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This paper deals with the approximate design of quantum unitary gates using perturbed harmonic oscillator dynamics. The harmonic oscillator dynamics is perturbed by a small time-varying electric field which leads to time-dependent Schrödinger equation. The corresponding unitary evolution after time T is obtained by approximately solving the time-dependent Schrödinger equation. The aim of this work is to minimize the discrepancy between a given unitary gate and the gate obtained by evolving the oscillator in the weak electric field over [0, T]. The proposed algorithm shows that the approximate design is able to realize the Hadamard gate and controlled unitary gate on three-qubit arrays with high accuracy. [ABSTRACT FROM AUTHOR]

The primary goal of this paper is to present a comprehensive study of the nonlinear Schrödinger equations with combined nonlinearities of the power-type and Hartree-type. Under certain structural conditions, the authors are able to provide a complete picture of how the nonlinear Schrödinger equations with combined nonlinearities interact in the given energy space. The method used in the paper is based upon the Morawetz estimates and perturbation principles. [ABSTRACT FROM AUTHOR]

An original approach to the solution of the nonlinear Schrodinger equation (NLSE) is pursued in this paper, following the regular perturbation (RP) method. Such an iterative method provides a closed-form approximation of the received field and is thus appealing for devising nonlinear equalization/compensation techniques for optical transmission systems operating in the nonlinear regime. It is shown that, when the nonlinearity is due to the Kerr effect alone, the order n RP solution coincides with the order 2n + 1 Volterra series solution proposed by Brandt-Pearce and co-workers. The RP method thus provides a computationally efficient way of evaluating the Volterra kernels, with a complexity comparable to that of the split-step Fourier method (SSFM). Numerical results on 10 Gb/s single-channel terrestrial transmission systems employing common dispersion maps show that the simplest third-order Volterra series solution is applicable only in the weakly nonlinear propagation regime, for peak transmitted power well below 5 dBm. However, the insight in the nonlinear propagation phenomenon provided by the RP method suggests an enhanced regular perturbation (ERP) method, which allows the first order ERP solution to be fairly accurate for terrestrial dispersion mapped systems up to launched peak powers of 10 dBm. [ABSTRACT FROM PUBLISHER]

This paper deals with the design of quantum unitary gate by matching the Hermitian generators. A given complicated quantum controlled gate is approximated by perturbing a simple quantum system with a small time-varying potential. The basic idea is to evaluate the generator $$H_\varphi $$ of the perturbed system approximately using first-order perturbation theory in the interaction picture. $$H_\varphi $$ depends on a modulating signal $$\varphi (t){:}\; 0\le t\le T$$ which modulates a known potential V. The generator $$H_\varphi $$ of the given gate $$U_\mathrm{g}$$ is evaluated using $$H_\mathrm{g}=\iota \log U_g$$ . The optimal modulating signal $$\varphi (t)$$ is chosen so that $$\Vert H_g - H_\varphi \Vert $$ is a minimum. The simple quantum system chosen for our simulation is harmonic oscillator with charge perturbed by an electric field that is a constant in space but time varying and is controlled externally. This is used to approximate the controlled unitary gate obtained by perturbing the oscillator with an anharmonic term proportional to $$q^3$$ . Simulations results show significantly small noise-to-signal ratio. Finally, we discuss how the proposed method is particularly suitable for designing some commonly used unitary gates. Another example was chosen to illustrate this method of gate design is the ion-trap model. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Schrödinger–Poisson system with perturbation: { - Δ ⁢ u + u + λ ⁢ ϕ ⁢ (x) ⁢ u = | u | p - 2 ⁢ u + g ⁢ (x) , x ∈ ℝ 3 , - Δ ⁢ ϕ = u 2 , x ∈ ℝ 3 , \left\{\begin{aligned} \displaystyle-\Delta u+u+\lambda\phi(x)u&\displaystyle=% |u|^{p-2}u+g(x),&&\displaystyle x\in\mathbb{R}^{3},\\ \displaystyle-\Delta\phi&\displaystyle=u^{2},&&\displaystyle x\in\mathbb{R}^{3% },\end{aligned}\right. where λ > 0 {\lambda>0} , p ∈ (3 , 6) {p\in(3,6)} and the radial general perturbation term g ⁢ (x) ∈ L p p - 1 ⁢ (ℝ 3) {g(x)\in L^{\frac{p}{p-1}}(\mathbb{R}^{3})}. By establishing a new abstract perturbation theorem based on the Bolle's method, we prove the existence of infinitely many radial solutions of the above system. Moreover, we give the asymptotic behaviors of these solutions as λ → 0 {\lambda\to 0}. Our results partially solve the open problem addressed in [Y. Jiang, Z. Wang and H.-S. Zhou, Multiple solutions for a nonhomogeneous Schrödinger–Maxwell system in ℝ 3 \mathbb{R}^{3} , Nonlinear Anal. 83 2013, 50–57] on the existence of infinitely many solutions of the Schrödinger–Poisson system for p ∈ (2 , 4 ] {p\in(2,4]} and a general perturbation term g. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fractional Schrödinger equation with steep potential well and sublinear perturbation. By virtue of variational methods, the existence criteria of infinitely many nontrivial high or small energy solutions are established. In addition, the phenomenon of the concentration of solutions is also explored. We also give some examples to demonstrate the main results. [ABSTRACT FROM AUTHOR]

We study functional and spectral properties of perturbations of the operator − ( ∂ s + i a ) 2 in L 2 ( S 1 ). This operator appears when considering the restriction to the unit circle of a two-dimensional Schrödinger operator with the Bohm-Aharonov vector potential. We prove a Hardy-type inequality on R 2 and, on S 1 , a sharp interpolation inequality and a sharp Keller-Lieb-Thirring inequality. [ABSTRACT FROM AUTHOR]

We propose an exponential wave integrator sine pseudospectral (EWI-SP) method for the nonlinear Schrödinger equation (NLS) with wave operator (NLSW), and carry out rigorous error analysis. The NLSW is NLS perturbed by the wave operator with strength described by a dimensionless parameter ε ∈ (0, 1]. As ε → 0+, the NLSW converges to the NLS and for the small perturbation, i.e., 0 < ε ≪ 1, the solution of the NLSW differs from that of the NLS with a function oscillating in time with O(ε²) wavelength at O(ε²) and O(ε4) amplitudes for ill-prepared and well-prepared initial data, respectively. This rapid oscillation in time brings significant difficulties in designing and analyzing numerical methods with error bounds uniformly in ε. In this work, we show that the proposed EWI-SP possesses the optimal uniform error bounds at O(τ²) and O(τ) in τ (time step) for well-prepared initial data and ill-prepared initial data, respectively, and spectral accuracy in h (mesh size) for both cases, in the L² and semi-H¹ norms. This result significantly improves the error bounds of the finite difference methods for the NLSW. Our approach involves a careful study of the error propagation, cut-off of the nonlinearity, and the energy method. Numerical examples are provided to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

This work is motivated by the desire to develop a method that allows for easy and accurate calculation of complex resonances of a one-dimensional Schröodinger's equation whose potential is a low-energy well surrounded by a thick barrier. The resonance is calculated as a perturbation of the bound state associated with a barrier of infinite thickness. We show that the corrector to the bound-state energy is exponentially small in the barrier thickness. A simple computational strategy that exploits this smallness is devised and numerically verified to be very accurate. We also provide a study of high-frequency resonances and show how they can be approximated. Numerical examples are given to illustrate the main ideas in this work. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a congruence transformation for studying the numerical stability condition of the nonlinear Shrödinger equation , subject to homogeneous Neumann boundary conditions, where and are real parameters. The congruence transformation takes advantage of a special structure of the matrices that result from Galerkin’s discretization with piecewise linear functions and product approximation for the nonlinear term by decoupling the system of perturbation into independent subsystems. This transformation can also be applied to solving any linear system if is of the same special structure. [Copyright &y& Elsevier]

An example of Schrödinger and Klein-Gordon equations with fast oscillating coefficients is used to show that they can be averaged by an adiabatic approximation based on V. P. Maslov's operator method. [ABSTRACT FROM AUTHOR]

A direct perturbation method for approximating dark soliton solutions of the nonlinear Schrödinger (NLS) equation under the influence of perturbations is presented. The problem is broken into an inner region, where the core of the soliton resides, and an outer region, which evolves independently of the soliton. It is shown that a shelf develops around the soliton that propagates with speed determined by the background intensity. Integral relations obtained from the conservation laws of the NLS equation are used to determine the properties of the shelf. The analysis is developed for both constant and slowly evolving backgrounds. A number of problems are investigated, including linear and nonlinear damping type perturbations. [ABSTRACT FROM AUTHOR]

We study the limit of the three-dimensional Schrödinger-Poisson system with a singular perturbation, to model a quantum electron gas that is strongly confined near an axis. For well-prepared data, which are polarized on the ground space of the transversal Hamiltonian, the resulting model is the cubic defocusing nonlinear Schrödinger equation. Our main tool is a refined analysis of the Poisson kernel when acting on strongly confined densities. In that direction, an appropriate scaling of the initial data is required, to avoid divergent integrals when the gas concentrates on the axis. [ABSTRACT FROM AUTHOR]

A new technique is presented which gives conditions under which perturbations of certain base potentials are uniquely determined from the location of eigenvalues and resonances in the context of a Schrödinger operator on a half line. The method extends to complex-valued potentials and certain potentials whose first moment is not integrable.[Research supported in part by the US National Science Foundation under grants DMS-9970299 and DMS-0107492 and by the UK EPSRC.] [ABSTRACT FROM AUTHOR]