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 $H_{\rm eff}$ 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 $H_{\rm eff}$ 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.