Skip to main content
SpringerLink
Log in

The Schrödinger equation on cylinders and the n-torus

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

In this paper, we study the solutions to the Schrödinger equation on some conformally flat cylinders and on the n-torus. First, we apply an appropriate regularization procedure. Using the Clifford algebra calculus with an appropriate Witt basis, the solutions can be expressed as multiperiodic eigensolutions to the regularized parabolic-type Dirac operator. We study their fundamental properties, give representation formulas of all these solutions in terms of multiperiodic generalizations of the elliptic functions in the context of the regularized parabolic-type Dirac operator. Furthermore, we also develop some integral representation formulas. In particular, we set up a Green type integral formula for the solutions to the homogeneous regularized Schrödinger equation on cylinders and n-tori. Then, we treat the inhomogeneous Schrödinger equation with prescribed boundary conditions in Lipschitz domains on these manifolds. We present an L p -decomposition where one of the components is the kernel of the first-order differential operator that factorizes the cylindrical (resp. toroidal) Schrödinger operator. Finally, we study the behavior of our results in the limit case where the regularization parameter tends to zero.

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. Artino, R., Barros-Neto, J.: Hypoelliptic Boundary-Value Problems, Lectures Notes in Pure and Applied Mathematics 53, Marcel Dekker: New York; (1980).

  2. Cerejeiras P., Vieira N.: Regularization of the non-stationary Schrödinger operator. Math. Meth. in Appl. Sc. 32(4), 535–555 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cerejeiras P., Kähler U., Sommen F.: Parabolic Dirac operators and the Navier-Stokes equations over time-varying domains. Math. Meth. in the Appl. Sc. 28, 1715–1724 (2005)

    Article  MATH  Google Scholar 

  4. Cerejeiras P., Kähler U.: Elliptic Boundary Value Problems of Fluid Dynamics over Unbounded Domains. Math. Meth. in Appl. Sc. 23, 81–101 (2000)

    Article  MATH  Google Scholar 

  5. Delanghe, R., Sommen, F., Souçek, V., Clifford algebras and spinor-valued functions, Kluwer Academic Publishers, 1992.

  6. Dix, D.: D., Application of Clifford analysis to inverse scattering for the linear hierarchy in several space dimensions, In: Ryan, J. (ed.), pp. 260–282, CRC Press, Boca Raton, FL (1995).

  7. Gürlebeck, K., Sprößig, W.: Quaternionic and Clifford calculus for physicists and engineers, John Wiley and Sons, (1997).

  8. Kraußhar R.S., Ryan J.: Some Conformally Flat Spin Manifolds, Dirac Operators and Automorphic Forms. Journal of Mathematical Analysis and its Applications 325(1), 359–376 (2007)

    Article  MATH  Google Scholar 

  9. Kraußhar R.S., Ryan J.: Clifford and harmonic analysis on cylinders and tori. Revista Matematica Iberoamericana 21, 87–110 (2005)

    MathSciNet  MATH  Google Scholar 

  10. Kraußhar, R.S.: Generalized Analytic Automorphic Forms in Hypercomplex Spaces, Frontiers in Mathematics, Birkhäuser, Basel, (2004).

  11. Kravchenko V.G., Kravchenko V.V.: Quaternionic Factorization of the Schrödinger operator and its applications to some first-order systems of mathematical physics. J. Phys. A: Math. Gen. 36(44), 11285–11297 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Shapiro, M., Kravchenko, V.V.: Integral Representation for spatial models of mathematical physics, Pitman research notes in mathematics series 351, Harlow, Longman, (1996).

  13. Tao, T.: Nonlinear Dispersive Equations, Local and Global Analysis, CBMS Regional Conference Series in Mathematics 106, American Mathematical Society: Providence, RI, (2006).

  14. Velo, V.: Mathematical Aspects of the nonlinear Schrödinger Equation, Proceedings of the Euroconference on nonlinear Klein-Gordon and Schrödinger systems: theory and applications, Vázquez, Luis et al.(ed.), pp. 39–67 World Scientific, Singapore (1996).

  15. Vieira, N.: Theory of the parabolic Dirac operator and its applications to non-linear differential equations, PhD Thesis, Univerity of Aveiro, (2009).

  16. Vieira N.: Powers of the parabolic Dirac operator. Adv. Appl. Clifford Algebr. 18, 1023–1032 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstraße 7, 64289, Darmstadt, Germany

    R. S. Kraußhar

  2. Department of Mathematic, Faculty of Science, University of Porto, Rua do Campo Alegre, 4169-007, Porto, Portugal

    N. Vieira

Authors
  1. R. S. Kraußhar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. Vieira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. Vieira.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kraußhar, R.S., Vieira, N. The Schrödinger equation on cylinders and the n-torus. J. Evol. Equ. 11, 215–237 (2011). https://doi.org/10.1007/s00028-010-0089-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00028-010-0089-4

Mathematics Subject Classification (2000)

Keywords

Search

Navigation