Startseite On separability of non-linear Schrodinger operators with matrix potentials
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

On separability of non-linear Schrodinger operators with matrix potentials

  • Hany A. Atia EMAIL logo , H. M. Abu-Donia und F. Mahmoud Ellaithy
Veröffentlicht/Copyright: 4. Mai 2021
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

In this paper, we studied the separability of the non-linear Schrodinger operator of the form S u x = Δ u x + V x , u u x ,  where  Δ u x = i = 1 n 2 u x x i 2 , with the non-linear matrix potential V x , u . We obtained the sufficient conditions for separability of this operator in the space L 2 R n and we established the suitable coercive inequalities.

AMS subject classification: 47F05; 58J99

Corresponding author: Hany A. Atia, Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

1. N. W. Everitt and M. Giertz, “Some properties of the domains of certain differential operators,” Proc. Lond. Math. Soc., vol. 3, no. 2, pp. 301–324, 1971.10.1112/plms/s3-23.2.301Suche in Google Scholar

2. W. N. Everitt and M. Giertz, “Inequalities and separation for Schrodinger type operators in L2(Rn),” Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci., vol. 79, no. 3–4, pp. 257–265, 1978. https://doi.org/10.1017/s0308210500019764.Suche in Google Scholar

3. W. N. Everitt and M. Giertz, “On some properties of the domains of powers of certain differential operators,” Proc. Lond. Math. Soc., vol. 3, no. 4, pp. 756–768, 1972. https://doi.org/10.1112/plms/s3-24.4.756.Suche in Google Scholar

4. W. N. Everitt and M. Giertz, “On some properties of the powers of a formally self-adjoint differential expression,” Proc. Lond. Math. Soc., vol. 3, no. 1, pp. 149–170, 1972. https://doi.org/10.1112/plms/s3-24.1.149.Suche in Google Scholar

5. K. H. Biomatov, “Separability theorem, weighted spaces and their applications to boundary value problems,” Dokl. Akad. Nauk SSSR, vol. 247, no. 3, pp. 532–536, 1979.Suche in Google Scholar

6. W. D. Evans and A. Zettl, “Dirichlet and separation results for Schrodinger-type operators,” Proc. - R. Soc. Edinburgh, Sect. A: Math. Phys. Sci., vol. 80, no. 1–2, pp. 151–162, 1978. https://doi.org/10.1017/s0308210500010180.Suche in Google Scholar

7. K. K. Biomatov, “Coercive estimates and separation for second order elliptic differential equations,” Soviet Math. Dokl., vol. 38, pp. 157–160, 1989.Suche in Google Scholar

8. A. S. Mohammed and H. A. Atia, “Separation of the Sturm–Liouville differential operator with an operator potential,” Appl. Math. Comput., vol. 156, no. 2, pp. 387–394, 2004. https://doi.org/10.1016/j.amc.2003.07.026.Suche in Google Scholar

9. A. Bergbaev, “Smooth solution of non-linear differential equation with matrix potential,” in The VII Scientific Conference of Mathematics and Mechanics, AlmaAta, 1989.Suche in Google Scholar

10. E. M. E. Zayed, A. S. Mohamed, and H. A. Atia, “Inequalities and separation for the Laplace Beltrami differential operator in Hilbert spacesî,” J. Math. Anal. Appl., vol. 336, pp. 81–92, 2007. https://doi.org/10.1016/j.jmaa.2006.07.031.Suche in Google Scholar

11. O. Milatovic, “A separation property for magnetic Schrodinger operators on Riemannian manifolds,” J. Geom. Phys., vol. 61, no. 1, pp. 1–7, 2011. https://doi.org/10.1016/j.geomphys.2010.09.001.Suche in Google Scholar

12. O. Milatovic, “Separation property for Schrodinger operators in L p-spaces on non-compact manifolds,” Complex Var. Elliptic Equ., vol. 58, no. 6, pp. 853–864, 2013. https://doi.org/10.1080/17476933.2011.625090.Suche in Google Scholar

13. H. A. Atia, “Magnetic bi-harmonic differential operators on Riemannian manifolds and the separation problem,” J. Contemp. Math. Anal., vol. 51, no. 5, pp. 222–226, 2016. https://doi.org/10.3103/s1068362316050022.Suche in Google Scholar

14. O. K. Karimov, “On coercive properties and separability of biharmonic operator with matrix potential,” Ufa Mathematical Journal, vol. 9, no. 1, pp. 54–61, 2017. https://doi.org/10.13108/2017-9-1-54.Suche in Google Scholar

Received: 2020-02-21
Accepted: 2021-03-29
Published Online: 2021-05-04
Published in Print: 2022-10-27

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 19.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ijnsns-2020-0036/html
Button zum nach oben scrollen