Near-Dirichlet quantum dynamics for a \(p^3\)-corrected particle on an interval

  • Jorma LoukoEmail author
Research Article


We study a nonrelativistic quantum mechanical particle on an interval of finite length with a Hamiltonian that has a \(p^3\) correction term, modelling potential low energy quantum gravity effects. We describe explicitly the \(U(3)\) family of the self-adjoint extensions of the Hamiltonian and discuss several subfamilies of interest. As the main result, we find a family of self-adjoint Hamiltonians, indexed by four continuous parameters and one binary parameter, whose spectrum and eigenfunctions are perturbatively close to those of the uncorrected particle with Dirichlet boundary conditions, even though the Dirichlet condition as such is not in the \(U(3)\) family. Our boundary conditions do not single out distinguished discrete values for the length of the interval in terms of the underlying quantum gravity scale.


Low energy quantum gravity Higher derivative quantum mechanics Quantum mechanics on an interval Boundary conditions in quantum mechanics Self-adjoint extensions 



I thank Saurya Das and Elias Vagenas for helpful correspondence and an anonymous referee for helpful comments. This work was supported in part by STFC (Theory Consolidated Grant ST/J000388/1).


  1. 1.
    Amelino-Camelia, G.: Quantum-spacetime phenomenology. Living Rev. Rel. 16, 5 (2013). arXiv:0806.0339 [gr-qc]Google Scholar
  2. 2.
    Kempf, A., Mangano, G., Mann, R.B.: Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D 52, 1108 (1995). arXiv:hep-th/9412167 CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Husain, V., Kothawala, D., Seahra, S.S.: Generalized uncertainty principles and quantum field theory. Phys. Rev. D 87, 025014 (2013). arXiv:1208.5761 [hep-th]CrossRefADSGoogle Scholar
  4. 4.
    Ali, A.F., Das, S., Vagenas, E.C.: Discreteness of space from the generalized uncertainty principle. Phys. Lett. B 678, 497 (2009). arXiv:0906.5396 [hep-th]CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness. Academic, New York (1975)zbMATHGoogle Scholar
  6. 6.
    Blank, J., Exner, P., Havlíček, M.: Hilbert Space Operators in Quantum Physics, 2nd edn. Springer, New York (2008)zbMATHGoogle Scholar
  7. 7.
    Bonneau, G., Faraut, J., Valent, G.: Selfadjoint extensions of operators and the teaching of quantum mechanics. Am. J. Phys. 69, 322 (2001). arXiv:quant-ph/0103153 CrossRefADSGoogle Scholar
  8. 8.
    Asorey, M., Ibort, A., Marmo, G.: Global theory of quantum boundary conditions and topology change. Int. J. Mod. Phys. A 20, 1001 (2005). arXiv:hep-th/0403048 CrossRefADSzbMATHMathSciNetGoogle Scholar
  9. 9.
    Ibort, A., Pérez-Pardo, J.M.: Numerical solutions of the spectral problem for arbitrary self-adjoint extensions of the one-dimensional Schrödinger equation. SIAM J. Numer. Anal. 51, 1254 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Asorey, M., Muñoz-Castañeda, J.M.: Attractive and repulsive Casimir vacuum energy with general boundary conditions. Nucl. Phys. B 874, 852 (2013). arXiv:1306.4370 [hep-th]CrossRefADSzbMATHGoogle Scholar
  11. 11.
    Asorey, M., Balachandran, A.P., Pérez-Pardo, J.M.: Edge states: topological insulators, superconductors and QCD chiral bags. JHEP 1312, 073 (2013). arXiv:1308.5635 [cond-mat.mtrl-sci]CrossRefADSGoogle Scholar
  12. 12.
    Muñoz-Castañeda, J.M., Kirsten, K., Bordag, M.: QFT over the finite line. Heat kernel coefficients, spectral zeta functions and selfadjoint extensions. Lett. Math. Phys. 105, 523 (2015)CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Balasubramanian, V., Das, S., Vagenas, E.C.: Generalized uncertainty principle and self-adjoint operators. arXiv:1404.3962 [hep-th]
  14. 14.
    Belchev, B., Walton, M.A.: Robin boundary conditions and the Morse potential in quantum mechanics. J. Phys. A 43, 085301 (2010). arXiv:1002.2139 [quant-ph]CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Simon, J.Z.: Higher-derivative Lagrangians, nonlocality, problems, and solutions. Phys. Rev. D 41, 3720 (1990)CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Louko, J., Marples, C.R.: Unpublished (2014)Google Scholar
  17. 17.
    Kochubei, A.N.: Extensions of symmetric operators and symmetric binary relations. Math. Notes 17, 25 (1975)CrossRefGoogle Scholar
  18. 18.
    Brüning, J., Geyler, V., Pankrashkin, K.: Spectra of self-adjoint extensions and applications to solvable Schrödinger operators. Rev. Math. Phys. 20, 1 (2008)CrossRefzbMATHMathSciNetGoogle Scholar

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of NottinghamNottinghamUK

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