Year: 2026
Type: article
Abstract: Every realistic quantum system is coupled to an environment. Including the environment restores a closed, Hermitian description, but at the cost of an intractably large Hilbert space. The Feshbach–Löwdin projection eliminates the unwanted degrees of freedom exactly, yielding an effective Hamiltonian H eff on the relevant subspace alone—which generically becomes non-Hermitian when the eliminated subspace contains an energetically accessible continuum and outgoing boundary conditions are imposed. We illustrate this through a minimal model: a particle in a box with a delta barrier at its center. Direct elimination of one half-well reduces all information about the discarded sector to a single boundary parameter β ; the same result is then rederived via the Green’s function and Dyson equation, offering a pedagogical illustration of how boundary data enter the Green’s-function formalism and relating β to a self-energy term. Whether β is real or complex determines whether H eff is Hermitian or not, and the full range of behavior—from Hermitian dynamics through finite-width resonances to the counterintuitive quantum Zeno effect—follows from this single parameter.
Source: European Journal of Physics
Authors: Ugo Lair, David Guéry-Odelin
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,325
Funder: Institut Universitaire de France
OpenAlex ID: W7204214140