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.
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. ...
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.
Qihong Shi, Yaqian Jia, Jianwei Yang (2024)
Applications of Mathematics
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We investigate the Cauchy problem of the one dimensional Maxwell-Schrödinger (MS) system under the Lorenz gauge condition. Different from the classical case, we consider the electromagnetic and electrostatic potentials which are growing at space infinity. More precisely, the electrostatic potential is allowed to grow linearly, while for the electromagnetic potential the growth is sublinear. Based on the energy estimates and the gauge transformation, we prove the global existence and...
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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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...
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...