One-dimensional Schrodinger Equation Research Articles

Poincaré-Dulac Normal Form Reduction for Unconditional Well-Posedness of the Periodic Cubic NLS

We implement an infinite iteration scheme of Poincare-Dulac normal form reductions to establish an energy estimate on the one-dimensional cubic nonlinear Schrodinger equation (NLS) in C_t L^2(T), without using any auxiliary function space. This allows us to construct weak solutions of NLS in C_t L^2(T)$ with initial data in L^2(T) as limits of classical solutions. As a consequence of our construction, we also prove unconditional well-posedness of NLS in H^s(T) for s \geq 1/6.

Class of invariants for a time dependent linear potential

General solution of the one-dimensional Schrodinger equation in presence of a time-dependent linear potential is reconsidered in the context of Lewis–Riesenfeld and unitary transformation approaches. Three invariant operators are constructed as limiting cases of a general Hermitian quadratic invariant and their instantaneous eigenfunctions are obtained. Then the corresponding solutions of Schrodinger equation for each invariant operator are derived. These solutions include all known solutions of the system. Furthermore, it is shown how different solutions can be related to each other.

Reconstructing the potential for the one-dimensional Schrodinger equation from boundary measurements

We consider the inverse problem of determining the potential in the dynamical Schrödinger equation on the interval by the measurement on the boundary. We use the boundary control method to recover the spectrum of the problem from the observation at either left or right endpoints. Using the specificity of the one-dimensional situation, we recover the spectral function, reducing the problem to the classical one which could be treated by known methods. We apply the algorithm to the situation when only a finite number of eigenvalues are known and prove the convergence of the method.

Some consequences of the generalized uncertainty principle induced ultraviolet wave-vector cutoff in one-dimensional quantum mechanics

A projection method is proposed to treat the one-dimensional Schrodinger equation for a single particle when the Generalized Uncertainty Principle (GUP) generates an ultraviolet (UV) wavevector cutoff. The existence of a unique coordinate representation called the naive one is derived from the one-parameter family of discrete coordinate representations. In this bandlimited Quantum Mechanics a continuous potential is reconstructed from discrete sampled values observed by means of a particle in maximally localized states. It is shown that bandlimitation modifies the speed of the center and the spreading time of a Gaussian wavepacket moving in free space. Indication is found that GUP accompanied by bandlimitation may cause departures of the low-lying energy levels of a particle in a box from those in ordinary Quantum Mechanics much less suppressed than commonly thought when GUP without bandlimitation is in work.

Resonances of a Potential Well with a Thick Barrier

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 Schrodinger'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.

Analysis of isolated attosecond pulse generation from a multi-cycle two-color pulse

We theoretically investigate the high-order harmonic and attosecond pulse generation by numerically solving the one-dimensional time-dependent Schrodinger equation for a hydrogen atom in a two-color laser field that is synthesized by adding a suitable multi-cycle infrared pulse to a multicycle 800-nm fundamental pulse. Our result clearly shows that with the addition of a suitable second optical field, the electric field profile of the synthesized pulse presents multi-segments and that in the first and the last segments, the amplitudes of the electric field are dramatically suppressed. If this synthesized pulse is adopted as an incident driving pulse, only the amplitude of the electric field in the middle segment makes a major contribution to the plateau and to the cutoff region of the harmonic spectrum. This will cause a wide quasicontinuous spectrum around the cutoff position. Thus, an isolated attosecond pulse can be obtained even if we adopt the synthesized pulse with a multi-cycle pulse duration.

A remark on normal forms and the “upside-down” I-method for periodic NLS: Growth of higher Sobolev norms

We study growth of higher Sobolev norms of solutions of the onedimensional periodic nonlinear Schrodinger equation (NLS). By a combination of the normal form reduction and the upside-down I-method, we establish $${\left\| {u(t)} \right\|_{{H^s}}} \le {(1 + \left| t \right|)^{a(s - 1) + }}$$ with α = 1 for a general power nonlinearity. In the quintic case, we obtain the above estimate with α = 1/2 via the space-time estimate due to Bourgain [4, 5]. In the cubic case, we compute concretely the terms arising in the first few steps of the normal form reduction and prove the above estimate with α = 4/9. These results improve the previously known results (except for the quintic case). In the Appendix, we also show how Bourgain’s idea in [4] on the normal form reduction for the quintic nonlinearity can be applied to other powers.

Regularity of the Rotation Number for the One-Dimensional Time-Continuous Schrödinger Equation

Starting from results already obtained for quasi-periodic co-cycles in \(SL(2, \mathbb R),\) we show that the rotation number of the one-dimensional time-continuous Schrodinger equation with Diophantine frequencies and a small analytic potential has the behavior of a \(\frac{1}{2}-\)Holder function. We give also a sub-exponential estimate of the length of the gaps which depends on its label given by the gap-labeling theorem.

Eigenvalues in spectral gaps of differential operators

Spectral problems with band-gap spectral structure arise in numerous applications, including the study of crystalline structure and the determination of transmitted frequencies in photonic waveguides. Numerical discretization of these problems can result in spurious results, a phenomenon known as spectral pollution . We present a method for calculating eigenvalues in the gaps of self-adjoint operators which avoids spectral pollution. The method perturbs the problem into a dissipative problem in which the eigenvalues to be calculated are lifted out of the convex hull of the essential spectrum, away from the spectral pollution. The method is analysed here in the context of one-dimensional Schrodinger equations on the half line, but is applicable in a much wider variety of contexts, including PDEs, block operator matrices, multiplication operators, and others.

On the Quantization of One-Dimensional Nonstationary Coulomb Potential System

Exact solutions of the one-dimensional Schrodinger equation with a time-dependent Coulomb potential [- z ( t )/| x |] are investigated using the invariant method (Lewis and Riesenfeld theorem) together with unitary transformation approach. The eigenfunctions and the corresponding eigenvalues of the system are obtained analytically. When the time dependence of all coefficients vanishes, our results exactly reduce to those known for stationary case.

Nonresonant transparency channels of a two-barrier nanosystem in an electromagnetic field with an arbitrary strength

The theory of the quasistationary spectrum of the system of electrons interacting with a high-frequency electromagnetic field with an arbitrary strength in a two-barrier resonant tunneling nanostructure has been proposed on the basis of the exact solution of the total one-dimensional Schrodinger equation. It has been shown that nonresonant transparency channels in the planar two-barrier resonant tunneling structure appear owing to the superposition of a certain quasistationary state with first field resonance harmonics of both nearest states; this superposition is due to the interaction of electrons with the electromagnetic field.

Asymmetric double quantum wells with smoothed interfaces

Abstract We have derived and analyzed the wavefunctions and energy states for an asymmetric double quantum well (ADQW), broadened due to interdiffusion or other static interface disorder effects, within a known discreet variable representative approach for solving the one-dimensional Schrodinger equation. The main advantage of this approach is that it yields the energy eigenvalues, and the eigenvectors, in semiconductor nanostructures of different shapes as well as the strengths of the optical transitions between them. The behaviour of ADQW states for the different mutual widths of coupled wells, for the different degree of broadening, and under increasing external electric field is investigated. We have found that interface broadening effects change and shift energy levels, not monotonously, but the resonant conditions near an energy of sub-band coupling regions do not strongly distort. Also, it is shown that an external electric field may help to achieve resonant conditions for inter-sub-band inverse population by intrawell emission of LO-phonons in diffuse ADQW.

Analyzing trap generation in silicon-nanocrystal memory devices using capacitance and current measurement

The combination of capacitance- and current-voltage (CV/IV) measurements is used to analyze trap generation in silicon-nanocrystal memory devices during Fowler-Nordheim (FN) programming/erasing cycling. CV and IV curves are measured after certain P/E cycles. The flatband voltage (Vfb) and the threshold voltage (Vth) are extracted from CV curves by solving one-dimensional Schrodinger and Poisson equations. Both hole and electron trappings are observed in the tunneling SiO2. They show up in the accumulation and the inversion, respectively. By fitting FN tunneling current, the area densities of cycling-induced electron traps in the blocking oxide and in the tunneling oxide are finally determined.

Gribov pendulum in the Coulomb gauge on curved spaces

In this paper the generalization of the Gribov pendulum equation in the Coulomb gauge for curved spacetimes is analyzed on static spherically symmetric backgrounds. A rigorous argument for the existence and uniqueness of solution is provided in the asymptotically AdS case. The analysis of the strong and weak boundary conditions is equivalent to analyzing an effective one-dimensional Schrodinger equation. Necessary conditions in order for spherically symmetric backgrounds to admit solutions of the Gribov pendulum equation representing copies of the vacuum satisfying the strong boundary conditions are given. It is shown that asymptotically flat backgrounds do not support solutions of the Gribov pendulum equation of this type, while on asymptotically AdS backgrounds such ambiguities can appear. Some physical consequences are discussed.

Electrochemical capacitance-voltage profiling of the free-carrier concentration in HEMT heterostructures based on InGaAs/AlGaAs/GaAs compounds

The depth distribution of free carriers over the HEMT structures with quantum-well layers is studied by electrochemical capacitance-voltage profiling. It is shown that the actual distribution of the concentration of free carriers and their energy spectrum in the HEMT structure channel can be obtained by numerical simulation of the results of profiling based on the self-consistent solution of one-dimensional Schrodinger and Poisson equations.

The Fourier grid Hamiltonian method for calculating vibrational energy levels of triatomic molecules

The Fourier grid Hamiltonian method for calculating vibrational energy levels of triatomic molecules is presented. We have used direct product basis functions obtained from the Fourier grid Hamiltonian method for one-dimensional Schrodinger equation for radial coordinates and the associated Legendre polynomial for angular coordinate. The resulting matrix is diagonalized by using Lanczos method. The method is tested for the calculation of vibrational energy levels of H2O and MgH2 molecules. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Compression of Bright Bound Solitons in the Bose-Einstein Condensates with Exponentially Time-Dependent Atomic Scattering Length by the Feshbach Resonance

We investigate compression of the bright bound solitons in the Bose-Einstein condensates (BECs) by exponentially increasing the absolute value of the atomic scattering length. Similarity transformation and Hirota bilinear method are used to symbolically solve the one-dimensional nonlinear Schrodinger equation with the time-dependent coefficients. We present types of the bright bound solitons in compression through manipulating their initial coherence. Results show that the improved quantity of the atomic density peaks can be observed before the collapse of the solitons when their coherence is increased. Furthermore, we discuss how those compressed bound solitons are influenced by the adjacent solitons. The bound structures in our study are illustrated to exist with the parameters within the current experimental capacity (the spatial and temporal ranges of the bound solitons are less than 56 μm and 50 ms in our investigation), which suggests a future observation in the BECs experiments.

一维非线性Schrödinger 方程的两个无条件收敛的守恒紧致差分格式

Two compact difference schemes for one-dimensional cubic nonlinear Schrodinger equation are pro- posed. Both of the schemes are proved to conserve the discrete mass and energy. Unconditional convergence in maximum norm of the difference solutions with forth order in space and second order in time is also proved by utilizing the discrete energy method and two techniques. For computing the nonlinear algebraical system gener- ated by the nonlinear compact scheme, an iterative algorithm is constructed and proved to be convergent. As a byproduct of the iterative algorithm, a linearized compact difference scheme with high accuracy is obtained. Nu- merical examples are given to support the theoretical analysis, and two Richardson extrapolations are introduced to achieve higher order numerical accuracy.

Complex WKB method for adiabatic perturbations of a periodic Schrödinger operator

This review is devoted to an analog of the complex WKB method for studying solutions of a one-dimensional periodic Schrodinger equation with an adiabatic perturbation. We illustrate with examples how to use the method. Bibliography: 30 titles.

On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations

We consider one-dimensional difference Schrodinger equations [H(x,ω)ϕ] (n) ≡ ―ϕ(n ―1) ― ϕ(n + 1) + V(x + nω)ϕ(n) = Eϕ(n), n ∈ Z, x, ω ∈ [0,1] with real-analytic potential function V (x). If L(E, w o ) is greater than 0 for all E ∈ (E', E) and some Diophantine ω 0 , then the integrated density of states is absolutely continuous for almost every ω close to ω 0 , as shown by the authors in earlier work. In this paper we establish the formation of a dense set of gaps in spec(H(x, w)) ∩ (E',E). Our approach is based on an induction on scales argument, and is therefore both constructive as well as quantitative. Resonances between eigenfunctions of one scale lead to at a larger scale. To pass to actual gaps in the spectrum, we show that these pre-gaps cannot be filled more than a finite (and uniformly bounded) number of times. To accomplish this, one relates a pre-gap to pairs of complex zeros of the Dirichlet determinants off the unit circle. Amongst other things, we establish a nonperturbative version of the co-variant parametrization of the eigenvalues and eigenfunctions via the phases in the spirit of Sinai's (perturbative) description of the spectrum via his function A. This allows us to relate the gaps in the spectrum with the graphs of the eigenvalues parametrized by the phase. Our infinite volume theorems hold for all Diophantine frequencies ω up to a set of Hausdorff dimension zero.