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We study its controllability around the ground state when the linearized system is not controllable and wonder whether the quadratic term can help to recover the directions lost at the first order. More precisely, in this paper, we formulate assumptions under which the quadratic term induces a drift which prevents the small-time local controllability (STLC) of the system in appropriate spaces. For finite-dimensional systems, quadratic terms induce coercive drifts in the dynamic, quantified by integer negative Sobolev norms, along explicit Lie brackets which prevent STLC.In the context of the bilinear Schr{\"o}dinger equation, the first drift, quantified by the $H^{-1}$-norm of the control, was already observed and used to deny STLC with controls small in $L^\infty$. In this article, we improve this result by denying STLC with controls small in $W^{-1, \infty}$. Furthermore, for any positive integer $n$, we formulate assumptions under which one may observe a quadratic drift quantified by the $H^{-n}$-norm of the control and we use it to deny STLC in suitable spaces.

An algorithm for solving exciton Schrodinger equation in an arbitrary 2D multiple quantum well structure under applied electric field is proposed, allowing one to model both the absorption and refraction spectra consistently and efficiently. It gives much better quantitative results than variational approximation, while being faster than more advanced methods. The algorithm is implemented using R language and tested by comparing to other published results. Possible improvements are discussed.

In this work, an efficient novel technique, namely, the q -homotopy analysis transform method ( q -HATM) is applied to obtain analytical solutions for a system of time-fractional coupled nonlinear Schrodinger (TF-CNLS) equations with the time-fractional derivative taken in the Caputo sense. This system of equations incorporate nonlocality behaviours which cannot be modeled under the framework of classical calculus. With numerous important applications in nonlinear optics, it describes interactions between waves of different frequencies or the same frequency but belonging to different polarizations. We first establish existence and uniqueness of solutions for the considered time-fractional problem via a fixed point argument. To demonstrate the effectiveness and efficiency of the q − HATM, two cases each of two time-fractional problems are considered. One important feature of the q − HATM is that it provides reliable algorithms which can be used to generate easily computable solutions for the considered problems in the form of rapidly convergent series. Numerical simulation are provided to capture the behaviour of the state variables for distinct values of the fractional order parameter. The results demonstrate that the general response expression obtained by the q − HATM contains the fractional order parameter which can be varied to obtain other responses. Particularly, as this parameter approaches unity, the responses obtained for the considered fractional equations approaches that of the corresponding classical equations.

Schrodinger equation is an indispensable model for quantum mechanics, used for modelling several fascinating complex nonlinear physical systems, such as quantum condensates, nonlinear optics, hydrodynamics, shallow-water waves, and the harmonic oscillator. The objective of this paper is to investigate and study the exact travelling wave solutions of nonlinear triple fractional Schrodinger equations involving a modified Riemann–Liouville fractional derivative. Using the Riccati-Bernoulli Sub-ODE technique, the Backlund transformation is employed to handle the posed system. The traveling wave solutions methodology lies in converting the fractional Schrodinger equations into a nonlinear system of fractional ODEs. An infinite sequence of solutions to the fractional partial differential equations can be obtained directly through solving the resulting nonlinear fractional system. Some graphical representations of the obtained solutions after selecting suitable values for fractional values and parameters are illustrated to test accuracy and verify the power, and effectiveness of the proposed method.

International audience; The paper presents a dynamic event-triggering mechanism for the constant and the localized damped linear Schrödinger equation. The following results are tackled: the existence of solution to the closed-loop event-triggered control system; the avoidance of the Zeno behavior due to the absence of any accumulation point of the sequence of time instants and the exponential stability based on energy estimate through the observability inequality. A simulation example based on the one-dimensional Schrödinger equation is presented to validate the theoretical results.

The research exploration will comprise of investigating the hyperpolarizability, frontier molecular orbital analysis and NBO, the data analyzed through theoretical methods using Density Functional Theories (DFT) involves the basic principle of solving the Schrodinger equation for many body systems. The results of the organic amine 2–4-difluoroaniline is analyzed and discussed using higher basis set 6-311++G(d, 2p). The energy gap between the HOMO and the LUMO of 2, 4-difluoroaniline is found to be 5.2186 eV. The chemical hardness of the 2, 4-difluoroaniline molecule is found to be 2.6093, indicating that the chemical hardness is very high and hence, the compound is chemically stable. The electrophilicity index of the title molecule using the B3LYP method is 2.3371 for 2, 4-difluoroaniline molecule. Thus the chosen molecule is a potential candidate for NLO applications.

In this paper we first establish a new critical point theorem which may produce a relationship between the sign-changing critical point, the Morse index, the upper bound of the critical value, the value of the type inf k ≤ dim ⁡ Y ∞ Y ⊂ E , ⁡ sup Y ⁡ G and the number of nodal domains. An unified approach is introduced for these composite properties. As one application of the abstract theorem, we study the existence of infinitely many sign-changing bound states for the following nonlinear stationary Schrodinger equation: − Δ u + a ( x ) u = | u | p − 2 u , u ∈ H 1 ( R N ) , where N ≥ 2 , p > 2 and p 2 N / ( N − 2 ) : = 2 ⁎ when N > 2 ; the potential a ( x ) ∈ C ( R N , R ) is a nonnegative function verifying suitable decay assumptions, but being not symmetric or periodic. We obtain the information on the precise estimates of the energies, the Morse indices and the number of nodal domains for these sign-changing bound states. We believe that the abstract theorem in this article will have more applications.

We present a definition of stochastic Hamiltonian process on finite graph via its corresponding density dynamics in Wasserstein manifold. We demonstrate the existence of stochastic Hamiltonian process in many classical discrete problems, such as the optimal transport problem, Schrodinger equation and Schrodinger bridge problem (SBP). The stationary and periodic properties of Hamiltonian processes are also investigated in the framework of SBP.

This paper introduces the energy-preserving scheme for the nonlinear fractional Klein–Gordon Schrodinger equation, which uses the scalar auxiliary variable approach. By a scalar variable, the system is transformed into a new equivalent system. Then applying the extrapolated Crank–Nicolson method on the temporal direction and Fourier pseudospectral method on space direction, we give a linear implicit energy-preserving scheme. Moreover, it proved that at each discrete time the scheme preserves the corresponding discrete mass and energy. The unique solvability and convergence of the numerical solution are also investigated. In particular, it shows the method has the second-order accuracy in time and the spectral accuracy in space. Finally, it gives the algorithm implementation. Several numerical examples illustrate the efficiency and accuracy of the numerical scheme.

Time-decaying harmonic oscillators yield dispersive estimates with weak decay, and change the threshold power of the nonlinearity between the short and the long range. In the non-critical case for the time-decaying harmonic oscillator, this threshold can be characterized by polynomial nonlinearities. However, in the critical case, it is difficult to characterize the threshold using only polynomial terms, and thus we use logarithmic nonlinear terms.

An efficient numerical method is proposed for the solution of the nonlinear cubic Schrodinger equation. The proposed method is based on the Frechet derivative and the meshless method with radial basis functions. An important characteristic of the method is that it can be extended from one-dimensional problems to multi-dimensional ones easily. By using the Frechet derivative and Newton–Raphson technique, the nonlinear equation is converted into a set of linear algebraic equations which are solved iteratively. Numerical examples reveal that the proposed method is efficient and reliable with respect to the accuracy and stability.

We consider the Schrodinger equations with arbitrary (large) power non-linearity on the three-dimensional torus. We construct non-trivial probability measures supported on Sobolev spaces and show that the equations are globally well-posed on the supports of these measures, respectively. Moreover, these measures are invariant under the flows that are constructed. Therefore, the constructed solutions are recurrent in time. Also, we show slow growth control on the time evolution of the solutions. A generalization to any dimension is given. Our proof relies on a new approach combining the fluctuation-dissipation method and some features of the Gibbs measures theory for Hamiltonian PDEs. The strategy of the paper applies to other contexts.

In this work, we consider the numerical computation for the two-dimensional generalized nonlinear Schrodinger equations with wave operator. Based on the scalar auxiliary variable (SAV) approach, the original problem is transformed into an equivalent one, which corresponds to the energy-conservation laws. We present an energy-conserving and linearly implicit three-level scheme for the equivalent system. The energy-conserving property, boundedness of the numerical solution and convergence analysis in the discrete maximum norm are derived, which has not restricted to specific forms of cubic nonlinear term f and not needed the sharply restriction on mesh size. Finally, numerical experiments on several models confirm our theoretical results.

Exact analytical expression for the Hulthen off-shell Jost solution is constructed by using the particular integral of the inhomogeneous Schrodinger equation. The Jost solution is exploited to calculate the off-shell physical solution and in turn the off-shell T-matrix. The off-shell T-matrices are computed for the (n-p) and (n-d) systems and are found in order with respect to its half-shell behavior.

In this paper, we consider the well-posedness of the inhomogeneous nonlinear biharmonic Schrodinger equation with spatial inhomogeneity coefficient K ( x ) behaves like | x | − b for 0 b min ⁡ { N 2 , 4 } . We show the local well-posedness in the whole H s -subcritical case, with 0 s ≤ 2 . The difficulties of this problem come from the singularity of K ( x ) and the lack of differentiability of the nonlinear term. To resolve this, we derive the bilinear Strichartz's type estimates for the nonlinear biharmonic Schrodinger equations in Besov spaces.

We derive the expressions for the collision-induced amplitude dynamics of two flat-top solitons in spatial dimensions of 1, 2, and 3 caused by the generic weak nonlinear loss under the framework of coupled cubic-quintic ( n + 1 )D nonlinear Schrodinger equations for n = 1 , 2 , 3 . We develop a new perturbative technique which is mainly based on the calculations for the fast collision-induced changes in the soliton envelope and the use of the single perturbed soliton solution for the calculations. These results quantify the energy dropdown due to a fast collision of two pulses ( n = 1 ) , or two optical beams ( n = 2 ), or two light bullets ( n = 3 ) in cubic-quintic nonlinear media with dissipation. The theoretical calculations are then confirmed by numerical simulations with the corresponding coupled nonlinear Schrodinger equations. Our approach can be used for studying the effects of dissipation on colliding solitons described by the coupled nonlinear Schrodinger models, where the unperturbed equation is nonintegrable.

We adopt an operatorial method, based on truncated bosons, to describe the dynamics of populations in a closed region with a non trivial topology. The main operator that includes the various mechanisms and interactions between the populations is the Hamiltonian, constructed with the density and transport operators. The whole evolution is derived from the Schrodinger equation, and the densities of the populations are retrieved from the normalized expected values of the density operators. We show that this approach is suitable for applications in very large domain, solving the computational issues that typically occur when using an Hamiltonian based on fermionic ladder operators.

In this paper, we make a comprehensive study for the orbital stability of standing waves for the fractional Schrodinger equation with combined power-type nonlinearities (FNLS) i ∂ t ψ − ( − Δ ) s ψ + a | ψ | p 1 ψ + | ψ | p 2 ψ = 0 . We prove that when p 2 = 4 s N and a ( p 1 − 4 s N ) 0 , there exist the standing waves of (FNLS), which are orbitally stable. When a = 0 and 4 s N p 2 4 s N − 2 s , we present a new, simpler method to study the strong instability of standing waves. When a = − 1 , 0 p 1 p 2 and 4 s N ≤ p 2 4 s N − 2 s , or a = 1 and 4 s N ≤ p 1 p 2 4 s N − 2 s , or a = 1 , 0 p 1 4 s N p 2 4 s N − 2 s and ∂ λ 2 S ω ( u ω λ ) | λ = 1 ≤ 0 , we deduce that the ground state standing waves of (FNLS) are strongly unstable by blow-up.

In terms of optimization algorithm improvement, researchers utilize some strategies to establish the right balance between local and global search phases to escape from a local minimum in some NP-hard problems. The approach of quantum-behaved particles, as a novel strategy, uses uncertainty law and a distinct formulation obtained from solving the time-independent Schrodinger differential equation in the delta-potential-well function to update the solution candidates' positions. New formulation defines the local attractors as probable solutions between the best solution and the others to explore all solution space domain. Also, mentioned formulation employs the difference between the average solution and other ones as a new step size. Both local attractors and new step sizes guarantee diversification besides the inherent intensification. In the present paper, the teacher phase of the Teaching-Learning-Based Optimization algorithm (TLBO), which has been introduced recently, is improved using the formulation obtained from solving the time-independent Schrodinger equation to predict the probable positions of optimal solutions. The results show that QTLBO, an acronym for the Quantum Teaching- Learning- Based Optimization, improves the stability and robustness of the TLBO by defining the quantum teacher phase. The two planner rigid frames and two trusses with strength and serviceability constraints are chosen to verify the quality and performance of QTLBO. Comparing the results obtained from the proposed algorithm with those of the standard version of the TLBO algorithm and other literature methods shows that QTLBO increases the chance of finding a better solution besides improving the statistical criteria compared to the current TLBO.

We study the uniform weighted resolvent estimates of Schrodinger operator with scaling-critical electromagnetic potentials which, in particular, include the Aharonov-Bohm magnetic potential and inverse-square potential. The potentials are critical due to the scaling invariance of the model and the singularities of the potentials, which are not locally integrable. In contrast to the Laplacian −Δ on R 2 , we prove some new uniform weighted resolvent estimates for this 2D Schrodinger operator and, as applications, we show local smoothing estimates for the Schrodinger equation in this setting.

This experiment investigated the theory of energy loss in charged particles in plasma using a semi-classical method based on quantum scattering theory. The energy loss was expressed in terms of the transport cross-section for electron–ion scattering, which is calculated by semiclassical approximation of the scattering phase shifts. Exact numerical integration of the Schrodinger equation is used to test the derived results. This approach provides a non-perturbative representation of the energy loss, which bridges the gap between classical and quantum descriptions. On the other hand, the semi-classical method shows the correct behaviour of all plasma conditions and projectile charges and velocities. All equations have been programmed to present results using FORTRAN-90, and a program has been written for numerical calculations.

In this paper, we consider two kinds of nonlinear Schrodinger equations with the fractional Laplacian and Hardy potential ( λ | x | s , 0 λ ⩽ λ ∗ , λ ∗ is a constant of the Hardy-Sobolev inequality), which represent the generalized form of Hartree and Pekar-Choquard type time dependent fractional Hardy-Schrodinger equations. Applying the direct method of moving planes, we obtain the radial symmetry and monotonicity of the standing waves for the given equations.

In this paper, we study the Hamiltonian structure and develop a novel energy-preserving scheme for the two-dimensional fractional nonlinear Schrodinger equation. First, we present the variational derivative of the functional with fractional Laplacian to derive the Hamiltonian formula of the equation and obtain an equivalent system by defining a scalar variable. An energy-preserving scheme is then presented by applying exponential time differencing approximations for time integration and Fourier pseudo-spectral discretization in space. The proposed scheme is a linear system and can be solved efficiently. Numerical experiments are displayed to verify the conservation, efficiency, and good performance at a relatively large time step in long time computations.

We study the large-time behavior of the solutions to the Schrodinger equation associated with a non-selfadjoint operator having zero energy eigenvalue and real resonances. Our results extend those of Jensen and Kato in the three dimensional selfadjoint case. We consider a model of Schrodinguer operator with a quickly decaying complex-valued potential in dimension three. We assume that the latter has a finite number of real resonances. We are interested in the expansions of the resolvent in the low and intermediate energy parts. In particular, we discuss, under different conditions, the three situations of zero energy: zero is an eigenvalue or a resonance, or both.

The paper deals with the Caldirola–Kanai Hamiltonian in which the potential energy is defined as a linear function of the position variable. The exact solution of the Schrodinger equation using this Hamiltonian is presented. It is shown that this solution can be utilized in the theory of electric conduction of metals and semiconductors. The main idea of this paper is that if the Caldirola–Kanai Hamiltonian is applied to conduction electrons , it is possible to calculate successfully the drift velocity of the electrons in a constant electric field. The alternative calculation of the drift velocity in the framework of the density-matrix theory which is also presented leads to the same result. A useful lesson is drawn: if the friction coefficient , the parameter defining the Caldirola–Kanai Hamiltonian, concerns the conduction electrons, it is precisely equal to the reciprocal value of the relaxation time used in the kinetic theory of the electron transport in solids.

In this paper, we prove a reducibility result for a relativistic Schrodinger equation on torus with time quasi-periodic unbounded perturbations of order 1/2. As far as we known, this is the first reducibility result for the relativistic Schrodinger equation.

In [14] Duca and Nersesyan proved a small-time controllability property of nonlinear Schrödinger equations on a d-dimensional torus $\mathbb{T}^d$. In this paper we study a similar property, in the linear setting, starting from a closed Riemannian manifold. We then focus on the 2-dimensional sphere $S^2$, which models the bilinear control of a rotating linear top: as a corollary, we obtain the approximate controllability in arbitrarily small times among particular eigenfunctions of the Laplacian of $S^2$., 10 pages, 1 figure

Semi oblate and semi prolate are among the most probable self-organized nanostructures shapes. The optoelectronic properties of such nanostructures are not just manipulated with the height and lateral size but also with the wetting layer element. The practical interest of derivatives of germanium and silicon has a great important role in optoelectronic devices. This study is a contribution to the analysis of linear and nonlinear optical properties of Si0.7Ge0.3/Si. In the framework of the effective mass approximation, we solve numerically the Schrodinger equation relative to one particle confined in Si0.7Ge0.3/Si semi prolate and semi oblate quantum dots by using the finite element method and by taking into consideration the effect of the wetting layer. The energy spectrum of the lowest states and the dipolar matrix for the fourth allowed transitions are determined and discussed. We also calculate the detailed optical properties, including absorption coefficients, refractive index changes, second and third harmonic generation as a function of the quantum dot sizes. We found that with the change in the size of prolate and oblate quantum dots, there is a shift in the resonance peaks for the absorption coefficient and refractive index. It is due to the modification in the energy levels with changing size. The study proves a redshift in the second harmonic generation and third harmonic generation coefficients with an increase in the height/radius of the oblate/prolate quantum dot, respectively. We also demonstrated the variation of wavefunction inside the quantum dot with the change in wetting layer thickness.

In this paper, a novel system of Maxwell–Schrodinger equations with nonlocal effect in metamaterials is derived from the Drude model, hydrodynamical model and Schrodinger equation. A leap-frog finite element scheme, which can be solved one by one efficiently, is constructed by presenting a group of initial values. This scheme is proved to be stable conditionally in energy norm. It is confirmed that the error convergent rate is O ( τ 2 + h r ) by splitting the proof into three parts, where τ is the time step, h is the mesh size and r is the maximum total degree of polynomials in finite element spaces. Finally, some numerical results are given to verify the theories.

In recent years, predictive computational modeling has become a cornerstone for the study of fundamental electronic, optical, and thermal properties in complex forms of condensed matter, including Dirac and topological materials. The simulation of quantum transport in realistic materials calls for the development of linear scaling, or order-$N$, numerical methods, which then become enabling tools for guiding experimental research and for supporting the interpretation of measurements. In this review, we describe and compare different order-$N$ computational methods that have been developed during the past twenty years, and which have been used extensively to explore quantum transport phenomena in disordered media. We place particular focus on the zero-frequency electrical conductivities derived within the Kubo-Greenwood and Kubo-Streda formalisms, and illustrate the capabilities of these methods to tackle the quasi-ballistic, diffusive, and localization regimes of quantum transport in the noninteracting limit. The fundamental issue of computational cost versus accuracy of various proposed numerical schemes is addressed in depth. We then illustrate the usefulness of these methods with various examples of transport in disordered materials, such as polycrystalline and defected graphene models, 3D metals and Dirac semimetals, carbon nanotubes, and organic semiconductors. Finally, we extend the review to the study of spin dynamics and topological transport, for which efficient approaches for calculating charge, spin, and valley Hall conductivities are described., 54 pages, 31 figures, Invited Review of Physics Reports

We consider the stochastic nonlinear Schrodinger equations on the half-line with mixed boundary conditions containing Brownian noise. The main novelty of this work is a convenient framework for the analysis of such problems. Our approach allows us to establish the global existence and uniqueness of a solution to initial-boundary value problem with values in H 1 .

In this paper, we focus on the standing waves with prescribed mass for the Schrodinger equations with van der Waals type potentials, that is, two-body potentials with different width. This leads to the study of the following nonlocal elliptic equation − Δ u = λ u + μ ( | x | − α ⁎ | u | 2 ) u + ( | x | − β ⁎ | u | 2 ) u , x ∈ R N under the normalized constraint ∫ R N u 2 = c > 0 , where N ≥ 3 , μ > 0 , α, β ∈ ( 0 , N ) , and the frequency λ ∈ R is unknown and appears as Lagrange multiplier. Compared with the well studied case α = β , the solution set of the above problem with different width of two body potentials α ≠ β is much richer. Under different assumptions on c, α and β, we prove several existence, multiplicity and asymptotic behavior of solutions to the above problem. In addition, the stability of the corresponding standing waves for the related time-dependent problem is discussed.

In this paper the space-fractional Schrodinger equations with singular potentials are studied. Delta like or even higher-order singularities are allowed. By using the regularising techniques, we introduce a family of ‘weakened’ solutions, calling them very weak solutions. The existence, uniqueness and consistency results are proved in an appropriate sense. Numerical simulations are done, and a particles accumulating effect is observed in the singular cases. From the mathematical point of view a “splitting of the strong singularity” phenomena is also observed.

In this paper, we investigate the following Schrodinger equation { − Δ u − μ | x | 2 u = g ( u ) in R N ∖ { 0 } , u ∈ H 1 ( R N ) , where N ≥ 3 , μ ( N − 2 ) 2 4 , 1 | x | 2 is called the Hardy potential (the inverse-square potential) and g satisfies the Berestycki-Lions type condition. If 0 μ ( N − 2 ) 2 4 , combining variational methods with analytical skills, we show that the above problem has a positive and radial ground state solution. At the same time, our results suggest that this solution together with its derivatives up to order 2 have exponential decay at infinity while this solution has the possibility of blow-up at the origin. Furthermore, we construct a family of ground state solutions which converges to a ground state solution of the limiting problem as μ → 0 + . If μ 0 , we prove that the mountain pass level in H 1 ( R N ) can not be achieved. Provided further assumption that the above problem in the radial space H r 1 ( R N ) , we obtain the ground state solutions whose energy is strictly greater than the mountain pass level in H 1 ( R N ) . We also construct a family of solutions which converges to the ground state solution of the limiting problem as μ → 0 − .

Bifurcation analysis of ion-acoustic wave (IAWs) solutions of the nonlinear Schrodinger equation (NLSE) is explored for the first time in an electron-ion (e-i) magnetized solar wind plasma. The existence of ion-acoustic (IA) periodic, superperiodic, kink, antikink, compressive and rarefactive solitary wave solutions are revealed. Special values of Solar wind plasma parameters at a normalized distance from the Sun are considered for numerical simulation. The IA wave solutions are derived analytically. These solutions are analyzed numerically considering the influence of parameters, namely, wave number (k), velocity (V) of traveling wave and nonextensive parameter (q). Computational simulation reveals that only IA periodic wave grows in amplitude as waves moves from the Sun.

We study the fourth order Schr\"odinger operator $H=(-\Delta)^2+V$ for a short range potential in three space dimensions. We provide a full classification of zero energy resonances and study the dynamic effect of each on the $L^1\to L^\infty$ dispersive bounds. In all cases, we show that the natural $|t|^{-\frac34}$ decay rate may be attained, though for some resonances this requires subtracting off a finite rank term, which we construct and analyze. The classification of these resonances, as well as their dynamical consequences differ from the Schr\"odinger operator $-\Delta+V$., Comment: 35 pages, submitted. arXiv admin note: text overlap with arXiv:1810.03678

Simulation studies of some of the 3D structures in respect of energy states and shape dependent optical properties of ZnS quantum dot has been carried out in the present work. ‘Quantum Dot Lab’ based on Schrodinger Equations have been employed to simulate and explore physical dynamics of this low dimensional structure of different geometries. Cone shaped quantum dot has been found to be of relatively higher energy than the other shapes whereas cuboid shape exhibits lowest energy. Furthermore, the integrated absorption parameters of cuboid were obtained to be the highest. The integrated absorption was also investigated as a function of temperature – a behaviour distinctly different from the GaAs quantum dot. These properties of ZnS quantum dot could be effectively used for high performance optoelectronic applications and in computing architecture.

The efficiency of Quantum Dots (QDs) devices, is strongly related to their distribution, density and size, and shape in a QD array. The small size of the QD structure leads to a three-dimensional confinement of the electronic energies. The energy levels of electrons in QDs depends on their size – the smaller the quantum dot, the greater the amount of energy required to give electrons transitions to the next level. In this paper, we report the simulation results of pyramidal InAs/GaAs QDs coupled to Wetting Layer (WL). To investigate the effects of the InAs/GaAs QDs size on the electronic states and the transition energy. In the first part, we calculate the strain distribution to find the band edges, while the second one concerns the variation of the electronic transition energy as a function of quantum dots size by solving the stationary Schrodinger equation. This work shows clearly that the structural properties of QDs have strong effects on optoelectronic properties devices based on these nanostructures.

In this paper we consider the linear quasi-periodic system x ˙ = ( A + ϵ P ( t ) ) x , x ∈ R d , where A is a d × d constant matrix of elliptic type and has multiple eigenvalues, P ( t ) is analytic quasi-periodic with respect to t with basic frequencies ω = ( 1 , α ) , where α is irrational, and ϵ is a small perturbation parameter. Under suitable non-resonant condition, non-degeneracy condition and 0 ≤ β ( α ) r ⁎ , where β ( α ) = lim sup n → ∞ ln ⁡ q n + 1 q n , q n is the sequence of denominations of the best rational approximations for α ∈ R ∖ Q , 0 r ⁎ r , r is the initial radius of analytic strip, it is proved that for most sufficiently small ϵ, this system can be reduced to a constant system x ˙ = A ⁎ x , where A ⁎ is a constant matrix close to A. As some applications, we apply our results to quasi-periodic Hill's equations, three dimensional skew symmetric systems and n coupled Schrodinger equations to study the Lyapunov stability of the equilibrium and the existence of quasi-periodic solutions.

The main objective of the present investigation is to find the solution for the fractional model of Klein-Gordon-Schrodinger system with the aid of q-homotopy analysis transform method (q-HATM). The projected solution procedure is an amalgamation of q-HAM with Laplace transform. More preciously, to elucidate the effectiveness of the projected scheme we illustrate the response of the q-HATM results, and the numerical simulation is offered to guarantee the exactness. Further, the physical behaviour has been presented associated with parameters present the method with respect fractional-order. The present study confirms that, the projected solution procedure is highly methodical and accurate to solve and study the behaviours of the system of differential equations with arbitrary order exemplifying the real word problems.

In this paper, the conservative Fourier spectral scheme is presented for higher order Klein-Gordon-Schrodinger (KGS) system with periodic boundary condition. First, using the Fourier collocation scheme in space, we obtain an conservative Fourier spectral scheme for the higher order KGS system. We prove that the proposed method satisfies the discrete mass and energy conservation laws exactly. The existence and uniqueness of the numerical solution is proved, and the stability and convergence of the scheme is discussed. Moreover, we find the scheme is decoupled and nonlinear. Then, we give linearized scheme of the higher order KGS system when p = 1 . The numerical experiments are given to show that verify the correctness of theoretical results and the efficiency of the scheme.

This paper aims to construct structure-preserving numerical schemes for the two-dimensional space fractional Klein-Gordon-Schrodinger equation, which are based on the newly developed partitioned averaged vector field methods. First, we derive an equivalent equation, and reformulate the equation as an infinite dimensional canonical Hamiltonian system by virtue of the variational derivative of the functional with fractional Laplacian. Then, we use the Fourier pseudo-spectral method to discrete the equation in space direction and obtain a semi-discrete conservative system, which can be reformulated a finite dimensional canonical Hamiltonian system. Further applying the partitioned averaged vector field methods to the semi-discrete system gives a class of fully-discrete schemes that can preserve the mass and energy exactly. Numerical examples are provided to confirm our theoretical analysis results at last.

In this work we are concerned with the following class of equations − Δ p u − λ h ( x ) | u | p − 2 u = f ( x ) | u | γ − 2 u , in R N , involving indefinite weight functions. The existence of solution may depend on the parameter λ. We analyze the extreme value λ ⁎ and we study its relation with the Nehari manifold. Our goal is to establish the existence of two solutions when λ > λ ⁎ . This work extends and complements the results obtained by Chabrowski and Costa (2008) [6] .

We study state estimation of the linearized Schrodinger equation within a prescribed terminal time. We make use of a time–varying, complex–valued observer gain and boundary measurements to construct the observer, where the gain is designed such that the estimate error converges to zero within the terminal time. The observer gain proposed herein is developed via the backstepping method by selecting a target error equation that stabilizes to zero within the terminal time. Our time–varying observer gain diverges as time approaches the terminal time. Nevertheless, we can guarantee prescribed–time stabilization of the estimator error equation by characterizing the growth–in–time of the observer gain and comparing it to the stability of the target error equation. We develop the full–state feedback dual result, and we combine the boundary estimation and control results to develop prescribed–time output regulation.

In this paper, an efficient linear compact alternating direction implicit (CADI) scheme with second order temporal accuracy and fourth order spatial accuracy is proposed for approximating solution to the initial–boundary problem for the two-dimensional Klein–Gordon–Schrodinger equations. Furthermore, maximum norm error estimates of numerical solutions are obtained by using the energy method and the mathematical induction method. Finally, some numerical experiments are presented to support the theoretical results.

We investigate extensions for the Schrodinger equation, by considering different non-linear dependencies on the kinetic term, which admit the standard probabilistic interpretation for the wave-function. For one of the cases, we show that the Heisenberg equation is invariant in form and a modified Hamilton-Jacobi equation emerges from the semi-classical approximation for the wave-function. Furthermore, we find solutions assuming time scaling functions different from the standard case and observe nonstandard relaxation processes for the wave-function.

In the present paper, we present the exact analytical solution of the time-fractional Schrodinger equation (TFSE) in the sense of conformable fractional derivative (Co-FD) based on the residual power series method (RPSM). Moreover, we make graphical comparisons between the solution of this problem in the sense of Co-FD and the obtained solutions in previous studied in the sense of Caputo fractional derivative (Ca-FD). Comparisons indicate that Co-FD is a suitable alternative to Ca-FD in the modeling of TFSE. The main advantage for employing the RPSM is the simplicity of computing the coefficients of the series solution by applying differential operators without having to use integrated operators. Finally, our proposed technique can be applied easily in the solution of higher dimension without any restriction on the type of equation and the Co-FD can be used to formulate other types of partial differential equation.

The present paper employs the space-time fractional nonlinear Bogoyavlenskii equation and Schrodinger equation. We perform a new method to take some new solitary wave phenomena for each equation. Those solutions can explain through hyperbolic, trigonometric and rational functions. The graphical design makes the dynamics of the equations noticeable. Herein, the intended approach is simplistic, conventional, and convenient in implementing many solitary wave phenomena of several nonlinear fractional wave equations occurring in mathematical physics and engineering as well.

Exactly-solvable confined model of the nonrelativistic quantum harmonic oscillator is proposed. Its position-dependent effective mass Hamiltonian is defined via the von Roos kinetic energy operator. The confinement effect to harmonic oscillator potential is included as a result of certain behaviour of the position-dependent effective mass. The corresponding Schrodinger equation in the canonical approach is exactly solved and it is shown that the discrete energy spectrum of the system under consideration depends on the confinement parameter a, von Roos parameters α, β, γ and has a non-equidistant form. Wave functions of the stationary states of the model are expressed through the Gegenbauer polynomials. The limit a → ∞ recovers both equidistant energy spectrum and wave functions of the stationary nonrelativistic harmonic oscillator expressed by Hermite polynomials.

We consider nonlinear elliptic equations that are naturally obtained from the elliptic Schrodinger equation − Δ u + V u = 0 in the setting of the calculus of variations, and obtain L q -estimates for the gradient of weak solutions. In particular, we generalize a result of Shen (1995) [39] in the nonlinear setting by using a different approach. This allows us to consider discontinuous coefficients with a small BMO semi-norm and non-smooth boundaries which might not be Lipschitz continuous.