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 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 ; 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 to a self-energy term. Whether is real or complex determines whether 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
@article{lair2026effective,
title = {{Effective Hamiltonians and non-Hermitian physics}},
author = {Ugo Lair and David Guéry-Odelin},
journal = {European Journal of Physics},
year = {2026},
month = sep,
doi = {10.1088/1361-6404/ae9e99},
url = {https://hal.science/hal-05786320v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Effective Hamiltonians and non-Hermitian physics
AU - Ugo Lair
AU - David Guéry-Odelin
PY - 2026
DA - 2026/09/16
JO - European Journal of Physics
DO - 10.1088/1361-6404/ae9e99
UR - https://hal.science/hal-05786320v1
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 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 ; 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 to a self-energy term. Whether is real or complex determines whether 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.
SN - 0143-0807
SN - 1361-6404
ER -