Skip to main content
SpringerLink
Log in

Explicit Solutions of Cylindrical Schrödinger Equation with Radial Potentials

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

A modified cylindrical Poisson–Schrödinger integral is constructed in terms of Nevanlinna norm associated with cylindrical Schrödinger equation, then with the help of Carleman–Schrödinger formula, explicit solutions of the equation mentioned above can be generated via the Schrödinger integral representation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Afrouzi, G.A., Mirzapour, M., Rădulescu, V.D.: Variational analysis of anisotropic Schrödinger equations without Ambrosetti-Rabinowitz-type condition. Z. Angew. Math. Phys. (2018). https://doi.org/10.1007/s00033-017-0900-y

    Article  MATH  Google Scholar 

  2. Anceschi, F., Polidoro, S., Ragusa, M.A.: Moser’s estimates for degenerate Kolmogorov equations with non-negative divergence lower order coefficients. Nonlinear Anal. (2019). https://doi.org/10.1016/j.na.2019.07.001

    Article  MathSciNet  MATH  Google Scholar 

  3. Ardila, A.: Orbital stability of standing waves for a system of nonlinear Schrödinger equations with three wave interaction. Nonlinear Anal. 167, 1–20 (2018)

    Article  MathSciNet  Google Scholar 

  4. Bahrouni, A., Ounaies, H., Rădulescu, V.D.: Bound state solutions of sublinear Schrödinger equations with lack of compactness. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 113(2), 1191–1210 (2019)

  5. Bisci, G.M., Rădulescu, V.D.: Ground state solutions of scalar field fractional Schrödinger equations. Calc. Var. Partial Differ. Equ. 54(3), 2985–3008 (2015)

    Article  Google Scholar 

  6. Colorado, E.: On the existence of bound and ground states for some coupled nonlinear Schrödinger-Korteweg-de Vries equations. Adv. Nonlinear Anal. 6(4), 407–426 (2017)

    Article  MathSciNet  Google Scholar 

  7. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1. Interscience Publishers, New York (1953)

    MATH  Google Scholar 

  8. Cranston, M.: Conditional Brownian Motion, Whitney Squares and the Conditional Gauge Theorm. Seminar on Stochastic Processes, 1988, pp. 109–119. Birkhäuser Verlag, Basel (1989)

  9. Escassut, A., Tutschke, W., Yang, C.C.: Some Topics on Value Distribution and Differentiability in Complex and P-adic Analysis. Science Press, Beijing (2008)

    MATH  Google Scholar 

  10. Finkelstein, M., Scheinberg, S.: Kernels for solving problems of Dirichlet typer in a half-plane. Adv. Math. 18(1), 108–113 (1975)

    Article  Google Scholar 

  11. Goubet, O., Hamraoui, E.: Blow-up of solutions to cubic nonlinear Schrödinger equations with defect: the radial case. Adv. Nonlinear Anal. 6(2), 183–197 (2017)

    Article  MathSciNet  Google Scholar 

  12. Goodrich, C.S., Ragusa, M.A.: Hölder continuity of weak solutions of \(p\)-Laplacian PDEs with VMO coefficients. Nonlinear Anal. 185, 336–355 (2019)

    Article  MathSciNet  Google Scholar 

  13. Guliyev, V.S., Guliyev, R.V., Omarova, M.N., Ragusa, M.A.: Schrödinger type operators on local generalized Morrey spaces related to certain nonnegative potentials. Discrete Contin. Dyn. Syst. Ser. B 25(2), 671–690 (2020)

    MathSciNet  MATH  Google Scholar 

  14. Jäger, W., Saitō, Y.: The uniqueness of the solution of the Schrödinger equation with discontinuous coefficients. Rev. Math. Phys. 10(7), 963–987 (1998)

    Article  MathSciNet  Google Scholar 

  15. Kassay, G., Rădulescu, V.D.: Equilibrium Problems and Applications. Mathematics in Science and Engineering. Elsevier/Academic Press, London (2018)

    MATH  Google Scholar 

  16. Miyamoto, I.: A type of uniqueness of solutions for the Dirichlet problem on a cylinder. Tohoku Math. J. 48(2), 267–292 (1996)

    Article  MathSciNet  Google Scholar 

  17. Mizohata, S.: On some Schrödinger type equations. Proc. Jpn. Acad. Ser. A Math. Sci. 57(2), 81–84 (1981)

    Article  Google Scholar 

  18. Polidoro, S., Ragusa, M.A.: Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term. Rev. Mat. Iberoam. 24(3), 1011–1046 (2008)

    Article  MathSciNet  Google Scholar 

  19. Qiao, L.: Weak solutions for the stationary Schrödinger equation and its application. Appl. Math. Lett. 63, 34–39 (2017)

    Article  MathSciNet  Google Scholar 

  20. Qiao, L.: Solutions of the Dirichlet-Sch problem and asymptotic properties of solutions for the Schrödinger equation. Appl. Math. Lett. 71, 44–50 (2017)

    Article  MathSciNet  Google Scholar 

  21. Ragusa, M.A., Tachikawa, A.: Boundary regularity of minimizers of \(p(x)\)-energy functionals. Ann. L’Inst. Poincaré Anal. Nonlineaire 33(2), 451–476 (2016)

    Article  MathSciNet  Google Scholar 

  22. Rădulescu, V.D., Repovš, D.D.: Partial Differential Equations with Variable Exponents. Variational Methods and Qualitative Analysis. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL (2015)

  23. Rozenblyum, G.V., Solomyak, M.Z., Shubin, M.A.: Spectral Theory of Differential Operators. (Russian) Current Problems in Mathematics. Fundamental Directions, vol. 64 (Russian), pp. 5–248. Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (1989)

  24. Rybalko, Y.: Initial value problem for the time-dependent linear Schrödinger equation with a point singular potential by the unified transform method. Opuscula Math. 38(6), 883–898 (2018)

    Article  MathSciNet  Google Scholar 

  25. Simon, B.: Schrödinger semigroups. Bull. Am. Math. Soc. (N.S.) 7, 447–526 (1982)

    Article  Google Scholar 

  26. Xue, Y., Tang, C.: Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth. Commun. Pure Appl. Anal. 17(3), 1121–1145 (2018)

    Article  MathSciNet  Google Scholar 

  27. Yoshida, H.: Harmonic majorization of a subharmonic function on a cone or on a cylinder. Pac. J. Math. 148(2), 369–395 (1991)

    Article  MathSciNet  Google Scholar 

  28. Yoshida, H.: A type of uniqueness for the Dirichlet problem on a half-space with continuous data. Pac. J. Math. 172(2), 591–609 (1996)

    Article  MathSciNet  Google Scholar 

  29. Yoshida, H., Miyamoto, I.: Solutions of the Dirichlet problem on a cone with continuous data. J. Math. Soc. Jpn. 50(1), 71–93 (1998)

    Article  MathSciNet  Google Scholar 

  30. Zhang, Y.: The Phragemén–Lindelöf principle of harmonic functions and conditions for Nevanlinna class in the half space. Math. Methods Appl. Sci. 42(12), 4360–4364 (2019)

    Article  MathSciNet  Google Scholar 

  31. Zhang, Y., Deng, G., Qian, T.: Integral representations of a class of harmonic functions in the half space. J. Differ. Equ. 260(2), 923–936 (2016)

    Article  MathSciNet  Google Scholar 

  32. Zhang, Y., Kou, K., Deng, G.: Integral representation and asymptotic behavior of harmonic functions in half space. J. Differ. Equ. 257(8), 2753–2764 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thanks the referees for their valuable comments. This work was supported by the Foundation for University Young Key Teacher Program of Henan Province (Grant No. 2017GGJS085) and Young Talents Fund of HUEL.

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450046, China

    Lei Qiao

Authors
  1. Lei Qiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lei Qiao.

Ethics declarations

Author contribution

LQ: Conceptualization, Writing-original draft, Writing-review & editing.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qiao, L. Explicit Solutions of Cylindrical Schrödinger Equation with Radial Potentials. J Geom Anal 31, 5437–5449 (2021). https://doi.org/10.1007/s12220-020-00498-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-020-00498-9

Keywords

Mathematics Subject Classification

Search

Navigation