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.
Giuseppe Maria Coclite (2002)
Annales Polonici Mathematici
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We prove the existence of a sequence of radial solutions with negative energy of the Schrödinger-Maxwell equations under the action of a negative potential.
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.
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.
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.
Antonio Ambrosetti (2008)
Bollettino dell'Unione Matematica Italiana
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We survey some recent results dealing with some classes of systems of nonlinear Schrödinger equations.
Chi-Hua Chan, Po-Chun Huang (2021)
Applications of Mathematics
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A special type of Jacobi matrices, discrete Schrödinger operators, is found to play an important role in quantum physics. In this paper, we show that given the spectrum of a discrete Schrödinger operator and the spectrum of the operator obtained by deleting the first row and the first column of it can determine the discrete Schrödinger operator uniquely, even though one eigenvalue of the latter is missing. Moreover, we find the forms of the discrete Schrödinger operators when their smallest...
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. ...
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.
Hitoshi Kitada (1988)
Mathematische Zeitschrift
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Arne Jensen (1986)
Mathematische Zeitschrift
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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...
Zhengping Wang, Huan-Song Zhou (2009)
Journal of the European Mathematical Society
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Arne Jensen (1994)
Mathematische Annalen
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M. Hoffmann-Ostenhof (1988)
Mathematische Zeitschrift
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Daniel Tataru (2012)
Journées Équations aux dérivées partielles
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The Schrödinger map equation is a geometric Schrödinger model, closely associated to the harmonic heat flow and to the wave map equation. The aim of these notes is to describe recent and ongoing work on this model, as well as a number of related open problems.