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 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 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.