Effective Hamiltonians and non-Hermitian physics
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 Heff on the relevant subspace alone---which \new{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 \new{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 Heff 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.
Versions
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BibTeX
@misc{lair2026effective,
title = {{Effective Hamiltonians and non-Hermitian physics}},
author = {Ugo Lair and David Guéry-Odelin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05268},
eprint = {2611.05268},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05268},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Effective Hamiltonians and non-Hermitian physics
AU - Ugo Lair
AU - David Guéry-Odelin
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05268
UR - https://arxiv.org/abs/2611.05268
AB - 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 \new{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 \new{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.
ER -