Perfect quantum reflection from a cliff: a potential unbounded from below

Abstract:We explain how a potential that is well-defined everywhere on the positive half-line, but is nowhere positive and diverges to $-\infty$ as $x\rightarrow 0^+$, can nevertheless confine a particle to the half-line and lead to well-defined dynamics. Such perfect reflection from a "cliff" potential is achieved by a purely dynamical mechanism, without the need to impose any boundary conditions at the edge. We discuss in detail the role of self-adjointness in ensuring dynamical closure at the quantum level, and advocate the principle that, when no boundary conditions or other physical data are given, the formal quantized Hamiltonian is most naturally defined on its maximal domain. We then construct an explicit cliff potential with the claimed properties, showing that the Hamiltonian is self-adjoint on this maximal domain and therefore defines a dynamically closed quantum system. Finally, we study its energy eigenstates, spectrum, and the phase shift associated with the reflection.

23 pages, 3 figures. Major rewriting of the earlier preprint "Quantum Bungee Jumping", with substantially strengthened physical and conceptual motivation and a more careful treatment of operator domains and self-adjointness

arXiv:2407.11137 [quant-ph]

(or arXiv:2407.11137v2 [quant-ph] for this version)