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In this paper, we are concerned with an averaging procedure, namely Stroboscopic averaging, for highly oscillatory evolution equations posed in a (possibly infinite dimensional) Banach space, typically partial differential equations in a high-frequency regime where only one frequency is present. We construct a high-order averaged system whose solution remains exponentially close to the exact one over long time intervals, possesses the same geometric properties (structure, invariants, ...) as compared to the original system and is non-oscillatory. We then apply our results to the nonlinear Schrodinger equation on the d-dimensional torus [T.sup.d], or in [R.sup.d] with a harmonic oscillator, for which we obtain a hierarchy of Hamiltonian averaged models. Our results are then illustrated numerically on several examples borrowed from the recent literature. Keywords Highly oscillatory evolution equation * Stroboscopic averaging * Hamiltonian PDEs * Invariants * Nonlinear Schrodinger * SAM, Mathematics Subject Classification 35A35 * 35J10 * 34K33 1 Introduction In this article, we are concerned with highly oscillatory evolution equations posed in a Banach space X d/dt [u.sup.[epsilon]](t) = [...]

The Schrodinger equation does not account for the 2[n.sup.2] degeneracy of the hydrogen atom, which it dismisses as an 'accidental' degeneracy. The factor of '2' in the 2[n.sup.2] degeneracy is well-accounted-for in the relativistic formulation by the two spin states of the electron. The [n.sup.2] degeneracy is nevertheless not quite an 'accident'; it is due to the SO(4), rather than SO(3), symmetry of the hydrogen atom. This result is well known, but is inadequately commented upon in most courses in quantum mechanics and atomic physics, leaving the student wondering about the origins of the [n.sup.2] degeneracy of the hydrogen atom. A pedagogical analysis of this interesting aspect, which highlights the fundamental principles of quantum mechanics, is presented in this article. While doing so, not only is the [n.sup.2] degeneracy of the hydrogen atom explained, but its energy spectrum and eigenfunctions are obtained without even using the Schrodinger equation, employing only the fundamental principles of quantum mechanics rather than the Schrodinger equation. PACS Nos.: 01.40.Fk, 01.40.G-, 02.20.Qs, 03.65.Ca, 03.65.Ge. L'equation de Schrodinger n'explique pas la degenerescence 2[n.sup.2] de l'atome d'hydrogene et l'assigne simplement a une degenerescence << accidentelle >>. On explique bien le facteur << 2 >> de la degenerescence 2[n.sup.2] dans la formulation relativiste qui tient compte des deux etats de spin de l'electron. Neanmoins, la degenerescence [n.sup.2] n'est pas vraiment << accidentelle >> ; elle est due fi la symetrie SO(4) plutot que SO(3) de l'atome d'hydrogene. Ce resultat est bien connu, mais n'est pas decrit de facon adequate dans la plupart des livres de mecanique quantique et de physique atomique. Nous presentons ici une analyse pedagogique de ce probleme interessant, qui souligne les principes fondamentaux de la mecanique quantique. Ce faisant, nous expliquons non seulement la degenerescence [n.sup.2] de l'atome d'hydrogene, mais nous obtenons le spectre en energie et les fonctions propres sans avoir a resoudre l'equation de Schrodinger, nous limitant aux principes premiers de la mecanique quantique plutot qu'a l'equation de Schrodinger. [Traduit par le Redaction], 1. Conservation principles from the physical law One can clearly demark physics before, and after, Albert Einstein [1]. One of his outstanding contributions is the recognition of symmetry-invariance as an [...]

FOR A FANTASTICALLY WELL-SUPPORTED THEORY that's been around for almost a century, quantum mechanics is embroiled in a surprising amount of controversy. Everyone seems to agree that quantum theory makes [...]

Basing on the priori assumption principle, the main goal of this paper is to propose the rational expansion method that can be used to handle the exact solution of the nonlinear partial differential equation. It is extension of tanh function method. As an application, three classes of exact solutions to Klein-Gordon-Schrodinger equation are obtained. Keywords Klein-Gordon-Schrodinger equation, exact solution, tanh function method, rational expansion method., §1. Introduction The investigation of exploring the exact traveling wave solutions of nonlinear mathematical physics equations plays an important role in the study of the solitary wave solution. Up to [...]

Different features of a potential in the form of a Gaussian well have been discussed extensively. Although the details of the calculation are involved, the general approach uses a variational method and WKB approximation, techniques that should be familiar to advanced undergraduates. A numerical solution of the Schrodinger equation through diagonalization has been developed in a self-contained way, and physical applications of the potential are mentioned. [c] 2010 American Association of Physics Teachers. [DOI: 10.1119/1.3474665]

The accuracy of different transfer matrix approaches, which are used for solving the stationary effective mass Schrodinger equation for arbitrary one-dimensional potentials, is examined. Analytical model potentials as well as self-consistent Schrodinger-Poisson simulations of a heterostructure device have shown that a symmetrized transfer matrix approach has produced a similar accuracy as the Airy function method at a reduced numerical cost, thus avoiding the numerical problems associated with Airy function.

The nature of a quantum computer is described in the concrete context of a quantum simulator of the single-particle Schrodinger equation. We show that a register of 6-10 qubits is sufficient to realize a useful quantum simulator capable of efficiently solving standard quantum mechanical problems. [DOI: 10.1119/1.2894532]

A system composed of an electron interacting with the electromagnetic field of a pulsed laser is analyzed. Findings reveal the suitability of the model for the interaction between electrons and laser beams.

An algorithm for the self-consistent solution of the Boltzmann transport equation (BTE) and Poisson-Schroinger (PS) equation in two dimensions is studied. Findings reveal that the proposed algorithm allows the full exploration of the entire range from drift-diffusion to ballistic regime in nanoscale field-effect transistors.

The initial value problem in quantum mechanics is most conveniently solved by the Green function method. Instead of the conventional methods of eigenfunction expansion and path integration, we present a new method for constructing the Green functions systematically. By using suitable elementary transformations, one of the conjugate variables in the Hamiltonian can be eliminated and the Green function for the simplified Hamiltonian can be easily derived. We then obtain the Green function for the original Hamiltonian by the reverse sequence of the elementary transformations. The method is illustrated for the linear potential, the harmonic oscillator, the centrifugal potential, and the centripetal barrier oscillator. [DOI: 10.1119/1.2186688]

The Einstein-Brillouin-Keller (EBK) quantization equation is used to determine the energy levels of a two-body system with an arbitrary central potential that allows for bound states. The treatment is based on the conservation laws and avoids both the Newtonian and Schrodinger differential equations. Because analytic solutions for the energy levels do not exist in general, the EBK condition is applied using the Newton-Raphson method and the radial probability density is computed. Potentials appropriate for a diatomic molecule are considered and the effect of the angular momentum on the radial distribution, the nature of the classical orbits, and the possibility of closed orbits is studied. [DOI: 10.1119/1.2192788]

One-dimensional Schrodinger-Poissen self-consistent solver is applied to calculate the leakage current through the oxide of an n-channel metal-oxide-semiconductor (MOS) structure with an open boundary on one side. The tunneling current is computed for a MOS structure with Si[O.sub.2] and [Si.sub.3][N.sub.4] gate dielectrics and the computational results are compared with those obtained experimentally for similar structures.

A self-consistent Poisson-Schrodinger solver is used to measure the current and the quantum-mechanical effects in trigate n-channel silicon-on-insulator transistors. The threshold voltage is higher, the minimum energy for the electrons in the conduction energy subbands increases with the electron concentration, and this effect reduces the current drive of the device and is not predicted by classical simulators.

An efficient and convenient method for determining the electrostatically confined electron and hole levels by using the plane-wave expansion method in conjunction with the k.p envelope function method is presented. The results indicate that by removing the defect-related current path, the threshold current density could be further reduce in GaInNAs quantum well (QW) lasers.

This article discusses Neutrosophic Logic interpretation of the Schrodinger's cat paradox. We argue that this paradox involves some degree of indeterminacy (unknown) which Neutrosophic Logic could take into consideration, whereas [...]

The mathematical conditions for incorporating the geometric phase effect in triatomic and tetraatomic hyperspherical harmonics are provided. A special pseudorotation for triatomic and tetraatomic is defined, which is useful for incorporating the geometric phase effects in the hyperspherical harmonics.

A new nonlinear Schrodinger equation is obtained explicitly from the (fractal) Brownian motion of a massive particle with a complex-valued diffusion constant. Real-valued energy plane-wave solutions and solitons exist in [...]

Theoretical developments in heavy-quarkonium physics from the point of view of effective-field theories of QCD are reviewed. In the weak coupling regime, the application of QCD effective-field theories to heavy quarkonia allows higher order calculations to be carried out in a systematic and simple manner.

A one-dimensional Poisson/Schrodinger program is used to investigate the effect of layer thicknesses, donor concentration, and band-gap offset on the electrical properties of transparent conducting modulation-doped ZnO/ZnMgO multilayer structures. The optimal thicknesses to simultaneously achieve high mobility and low resistivity are 2-5 nm for both the pure ZnO and ZnMgO:Al layers.

Energy levels and oscillator strengths of an electron are calculated for a finite rectangular quantum wire in a transverse magnetic field. The results reveal that the decoupled approximation, a numerical method for solving the Schrodinger equation, is not suitable for a quantum wire with a smaller cross section when the quantum wire is given width in one direction.

The requirements for the detailed physical modeling of nanoscale MOSFETs are analyzed and it is showed that the Schrodigner drift-diffusion per subband simulations are adequate for the inverse modeling of bulk-Si MOSFETs with gate length down to 40nm. A proper treatment of quantum effects in the channel and in the polysilicon gate through the direct solution of Schrodinger equation and a transport model based on two-dimensional subbands are required for accurate modeling.

The impact of image and many-body exchange-correlation effects on electron transport are investigated for nanoscale double-scale MOSFETs using the nonequilibrim Green function method. The simulation is performed for metal gate and polysilicon gate MOSFETs.

Byline: Metin AktaA (1), Ramazan Sever (1) Keywords: deformed ring-shaped potential; the Nikiforov--Uvarov method Abstract: Exact solution of the Schrodinger equation with deformed ring-shaped potential is obtained in the parabolic and spherical coordinates. The Nikiforov--Uvarov method is used in the solution. Eigenfunctions and corresponding energy eigenvalues are calculated analytically. The agreement of our results is good. Author Affiliation: (1) Department of Physics, Middle East Technical University, 06531, Ankara, Turkey Article History: Registration Date: 06/12/2004 Received Date: 13/02/2004 Article note: AMS Subject Classification: 03.65.-w, 12.39.Jh, 21.10.-k

Byline: H. TaAeli (1) Keywords: Schrodinger equation; Exactly solvable hamiltonians; special functions; classical orthogonal polynomials Abstract: A new subclass of the Jacobi polynomials arising in the exact analytical solution of the one-dimensional Schrodinger equation with a trigonometric potential has been introduced. The polynomials which consist of a free parameter are not ultraspherical polynomials and have been simply named the T -polynomials since they are generated by a trigonometric Hamiltonian. In certain sense, it is shown that the T -polynomials can be regarded as a generalisation of the airfoil polynomials or the Chebyshev polynomials of the third kind. This paper is intended to discuss the basic properties of the polynomials so defined. Author Affiliation: (1) Department of Mathematics, Middle East Technical University, 06531, Ankara, Turkey Article History: Registration Date: 26/09/2004

Byline: T.E. Simos (1) Keywords: multiderivative methods; multistep methods; numerical methods; schrodinger equation Abstract: In this paper exponentially fitted multiderivative methods are developed for the numerical solution of the one-dimensional Schrodinger equation. The methods are called multiderivative since uses derivatives of order two and four. An application to the the resonance problem of the radial Schrodinger equation indicates that the new method is more efficient than other similar well known methods of the literature. Author Affiliation: (1) Department of Computer Science and Technology, Faculty of Sciences and Technology, University of Peloponnese, GR-221 00, Tripolis, Greece Article History: Registration Date: 26/09/2004

Byline: Kostas Tselios (1), T.E. Simos (2) Keywords: radial Shrodinger equation; symplectic schemes; fifth order Abstract: In this paper the numerical solution of the radial Shrodinger equation via new proposed symplectic-schemes is investigated. In particular, the radial Schodinger equation is transformed into Hamiltonian canonical form and is solved via symplectic integrators. Based on this approach, fifth-order methods are proposed. We compare these methods with well-known existing symplectic methods. The numerical results show the efficiency of the proposed method. Author Affiliation: (1) Department of International Trade, Technological Institute of Western Macedonia at Kastoria, GR-521 00, Kastoria, Greece (2) Department of Computer Science and Technology, Faculty of Sciences and Technology, University of Peloponnese, GR-221 00, Tripolis, Greece Article History: Registration Date: 29/09/2004

I present a method in the spirit of Green's functions for generating bound state solutions to the Schrodinger equation. The method is demonstrated by generating ground state and excited state solutions to the one-dimensional, time-independent Schrodinger equation. Its applicability to the radial equation for two and three dimensions is also discussed. [DOI: 10.1119/1.1466816] (Received 30 July 2001; accepted 7 February 2002)

The quantum pendulum finds application in surprising contexts. We use commercially available software to numerically solve the Schrodinger equation for a microscopic pendulum subject to molecular (electromagnetic) restoring forces, and a macroscopic pendulum subject to a gravitational restoring force. The dynamics of the microscopic quantum pendulum are closely related to molecular motions known as hindered rotations. We use standard probabilistic methods to predict whether this motion is weakly or strongly hindered at ambient temperature and test the prediction against experimental data for [C.sub.2][H.sub.6] and [K.sub.2]Pt[Cl.sub.6]. For the macroscopic gravitational pendulum, we examine the uncertainty in position and find, not surprisingly, that it is too small to measure physically, but is nevertheless relatively large compared to present-day limits in computation. The latter juxtaposition of computational precision with quantum uncertainty has consequences for the study of chaotic dynamics. [DOI: 10.1119/1.1456069] (Received 14 June 2000; accepted 8 January 2002)

Byline: Z. Kalogiratou (1), T.E. Simos (2) Keywords: trigonometrically-fitted; exponentially-fitted; Runge--Kutta--Nystrom; Schrodinger equation; scattering problems; resonance problem; periodic solutions; oscillating solutions Abstract: General conditions are presented for an m-stage Runge--Kutta--Nystrom fitting to exponential and trigonometric functions. As an example an 8th order Runge--Kutta--Nystrom method is constructed. Numerical results on the numerical solution of the Schrodinger equation and related problems indicate that the new method is more accurate than the classical one (we call classical Runge--Kutta--Nystrom method the corresponding method with constant coefficients). Author Affiliation: (1) Department of International Trade, Technological Educational Institute of Western Macedonia at Kastoria, P.O. Box 30, GR-521 00, Kastoria, Greece (2) Section of Mathematics, Department of Civil Engineering, School of Engineering, Democritus University of Thrace, GR-671 00, Xanthi, Greece Article History: Registration Date: 09/10/2004

Byline: T.E. Simos (1), Jesus Vigo-Aguiar (2) Keywords: symmetric methods; multistep methods; radial Schrodinger equation; resonance problems; scattering problems; phase shift problems; phase-lag Abstract: In this paper some new eighth algebraic order symmetric eight-step methods are introduced. For these methods a direct formula for the computation of the phase-lag is given. Based on this formula, the calculation of free parameters is done in order the phase-lag to be minimal. The new methods have better stability properties than the classical one. Numerical illustrations on the radial Schrodinger equation indicate that the new method is more efficient than older ones. Author Affiliation: (1) Section of Mathematics, School of Engineering, Department of Civil Engineering, University of Thrace, GR-67 100, Xanthi, Greece (2) Departamento de Matematica Aplicada, Facultad de Ciencias, Universidad de Salamanca, E-37008, Salamanca, Spain Article History: Registration Date: 09/10/2004

Byline: Gregory S. Tschumper (1), Mark R. Hoffmann (2) Keywords: perturbation theory; time-independent Schrodinger equation; quantum mechanical oscillators; superconvergence Abstract: A computationally facile superconvergent perturbation theory for the energies and wavefunctions of the bound states of one-dimensional anharmonic oscillators is suggested. The proposed approach uses a Kolmogorov repartitioning of the Hamiltonian with perturbative order. The unperturbed and perturbed parts of the Hamiltonian are defined in terms of projections in Hilbert space, which allows for zero-order wavefunctions that are linear combinations of basis functions. The method is demonstrated on quartic anharmonic oscillators using a basis of generalized coherent states and, in contrast to usual perturbation theories, converges absolutely. Moreover, the method is shown to converge for excited states, and it is shown that the rate of convergence does not deteriorate appreciably with excitation. Author Affiliation: (1) Department of Chemistry, University of North Dakota, Grand Forks, North Dakota, USA (2) Department of Chemistry and Biochemistry, University of Mississippi, University, MS, 38677, USA Article History: Registration Date: 09/10/2004

We solve the one-dimensional nonlinear Schrodinger equation for an attractive delta-function potential at the origin, [([p.sup.2]/2m)-[OMEGA][delta](x)|[[phi] (x)|sup.[alpha]][phi](x) = E [phi](x),= E [phi] (x), and obtain the bound state in closed form as a function of the nonlinear exponent [alpha]. The bound state probability profile decays exponentially away from the origin, with a profile width that increases monotonically with [alpha], becoming an almost completely extended state when [alpha] [right arrow] [2.sup.-]. At [alpha] = 2, the bound state suffers a discontinuous change to a delta function-like profile. A further increase of [alpha] increases the width of the probability profile, although the bound state is stable only for [alpha]

Byline: Ya. V. Mykytyuk (1), U. S. Djala (1) Abstract: Singular relatively compact perturbations of self-adjoint operators are studied. The results obtained are applied to the Schrodinger operator with a singular potential. Author Affiliation: (1) I. Franko Lviv State University, Ukraine Article History: Registration Date: 20/10/2004

The electronic structure of a hydrogenic impurity atom located at the center of a multilayer coated quantum dot (CQD) is investigated. The electronic eigenstates of the CQD system are expressed in terms of the geometry and the material parameters by solving the Schrodinger equations analytically.

Byline: T.E. Simos (1), Jesus Vigo Aguiar (2) Keywords: Runge--Kutta methods; phase-lag; phase-fitted; explicit methods Abstract: A modified phase-fitted Runge--Kutta method (i.e., a method with phase-lag of order infinity) for the numerical solution of periodic initial-value problems is constructed in this paper. This new modified method is based on the Runge--Kutta fifth algebraic order method of Dormand and Prince [33]. The numerical results indicate that this new method is more efficient for the numerical solution of periodic initial-value problems than the well known Runge--Kutta method of Dormand and Prince [33] with algebraic order five. Author Affiliation: (1) Section of Mathematics, School of Engineering, Department of Civil Engineering, University of Thrace, GR-67 100, Xanthi, Greece (2) Departamento de Matematica Aplicada, Facultad de Ciencias, Universidad de Salamanca, E-37008, Salamanca, Spain Article History: Registration Date: 08/10/2004

Byline: G. Avdelas (1,2), A. Konguetsof (1), T.E. Simos (1) Keywords: hybrid methods; explicit methods; algebraic order; periodic problems; initial-value problems; Schrodinger equation Abstract: In this paper we present the development of a generator of hybrid explicit methods for the numerical solution of the Schrodinger equation. The methods are of algebraic order ten. The coefficients of the generator are calculated appropriately. Author Affiliation: (1) Section of Mathematics, Department of Civil Engineering, School of Engineering, Democritus University of Thrace, GR-671 00, Xanthi, Greece (2) Laboratory of Applied Mathematics and Computers, Department of Sciences, Technical University of Crete, GR-731 00, Chania, Crete Article History: Registration Date: 08/10/2004

Byline: G. Avdelas (1,2), A. Konguetsof (1), T.E. Simos (1) Keywords: phase-lag; hybrid methods; explicit methods; algebraic order; intervals of stability; Schrodinger equation; periodic problems; initial-value problems Abstract: The generator of tenth-order hybrid explicit methods, the basic method of which has been developed in part 1, is constructed and also optimized, by maximization of the intervals of periodicity. The efficiency of the new methods is shown by their application to the coupled differential equations of the Schrodinger type. Author Affiliation: (1) Section of Mathematics, Department of Civil Engineering, School of Engineering, Democritus University of Thrace, GR-671 00, Xanthi, Greece (2) Laboratory of Applied Mathematics and Computers, Department of Sciences, Technical University of Crete, GR-731 00, Chania, Crete Article History: Registration Date: 08/10/2004

Byline: N. Slavnov (1) Abstract: The correlation functions of the quantum nonlinear Schrodinger equation can be presented in terms of the Fredholm determinant. An explicit expression for this determinant is found for large time and long distance. Bibliography: 6 titles. Author Affiliation: (1) Steklov Mathematical Institute, Moskow Article History: Registration Date: 06/10/2004