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In this article we prove that the fractional integral operator associated to the Schrödinger second order differential operator L-α/2=(-Δ + V)-α/2maps with continuity weak Lebesgue space Lp,∞(v) into weighted Campanato-Hölder type spaces BMOβL(w), thus improving regularity under appropriate conditions on the pair of weights (v,w) and the parameters p, α and β. We also prove the continuous mapping from BMOβL(v) to BMOγL(w) for adequate pair of weights. Our results improve those known for the same weight in both sides of the inequality and they also enlarge the families of weights known for the classical fractional integral associated to the Laplacian operator L = -Δ. Fil: Crescimbeni, Raquel Liliana. Universidad Nacional del Comahue; Argentina Fil: Hartzstein, Silvia Inés. Universidad Nacional del Litoral; Argentina Fil: Salinas, Oscar Mario. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina

The one-dimensional Schr&ouml, dinger equation, applied to the H2 intramolecular stretch coordinate in singly to quadruply occupied large cages in extended Type II (sII) hydrogen clathrate hydrate, was solved numerically herein via potential-energy scans from classical molecular dynamics (MD), employing bespoke force-matched H2&ndash, water potential. For both occupation cases, the resultant H&ndash, H stretch spectra were redshifted by ~350 cm&minus, 1 vis-&agrave, vis their classically sampled counterparts, yielding semi-quantitative agreement with experimental Raman spectra. In addition, ab initio MD was carried out systematically for different cage occupations in the extended sII hydrate to assess the effect of differing intra-cage intrinsic electric field milieux on H&ndash, H stretch frequencies, we suggest that spatial heterogeneity of the electrostatic environment is responsible for some degree of peak splitting.

The generalized Fourier transforms (GFTs) for the hierarchies of multi-component models of Manakov type are revisited. Our aim is to adapt GFT so that one could treat the whole hierarchy of nonlinear evolution equations simultaneously. To this end we consider the potential Q of the Lax operator L as local coordinate on some symmetric space depending on an infinite number of variables t, and $$z_k$$ , $$k=1, 2, \ldots $$ . The dependence on $$z_k$$ is determined by the k-th higher flow (conserved density) of the hierarchy. Thus we have an infinite set of commuting operators with common fundamental analytic solution. We analyze the properties of the resolvent and thus determine the spectral properties of L. Then we derive the generalized Fourier transforms that linearize this hierarchy of NLEE and establish their fundamental properties as well as dynamical compatibility of each two pairs of such flows. Using the classical R-matrix approach we derive the Poisson brackets between all conserved quantities first assuming that Q is a quasi-periodic function of t. Next taking the limit when the period tends to $$\infty $$ , we derive the Poisson brackets between the scattering data of L. In addition we analyze a possible relation of our approach to the one based on the $$\tau $$ -function that could lead to multi-dimensional integrable equations.

In this note, we give a rigorous proof that the NLS periodic Akhmediev breather is unstable. The proof follows the ideas in [17], in the sense that a suitable modification of the Stokes wave is the global attractor of the local Akhmediev dynamics for sufficiently large time, and therefore the latter cannot be stable in any suitable finite energy periodic Sobolev space., Comment: Submitted to the Proceedings of the \emph{Third Workshop on Nonlinear Dispersive Equations}, held in Campinas, Brazil, during November 8-10 2017. Revised version. Accepted in SPJM

We present some lower bounds for regular solutions of Schr\"odinger equations with bounded and time dependent complex potentials. Assuming that the solution has some positive mass at time zero within a ball of certain radius, we prove that this mass can be observed if one looks at the solution and its gradient in space-time regions outside of that ball.

In this work, we obtained an exact solution to Schrodinger equation using q-deformed Woods-Saxon plus modified Coulomb potential Using conventional Nikiforov-Uvarov method. We also obtained the energy eigen value and its associated total wave function . This potential with some suitable conditions reduces to two well known potentials namely: the Yukawa and coulomb potential. Finally, we obtained the numerical results for energy eigen value with different values of q as dimensionless parameter. The result shows that the values of the energies for different quantum number(n) is negative(bound state condition) and increases with an increase in the value of the dimensionless parameter(arbitrary constant). The graph in figure (1) shows the different energy levels for a particular quantum number.

We study standing waves for nonlinear Schrodinger equations with the gauge field. Some existence results of standing waves are established by applying variational methods to the functional which is obtained by representing the gauge field A μ in terms of complex scalar field ϕ. We also show that there exists no standing wave for certain range of parameters by establishing a new inequality of Sobolev type.

We consider nontrivial solutions of − Δ u ( x ) = V ( x ) u ( x ) , where u ≡ 0 on the boundary of a bounded open region D ⊂ R n , and V ( x ) ∈ L ∞ ( D ) . We prove a sharp relationship between ‖ V ‖ ∞ and the measure of D , which generalizes the well-known Faber–Krahn theorem. We also prove some geometric properties of the zero sets of the solution of the Schrodinger equation − Δ u ( x ) = V ( x ) u ( x ) .

Let H ( λ ) = − Δ + λ b be a discrete Schrodinger operator on l 2 ( Z d ) with a potential b and a non-negative coupling constant λ. When b ≡ 0 , it is well known that σ ( − Δ ) = [ 0 , 4 d ] . When b ≢ 0 , let s ( − Δ + λ b ) : = inf σ ( − Δ + λ b ) and M ( − Δ + λ b ) : = sup σ ( − Δ + λ b ) be the bounds of the spectrum of the Schrodinger operator. One of the aims of this paper is to study the influence of the potential b on the bounds 0 and 4d of the spectrum of −Δ. More precisely, we give a necessary and sufficient condition on the potential b such that s ( − Δ + λ b ) is strictly positive for λ small enough. We obtain a similar necessary and sufficient condition on the potential b such that M ( − Δ + λ b ) is lower than 4d for λ small enough. In dimensions d = 1 and d = 2 , the situation is more precise. The following result was proved by Killip and Simon (2003) (for d = 1 ) in [5] , then by Damanik et al. (2003) (for d = 1 and d = 2 ) in [3] : If σ ( − Δ + b ) ⊂ [ 0 , 4 d ] , then b ≡ 0 . Our study on the bounds of the spectrum of ( − Δ + b ) allows us to give a different and easy proof to this result.

In this work we obtain boundedness on suitable weighted BMO type spaces of Riesz transforms, and their adjoints, associated to the Schrödinger operator - Δ + V, where V satisfies a reverse Hölder inequality. Our results are new even in the unweighted case. Fil: Bongioanni, Bruno. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina Fil: Harboure, Eleonor Ofelia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina Fil: Salinas, Oscar Mario. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina

This paper presents Erwin Schrödinger’s Wave Interpretation of Quantum Mechanics, with the main goal of comparing it to the so called Many Worlds Interpretation, of which Bryce DeWitt is the most important figure. It is commonly said that DeWitt’s and Everett’s Interpretations are equivalent, and both would have been inspired by Schrödinger’s wave-like approach. This paper claims those stances to be superficial, requiring a more detailed exam of the philosophical grounds held by the authors, besides other distinguishing details. A connexion may be established concerning the rejection of the quantum collapse, although one should be careful about the meaning of the term in each standpoint. This article also shows the balance of Schrödinger’s Interpretation of Quantum Mechanics in regard to his philosophy in a wider sense. Finally, it concludes that detaching Schrödinger’s interpretation from Many Worlds is more coherent with his philosophical assumptions; he conceives a single world containing infinite possibilities. Este artigo apresenta a interpretação ondulatória de Erwin Schrödinger da mecânica quântica, com o intuito principal de compará-la àquela conhecida como “muitos mundos”, cujo principal expoente é Bryce DeWitt. É comum dizer-se que a Interpretação de DeWitt e de Hugh Everett são equivalentes, e que ambas teriam se inspirado na abordagem ondulatória schrödingeriana. Defendemos que essas visões são superficiais, exigindo um exame mais detalhado dos pressupostos filosóficos envolvidos e outros detalhes que diferem nas três interpretações. Uma conexão pode ser feita a partir da rejeição do colapso, mas mesmo assim é preciso avaliar com cuidado o significado desse termo para os autores. Assim, este artigo evidencia a harmonia da Interpretação de Schrödinger com a sua filosofia, em sentido mais geral. Além disso, o seu afastamentocom relação à interpretação de muitos mundos é mais condizente com os pressupostos filosóficos sustentados por Schrödinger, que concebe um único mundo contendo infinitas possibilidades.

Erwin Schrödinger, o físico austríaco, que foi um dos principais autores da física quântica, realizou em 1950 uma série de conferências intituladas Ciência e humanismo, que estão traduzidas em português. Analisamos aqui a sua visão da ciência como parte do esforço do ser humano em conhecer-se. Debatemos a sua perspectiva da unidade das ciências, a relação entre ciência e técnica, as raízes profundas do pensamento científico na Antiguidade Grega e, além disso, a interacção da ciência com a filosofia e a religião. Expressamos a opinião de que uma boa parte das suas reflexões são relevantes nos dias de hoje, quando se fala da crise do humanismo. Mais humanismo significa mais e melhor ciência, o que significa progresso na integração de diferentes ramos do conhecimento humano., In 1950, Erwin Schrödinger, the Austrian physicist who was one of principal proponents of quantum physics, gave a series of lectures entitled Science and Humanism. It has been translated into Portuguese. This essay analyzes his perpective of science as a part of the effort of the human being to know him/ herself. We discuss his vision of the unity of sciences, the relation between science and technique, the deep roots of scientific thinking in Greek Antiquity and, in addition, the interplay of science with philosophy and religion. We argue that a great part of Schrödinger’s reflections are still relevant today, at a time when it has become usual to speak of the crisis of humanism. More humanism means more and better science, and that means progress in the integration of different branches of human knowledge.

We consider a nonrelativistic charged particle in a 1D moving potential well. This quantum system is subject to a control, which is the acceleration of the well. It is represented by a wave function solution of a Schrödinger equation, the position of the well together with its velocity. We prove the following controllability result for this bilinear control system: given ψ0 close enough to an eigenstate and ψf close enough to another eigenstate, the wave function can be moved exactly from ψ0 to ψf in finite time. Moreover, we can control the position and the velocity of the well. Our proof uses moment theory, a Nash–Moser implicit function theorem, the return method and expansion to the second order.

The two-dimensional multi-layered HEMT is modelled isothermally by solving the Poisson and current continuity equations consistently with the Schrödinger equation. A multigrid method is used on the Poisson and current continuity equations while the electron density is calculated at each level by solving the Schrödinger equation in onedimensional slices perpendicular to the layer structure. A correction factor is introduced which enables relatively accurate solutions to be obtained using a low number of eigensolutions. A novel method for discretising the current density which can be generalised to the non-isothermal case is described. Results are illustrated using a two layer AlGaAs-GaAs HEMT.

A self-consistent numerical simulation model for a pin single-cell solar cell is formulated. The solar cell device consists of a p–AlGaAs region, an intrinsic i–AlGaAs/GaAs region with several quantum wells, and a n–AlGaAs region. Our simulator solves a field-dependent Schrödinger equation self-consistently with Poisson and drift-diffusion equations. The field-dependent Schrödinger equation is solved using the transfer matrix method. The eigenfunctions and eigenenergies obtained are used to calculate the escape rate of carriers from the quantum wells, the capture rates of carriers by the wells, the absorption spectra in the wells, and the non-radiative recombination rates of carriers in the quantum wells. These rates are then used in a self-consistent finite-difference numerical Poisson-drift-diffusion solver. We believe this is the first such comprehensive model ever reported.

A self-consistent Boltzmann-Poisson-Schrödinger Solver is used to study the transconductance degradation in high electron mobility transistor (HEMT), which has extensively been reported by both experimental [1]-[8] and computational [9]-[ 13] researchers. As the gate voltage of a HEMT device is increased, its transconductance increases until it reaches a peak value, beyond which, the transconductance is degraded rather sharply with further increase in applied gate bias. We previously reported a two-subband self-consistent Boltzmann-Poisson- Schrödinger Solver for HEMT. [14] We further incorporated an additional self-consistency by calculating field-dependent, energy-dependent intersubband and intrasubband scattering rates due to ionized impurities and polar optical phonons.[15] In this work, we have used our Boltzmann-Poisson-Schrödinger Solver and studied the effects of the intersubband and intrasubband scatterings of electrons, on the transconductance of a single quantum well HEMT device. The results of our simulations exhibit the same pattern reported by others [1]-[13]. We concluded that the degradation of transconductance of the device with applied gate bias is attributed to the increased intersubband and intrasubband scattering of electrons, and hence to the reduction of electrons velocity in the channel.

In this paper we establish the Lp-boundedness properties of the variation operators associated with the heat semigroup, Riesz transforms and commutator between Riesz transforms and multiplication by BMO(ℝn)-functions in the Schrödinger setting. Fil: Betancor, J. J.. Universidad de la Laguna; España Fil: Fariña, J. C.. Universidad de la Laguna; España Fil: Harboure, Eleonor Ofelia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina Fil: Rodriguez Mesa, L.. Universidad de la Laguna; España

Noether’s Theorem relates the Action Integral of a Lagrangian with symmetries which leave it invariant and the first integrals consequent upon the variational principle and the existence of the symmetries. These each have an equivalent in the Schrödinger Equation corresponding to the Lagrangian and by extension to linear evolution equations in general. The implications of these connections are investigated.

By virtue of numerical arguments we study a bifurcation phenomenon occurring for a class of minimization problems associated with the quasi-linear Schrodinger equation., 9 pages

This article deals with the numerical calculation of eigenvalues of perturbed periodic Schr\"odinger operators located in spectral gaps. Such operators are encountered in the modeling of the electronic structure of crystals with local defects, and of photonic crystals. The usual finite element Galerkin approximation is known to give rise to spectral pollution. In this article, we give a precise description of the corresponding spurious states. We then prove that the supercell model does not produce spectral pollution. Lastly, we extend results by Lewin and S\'er\'e on some no-pollution criteria. In particular, we prove that using approximate spectral projectors enables one to eliminate spectral pollution in a given spectral gap of the reference periodic Sch\"odinger operator., Comment: 23 pages, 5 figures

We consider the Schrödinger equation for the harmonic oscillator i∂tu=Hu, where H=-Δ+{pipe}x{pipe}2, with initial data in the Hermite-Sobolev space H-s/2L2(ℝn). We obtain smoothing and maximal estimates and apply these to perturbations of the equation and almost everywhere convergence problems. Fil: Bongioanni, Bruno. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina Fil: Rogers, Keith M.. Instituto de Ciencias Matemáticas; España

The piecewise perturbation methods (PPM) have proven to be very efficient for the numerical solution of the linear time-independent Schrodinger equation. The underlying idea is to replace the potential function piecewisely by simpler approximations and then to solve the approximating problem. The accuracy is improved by adding some perturbation corrections. Two types of approximating potentials were considered in the literature, that is piecewise constant and piecewise linear functions, giving rise to the so-called CP methods (CPM) and LP methods (LPM). Piecewise polynomials of higher degree have not been used since the approximating problem is not easy to integrate analytically. As suggested by Ixaru (Comput Phys Commun 177:897–907, 2007), this problem can be circumvented using another perturbative approach to construct an expression for the solution of the approximating problem. In this paper, we show that there is, however, no need to consider PPM based on higher-order polynomials, since these methods are equivalent to the CPM. Also, LPM is equivalent to CPM, although it was sometimes suggested in the literature that an LP method is more suited for problems with strongly varying potentials. We advocate that CP schemes can (and should) be used in all cases, since it forms the most straightforward way of devising PPM and there is no advantage in considering other piecewise polynomial perturbation methods.

The propagation in time of a wavepacket is a conceptually rich problem suitable to be studied in any introductory quantum mechanics course. This subject is covered analytically in most of the standard textbooks. Computer simulations have become a widespread pedagogical tool, easily implemented in computer labs and in classroom demonstrations. However, we have detected issues raising difficulties in the practical effectuation of these codes which are especially evident when discrete grid methods are used. One issue—relatively well known—appears at high incident energies, producing a wavepacket slower than expected theoretically. The other issue, which appears at low wavepacket energies, does not affect the time evolution of the propagating wavepacket proper, but produces dramatic effects on its spectral decomposition. The origin of the troubles is investigated, and different ways to deal with these issues are proposed. Finally, we show how this problem is manifested and solved in the practical case of the electronic spectra of a metal surface ionized by an ultrashort laser pulse. Fil: Garriz, Abel Esteban. Consejo Nacional de Investigaciónes Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Astronomía y Física del Espacio. - Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Astronomía y Física del Espacio; Argentina Fil: Sztrajman, Alejandro. Consejo Nacional de Investigaciónes Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Astronomía y Física del Espacio. - Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Astronomía y Física del Espacio; Argentina Fil: Mitnik, Dario Marcelo. Consejo Nacional de Investigaciónes Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Astronomía y Física del Espacio. - Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Astronomía y Física del Espacio; Argentina

We prove the invariance of the Gibbs measure for the periodic Schrodinger-Benjamin-Ono system (when the coupling parameter |\gamma| \ne 0, 1) by establishing a new local well-posedness in a modified Sobolev space and constructing the Gibbs measure (which is in the sub-L^2 setting for the Benjamin-Ono part.) We also show the ill-posedness result in H^s(\mathbb{T}) \times H^{s-{1/2}}(\mathbb{T}) for s < {1/2} when |\gamma| \ne 0, 1 and for any s \in \mathbb{R} when |\gamma| =1., Comment: 22 pages

International audience; We prove global weighted Strichartz estimates for radial solutions of linear Schrödinger equation on a class of rotationally symmetric noncompact manifolds, generalizing the known results on hyperbolic and Damek-Ricci spaces. This yields classical Strichartz estimates with a larger class of exponents than in the Euclidian case and improvements for the scattering theory. The manifolds, whose volume element grows polynomially or exponentially at infinity, are characterized essentially by negativity conditions on the curvature, which shows in particular that the rich algebraic structure of the Hyperbolic and Damek-Ricci spaces is not the cause of the improved dispersive properties of the equation. The proofs are based on known dispersive results for the equation with potential on the Euclidean space, and on a new one, valid for C^1 potentials decaying like 1/r^2 at infinity.

Charge transport has been simulated in a novel two quantum well InGaN/GaN light emitting diode. Asymmetric tunnelling for holes and electrons has been used to enhance the quantum efficiency of the diode. A self-consistent solution of Poisson and Schrödinger equations has been used to obtain the band profile, energy levels and wave functions. Transport in the bulk nitride has been simulated by a drift diffusion model. Lattice strain and the resulting piezoelectric field effects have been shown to influence the device characteristics.

The hot-electron two-dimensional HEMT with recessed gate is modelled by solving the Poisson, current continuity and energy transport equations consistently with the Schrödinger equation using a finite difference scheme. New expressions are used for the energy densities inside and outside the quantum wells. A method is described for pinning the conduction band at the contact edge to produce an extremely stable numerical solution. Results are presented for an eight layer GaAs-ALGaAs-InGaAs device.

In a previous article we have shown how one can employ Artificial Neural Networks (ANNs) in order to solve non-homogeneous ordinary and partial differential equations. In the present work we consider the solution of eigenvalue problems for differential and integrodifferential operators, using ANNs. We start by considering the Schr\"odinger equation for the Morse potential that has an analytically known solution, to test the accuracy of the method. We then proceed with the Schr\"odinger and the Dirac equations for a muonic atom, as well as with a non-local Schr\"odinger integrodifferential equation that models the $n+\alpha$ system in the framework of the resonating group method. In two dimensions we consider the well studied Henon-Heiles Hamiltonian and in three dimensions the model problem of three coupled anharmonic oscillators. The method in all of the treated cases proved to be highly accurate, robust and efficient. Hence it is a promising tool for tackling problems of higher complexity and dimensionality., Comment: Latex file, 29pages, 11 psfigs, submitted in CPC

The author extends his alternative theory for Schrödinger quantum mechanics by introducing the idea of energy reference strata over configuration space. It is then shown that the view from various such strata defines, the content of the system of interest and enables a variety of different descriptions of events in the same space time region. Thus according to “the point of view” or energy stratum chosen so the type of Schrödinger equation, linear or otherwise, appropriate to describe the system is determined. A nonlinear information channel between two dimensional fluid action in hyperspace into two dimensional energy hyperspace is shown to exist generally as a background to nonlinear Schrödinger structures. In addition it is shown how soliton solutions of the one dimensional Schrödinger equation are related to two dimensional vortex fields in hyperspace.

A two-dimensional classical hyperspace fluid theory of the vacuum which is an alternative to Schrödinger quantum theory involves a “stirring” energy density. This energy density, like entropy density, measures the local “disturbance” involved by the time evolution of a state of the system. The stirring energy density is shown to be the local source term for the rate of generation of action density with time. A classical fluid action principle is solved using the “action” of this source term without assuming quantum time evolution. Schrödinger quantum theory is recovered from this new action principle which is then discussed. Finally, the quantity made stationary by the variational principle measures disturbance in the environment as an angular change in the orientation of the whole system.