Proposition: Bound-state failure test

An infinite half-line reservoir does not, by itself, guarantee source release. For one chain, replace the source onsite energy by ${\omega }_S>{\omega }_0+2c$, retain a nonzero source–reservoir bond, and keep the reservoir unchanged. The resulting one-particle operator has a bound state above the reservoir band with nonzero source overlap.

Proof. The modified operator is a finite-rank perturbation of ${\omega }_S$ on the source plus $h_B$ on the reservoir. Its essential spectrum is the reservoir band $[{\omega }_0-2c,{\omega }_0+2c]$.

The source unit vector has Rayleigh quotient ${\omega }_S$, strictly above the upper edge of that band. The variational characterization of the spectral supremum therefore gives spectrum above the essential band; such spectrum is discrete and contains an eigenvalue.

If an eigenvector above the band had zero source component, its reservoir component would be an eigenvector of $h_B$ above its spectrum. That is impossible. Thus its source overlap is nonzero.