zbMATH — the first resource for mathematics

Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials. (English) Zbl 1246.35189
Summary: The wellposedness of nonlinear Schrödinger equations (NLS) with inverse-square potentials is discussed in this article. The usual (NLS) is regarded as the potential-free case. The wellposedness of the usual (NLS) is well-known for a long time. In fact, several methods have been developed up to now. Among others, the Strichartz estimates seem to be essential in addition to the restriction on the nonlinear term caused by the Gagliardo-Nirengerg inequality. However, a parallel argument is not available when we apply such estimates to (NLS) with inverse-square potentials. Thus, we shall give only partial answer to the question in this article.

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q40 PDEs in connection with quantum mechanics
81Q15 Perturbation theories for operators and differential equations in quantum theory
Full Text: DOI
[1] DOI: 10.1016/0022-1236(79)90076-4 · Zbl 0396.35028
[2] Kato T, Ann. Inst. H. Poincaré, Phys. Théor. 46 pp 113– (1987)
[3] Tsutsumi Y, Funkcial. Ekvac. 30 pp 115– (1987)
[4] Kato T, Schrödinger Operators, Lecture Notes in Physics 345 pp 218– (1989)
[5] Cazenave T, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10 (2003)
[6] Sulem C, Nonlinear Schrödinger Equation, Applied Mathematical Sciences 139 (1999)
[7] Planchon F, Discr. Contin. Dyn. Syst. 9 pp 427– (2003)
[8] DOI: 10.1006/jfan.1999.3556 · Zbl 0953.35053
[9] Tanabe H, Equations of Evolution, Monographs and Studies in Mathematics 6 (1979)
[10] Cazenave T, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications 13 (1998) · Zbl 0926.35049
[11] DOI: 10.1016/S0022-1236(03)00238-6 · Zbl 1030.35024
[12] DOI: 10.3934/dcds.2010.28.311 · Zbl 1198.47089
[13] DOI: 10.1007/s00220-003-1004-4 · Zbl 1072.35171
[14] Bergh J, Interpolation Spaces, An Introduction, Grundlehren der Mathematics Wissenschaften 223 (1976)
[15] DiBenedetto E, Birkhäuser Advanced Texts: Basler Lehrbücher (2002)
[16] DOI: 10.1006/jfan.2000.3687 · Zbl 0974.47025
[17] Tao, T.Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, Vol. 106, AMS, Providence, RI, 2006 · Zbl 1106.35001
[18] DOI: 10.1007/s00526-005-0349-2 · Zbl 1089.35071
[19] Okazawa N, Japan. J. Math. 22 pp 199– (1996)
[20] DOI: 10.5802/aif.1463 · Zbl 0818.35021
[21] DOI: 10.1007/s00440-005-0437-4 · Zbl 1094.53034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.