Exceptional points of infinite order give a continuous spectrum

Abstract

The statement in the title discussed earlier in association with the Pais-Uhlenbeck oscillator with equal frequencies is illustrated for an elementary matrix model. In the limit when the order of the exceptional point N tends to infinity, an infinity of nontrivial states that do not change their norm during evolution appear. These states have real energies lying in a continuous interval. The norm of the "precursors" of these states at large finite N is not conserved, but the characteristic time scale where this nonconservation shows up grows linearly with N.