity under a Hölder continuity assumption of t → a ( t , ⋅ , ⋅ ) . In the non-Hilbert space situation we focus on Schrödinger type operators A ( t ) : = − Δ + m ( t , ⋅ ) and prove L p − L q estimates for a wide class of time and space dependent potentials m.

Formula omitted]-estimates of solutions for damped wave equations with space–time dependent damping term

In this paper, we consider the damped wave equation with space–time dependent potential b ( t , x ) and absorbing semilinear term | u | ρ − 1 u . Here, b ( t , x ) = b 0 ( 1 + | x | 2 ) − α 2 ( 1 + t ) − β with b 0 > 0 , α , β ⩾ 0 and α + β ∈ [ 0 , 1 ) . Based on the local existence theorem, we obtain the global existence and the L 2 decay rate of the solution by using the weighted energy method. The decay rate coincides with the result of Nishihara [K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, preprint] in the case of β = 0 and coincides with the result of Nishihara and Zhai [K. Nishihara, J. Zhai, Asymptotic behaviors of time dependent damped wave equations, preprint] in the case of α = 0 .

Path integration in QCD with arbitrary space-dependent static color potential

We perform path integral for a quark (antiquark) in the presence of an arbitrary space-dependent static color potential A^a_0(x)(=-\int dx E^a(x)) with arbitrary color index a=1,2,...8 in SU(3) and obtain an exact non-perturbative expression for the generating functional. We show that such a path integration is possible even if one can not solve the Dirac equation in the presence of arbitrary space-dependent potential. It may be possible to further explore this path integral technique to study non-perturbative bound state formation.

Space-dependent step features: Transient breakdown of slow-roll, homogeneity, and isotropy during inflation

A step feature in the inflaton potential can model a transient breakdown of slow-roll inflation. Here we generalize the step feature to include space-dependence, allowing it also to model a breakdown of homogeneity and isotropy. The space-dependent inflaton potential generates a classical curvature perturbation mode characterized by the wave number of the step inhomogeneity. For inhomogeneities small compared with the horizon at the step, space-dependence has a small effect on the curvature perturbation. Therefore, the smoothly oscillating quantum power spectrum predicted by the homogeneous step is robust with respect to subhorizon space-dependence. For inhomogeneities equal to or greater than the horizon at the step, the space-dependent classical mode can dominate, producing a curvature perturbation in which modes of wave number determined by the step inhomogeneity are superimposed on the oscillating power spectrum. Generation of a space-dependent step feature may therefore provide a mechanism to introduce primordial anisotropy into the curvature perturbation. Space-dependence also modifies the quantum fluctuations, in particular, via resonancelike features coming from mode coupling to amplified superhorizon modes. However, these effects are small relative to the classical modes.

Driving Weiss oscillations to zero resistance states by microwave Radiation

We present a theoretical model to study the effect of microwave radiation on Weiss oscillations. In our proposal Weiss oscillations, produced by a spatial periodic potential, are modulated by microwave radiation due to an interference effect between both, space and time-dependent, potentials. The final magnetoresistance depends mainly on the spatial period of the spatial potential and the frequency of radiation. Depending on the values of these parameters, we predict that Weiss oscillations can reach zero resistance states. On the other hand, these dissipationless transport states, created just by radiation, can be destroyed by the presence of a space-dependent potential.

Mesoscopic Aharonov-Bohm loops in a time-dependent potential: Quasistationary electronic states and quantum transitions

Motivated by the interest to testable, exactly solvable models of quantum behavior in the time-dependent potentials, which may be important in studying (and in proof) basic quantum mechanical laws, as well as in considering the possibility of using the electronic components suggested to hypothetical ``quantum computers,'' we develop a method of solving the Schr\"odinger equation in a certain class of time-periodic and space-dependent one-dimensional potentials. In particular, it is shown that the quasistationary electronic state in the one-dimensional cyclic mesoscopic metallic ring in a rotating-potential field displays periodic variation of quasienergy in the function of magnetic flux threading the ring (the Aharonov-Bohm effect) and oscillation, superposed on the monotonous dependence, in the function of angular velocity of rotating potential (the effect similar to Rabi and/or Bloch oscillation). At large speed of rotation, quasienergy decreases rather than increases with the increase of angular velocity. The dependence of quasienergy on flux in space periodic potential displays standard $hc∕e$ periodicity as well as the periodicity with a larger period, $Nhc∕e$, where $N$ is the number of sites in the loop, corresponding to one flux quantum per lattice site. This is an interference effect similar to one observed in the fractional quantum Hall effect but, unlike in the latter, not requiring the concept of ``fractional electron charge,'' $e∕N$. The physical significance of the quasienergy states is clarified by studying the quantum transitions between the states as well as by investigating the energy flow between the ring and the external source of the potential.

Boundary-value problems for linear PDEs with variable coefficients

A new method is introduced for studying boundary-value problems for a class of linear partial differential equations (PDEs) with variable coefficients. This method is based on ideas recently introduced by the author for the study of boundary-value problems for PDEs with constant coefficients. As illustrative examples the following boundary-value problems are solved: a Dirichlet and a Neumann problem on the half line for the time-dependent Schrodinger equation with a space-dependent potential; and a Poincare problem on the quarter plane for a variable coefficient generalization of the Laplace equation.

A Note on the Symplectic Integration of the Nonlinear Schrödinger Equation

Numerically solving the nonlinear schroedinger equation and being able to treat arbitrary space dependent potentials permits many application in the realm of quantum mechanics. The long-term stability of a numerical method and its conservation properties is an important feature since it assures that the underlying physics of the solution are respected and it ensures that the numerical result is correct also for small time spans. In this paper we describe symplectic integrators for the nonlinear schroedinger equation with arbitrary potentials and perform numerical experiments comparing different approaches and highlighting their respective advantages and disadvantages.

Unique continuation for the linearized Benjamin-Bona-Mahony equation with space-dependent potential

In this paper, we analyze a unique continuation problem for the linearized Benjamin-Bona-Mahony equation with space-dependent potential in a bounded interval with Dirichlet boundary conditions. The underlying Cauchy problem is a characteristic one. We prove two unique continuation results by means of spectral analysis and the (generalized) eigenvector expansion of the solution, instead of the usual Holmgren-type method or Carleman-type estimates. It is found that the unique continuation property depends very strongly on the nature of the potential and, in particular, on its zero set, and not only on its boundedness or integrability properties.