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.
Arne Jensen (1994)
Mathematische Annalen
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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.
Jacek Zienkiewicz (2004)
Studia Mathematica
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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). ...
Veronica Felli, Alberto Ferrero, Susanna Terracini (2011)
Journal of the European Mathematical Society
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Asymptotics of solutions to Schrödinger equations with singular magnetic and electric potentials is investigated. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis–Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order −1.
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.
Zhengping Wang, Huan-Song Zhou (2009)
Journal of the European Mathematical Society
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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.
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.
Nakao Hayashi, Tohru Ozawa (1988/89)
Mathematische Zeitschrift
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G. Nakamura, Z. Sun, G. Uhlmann (1995)
Mathematische Annalen
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Pierre Duclos, Markus Klein (1985)
Journées équations aux dérivées partielles
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M. Hoffmann-Ostenhof (1988)
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...
Arne Jensen (1986)
Mathematische Zeitschrift
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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.
Changxing Miao, Youbin Zhu (2006)
Annales Polonici Mathematici
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We consider the Cauchy problem for a generalized Klein-Gordon-Schrödinger system with Yukawa coupling. We prove the existence of global weak solutions by the compactness method and, through a special choice of the admissible pairs to match two types of equations, we prove the uniqueness of those solutions by an approach similar to the method presented by J. Ginibre and G. Velo for the pure Klein-Gordon equation or pure Schrödinger equation. Though it is very simple in form, the method...
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.