Skip to main content
SpringerLink
Log in

Notes on the Cauchy problem for the self-adjoint and non-self-adjoint Schrödinger equations with polynomially growing potentials

  • Published:
Journal of Pseudo-Differential Operators and Applications Aims and scope Submit manuscript

Abstract

The Cauchy problem is studied for the self-adjoint and non-self-adjoint Schrödinger equations. We first prove the existence and uniqueness of solutions in the weighted Sobolev spaces. Secondly we prove that if potentials are depending continuously and differentiably on a parameter, so are the solutions, respectively. The non-self-adjoint Schrödinger equations that we study are those used in the theory of continuous quantum measurements. The results on the existence and uniqueness of solutions in the weighted Sobolev spaces will play a crucial role in the proof for the convergence of the Feynman path integrals in the theories of quantum mechanics and continuous quantum measurements.

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. Albeverio, S.A., Høegh-Krohn, R.J., Mazzucchi, S.: Mathematical Theory of Feynman Path Integral. An Introduction. Lecture Notes in Mathematics, vol. 523. Springer, Berlin (2008)

    Book  Google Scholar 

  2. Berezin, F.A., Shubin, M.A.: The Schrödinger Equation. Kluwer Academic Publishers, Dordrecht (1991)

    Book  Google Scholar 

  3. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Springer, Berlin (1987)

    Book  Google Scholar 

  4. Davies, E.B.: Non-self-adjoint differential operators. Bull. Lond. Math. Soc. 34, 513–532 (2002)

    Article  MathSciNet  Google Scholar 

  5. Ichinose, W.: A note on the existence and \(\hbar \)-dependency of the solution of equations in quantum mechanics. Osaka J. Math. 32, 327–345 (1995)

    MathSciNet  MATH  Google Scholar 

  6. Ichinose, W.: On convergence of the Feynman path integral formulated through broken line paths. Rev. Math. Phys. 11, 1001–1025 (1999)

    Article  MathSciNet  Google Scholar 

  7. Ichinose, W.: The continuity and the differentiability of solutions on parameters to the Schrödinger equations and the Dirac equations. J. Pseudo-Differ. Oper. Appl. 3, 399–419 (2012)

    Article  MathSciNet  Google Scholar 

  8. Ichinose, W.: On the Feynman path integral for the Dirac equation in the general dimensional spacetime. Commun. Math. Phys. 329, 483–508 (2014)

    Article  MathSciNet  Google Scholar 

  9. Ichinose, W.: On the Feynman path integral for the Schrödinger equations with potentials growing polynomially in the spatial direction (2019). arXiv:1901.05677

  10. Ichinose, W.: Mathematical theory of the Feynman path integrals for continuous quantum measurements of positions of particles with spin (in preparation)

  11. Kumano-go, H.: Pseudo-Differential Operators. MIT Press, Cambridge (1981)

    MATH  Google Scholar 

  12. Leinfelder, H., Simader, C.G.: Schrödinger operators with singular magnetic vector potentials. Math. Z. 176, 1–19 (1981)

    Article  MathSciNet  Google Scholar 

  13. Mensky, M.B.: Continuous Quantum Measurements and Path Integrals. IOP Publishing, Bristol (1993)

    MATH  Google Scholar 

  14. Mensky, M.B.: Quantum Measurements and Decoherence. Kluwer Academic Publishers, Dordrecht (2000)

    Book  Google Scholar 

  15. Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis, Revised and, Enlarged Edition. Academic Press, San Diego (1980)

    Google Scholar 

  16. Reed, M., Simon, B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness. Academic Press, San Diego (1975)

    MATH  Google Scholar 

  17. Sambou, D.: Lieb-Thirring type inequalities for non-self-adjoint perturbations of magnetic Schrödinger operators. J. Funct. Anal. 266, 5016–5044 (2014)

    Article  MathSciNet  Google Scholar 

  18. Yajima, K.: Schrödinger evolution equations with magnetic fields. J. Anal. Math. 56, 29–76 (1991)

    Article  Google Scholar 

  19. Yajima, K.: Schrödinger equations with time-dependent unbounded singular potentials. Rev. Math. Phys. 23, 823–838 (2011)

    Article  MathSciNet  Google Scholar 

  20. Yajima, K.: Existence and regularity of propagators for multi-particle Schrödinger equations in external fields. Commun. Math. Phys. 347, 103–126 (2016)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shinshu University, Matsumoto, 390-8621, Japan

    Wataru Ichinose & Takayoshi Aoki

Authors
  1. Wataru Ichinose
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takayoshi Aoki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wataru Ichinose.

Additional information

Publisher's Note

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

Wataru Ichinose: This research is partially supported by JSPS KAKENHI Grant No. 26400161, JP18K03361 and Shinshu University. Takayoshi Aoki: This research is partially supported by Shinshu University RA Grant No. 25204513.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ichinose, W., Aoki, T. Notes on the Cauchy problem for the self-adjoint and non-self-adjoint Schrödinger equations with polynomially growing potentials. J. Pseudo-Differ. Oper. Appl. 11, 703–731 (2020). https://doi.org/10.1007/s11868-019-00301-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11868-019-00301-6

Keywords

Mathematics Subject Classification

Search

Navigation