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

  • Jorma Louko
Research Article
  • 68 Downloads

Abstract

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.

Keywords

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

Notes

Acknowledgments

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).

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of NottinghamNottinghamUK

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