Equivalence of the effective Hamiltonian approach and the Siegert boundary condition for resonant states

TL;DR

This paper demonstrates the algebraic equivalence between the Siegert boundary condition and the Feshbach formalism in identifying resonant states in open quantum systems, highlighting the implicit non-Hermitian nature of the effective Hamiltonian.

Contribution

It shows that two different theoretical approaches for resonant states are mathematically equivalent and clarifies the non-Hermitian characteristics of the effective Hamiltonian.

Findings

The Siegert and Feshbach methods are algebraically equivalent.

Open quantum systems have an implicitly non-Hermitian Hamiltonian.

Effective Hamiltonians are explicitly non-Hermitian in a contracted space.

Abstract

Two theoretical methods of finding resonant states in open quantum systems, namely the approach of the Siegert boundary condition and the Feshbach formalism, are reviewed and shown to be algebraically equivalent to each other for a simple model of the T-type quantum dot. It is stressed that the seemingly Hermitian Hamiltonian of an open quantum system is implicitly non-Hermitian outside the Hilbert space. The two theoretical approaches extract an explicitly non-Hermitian effective Hamiltonian in a contracted space out of the seemingly Hermitian (but implicitly non-Hermitian) full Hamiltonian in the infinite-dimensional state space of an open quantum system.