Displaying similar documents to “New method to solve certain differential equations”

Solutions of the time-independent Schrödinger equation by uniformization on the unit circle

Kazimierz Rajchel (2019)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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The idea presented here of a general quantization rule for bound states is mainly based on the Riccati equation which is a result of the transformed, time-independent, one-dimensional Schrödinger equation. The condition imposed on the logarithmic derivative of the ground state function W0 allows to present the Riccati equation as the unit circle equation with winding number equal to one which, by appropriately chosen transformations, can be converted into the unit circle equation with...

Stationary solutions of semilinear Schrödinger equations with trapping potentials in supercritical dimensions

Filip Ficek (2023)

Archivum Mathematicum

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Nonlinear Schrödinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of existence of the ground states in critical and supercritical cases. We formulate the assumptions on the system that are sufficient for this method to work. As examples, we consider Schrödinger-Newton and Gross-Pitaevskii equations with harmonic potentials. ...

Weighted Dispersive Estimates for Solutions of the Schrödinger Equation

Cardoso, Fernando, Cuevas, Claudio, Vodev, Georgi (2008)

Serdica Mathematical Journal

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2000 Mathematics Subject Classification: 35L15, 35B40, 47F05. Introduction and statement of results. In the present paper we will be interested in studying the decay properties of the Schrödinger group. The authors have been supported by the agreement Brazil-France in Mathematics – Proc. 69.0014/01-5. The first two authors have also been partially supported by the CNPq-Brazil.

Dispersion Phenomena in Dunkl-Schrödinger Equation and Applications

Mejjaoli, H. (2009)

Serdica Mathematical Journal

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2000 Mathematics Subject Classification: 35Q55,42B10. In this paper, we study the Schrödinger equation associated with the Dunkl operators, we study the dispersive phenomena and we prove the Strichartz estimates for this equation. Some applications are discussed.

Weak Asymptotics for Schrödinger Evolution

S. A. Denisov (2010)

Mathematical Modelling of Natural Phenomena

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In this short note, we apply the technique developed in [Math. Model. Nat. Phenom., 5 (2010), No. 4, 122-149] to study the long-time evolution for Schrödinger equation with slowly decaying potential.

Introduction to algorithms for molecular simulations

Kramář, Martin

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In the first part of the paper we survey some algorithms which describe time evolution of interacting particles in a bounded domain. Applications to macroscale as well as microscale are presented on two examples: motion of planets and collision of two bodies. In the second part of the paper we present solution to stationary Schrödinger equation for simple molecular models.

Hardy's uncertainty principle, convexity and Schrödinger evolutions

Luis Escauriaza, Carlos E. Kenig, G. Ponce, Luis Vega (2008)

Journal of the European Mathematical Society

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We prove the logarithmic convexity of certain quantities, which measure the quadratic exponential decay at infinity and within two characteristic hyperplanes of solutions of Schrödinger evolutions. As a consequence we obtain some uniqueness results that generalize (a weak form of) Hardy’s version of the uncertainty principle. We also obtain corresponding results for heat evolutions.

Numerical study of self-focusing solutions to the Schrödinger-Debye system

Christophe Besse, Brigitte Bidégaray (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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In this article we implement different numerical schemes to simulate the Schrödinger-Debye equations that occur in nonlinear optics. Since the existence of blow-up solutions is an open problem, we try to compute such solutions. The convergence of the methods is proved and simulations seem indeed to show that for at least small delays self-focusing solutions may exist.

Resonances of two-dimensional Schrödinger operators with strong magnetic fields

Tuan Duong, Anh (2012)

Serdica Mathematical Journal

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2010 Mathematics Subject Classification: 81Q20 (35P25, 81V10). The purpose of this paper is to study the Schrödinger operator P(B,w) = (Dx-By^2+Dy^2+w^2x^2+V(x,y),(x,y) О R^2, with the magnetic field B large enough and the constant w № 0 is fixed and proportional to the strength of the electric field. Under certain assumptions on the potential V, we prove the existence of resonances near Landau levels as B®Ґ. Moreover, we show that the width of resonances is of size O(B^-Ґ). ...

Semiclassical states for weakly coupled nonlinear Schrödinger systems

Eugenio Montefusco, Benedetta Pellacci, Marco Squassina (2008)

Journal of the European Mathematical Society

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We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.