Near-Dirichlet quantum dynamics for a $p^3$-corrected particle on an interval

TL;DR

This paper investigates a quantum particle with a $p^3$ correction term on an interval, exploring self-adjoint extensions and boundary conditions that approximate standard spectra without fixing the interval length.

Contribution

It explicitly characterizes the $U(3)$ family of self-adjoint Hamiltonians with a $p^3$ correction, revealing new boundary conditions close to Dirichlet without fixing interval length.

Findings

Identifies a family of self-adjoint Hamiltonians with spectra close to uncorrected cases.

Shows boundary conditions do not determine a specific interval length related to quantum gravity.

Provides explicit parametrization of boundary conditions for the corrected Hamiltonian.

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.