Topological directed amplification

A phenomenon of topological directed amplification certain initial perturbations is revealed theoretically to emerge in a class asymptotically stable skin-effect lattices described by non-normal Toeplitz operators ${H}_{g}$ with positive ``numerical ordinate'' $\ensuremath{\omega}({H}_{g})>0$. Non-normal temporal evolution, even the presence global dissipation, shown manifest counterintuitive transient phase edge-state amplification---a behavior, drastically different from asymptote, that spectral analysis fails directly reveal. consistent description effect provided general tool ``pseudospectrum,'' and quantitative estimation maximum power Kreiss constant. recipe determine an optimal condition will attain given singular value decomposition propagator ${e}^{\ensuremath{-}i{H}_{g}t}$. It further predicted interplay between non-normality nonlinearity laser array can facilitate narrow-emission spectra scalable stable-output power.