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.

7 pages, 1 figure

References in corpus (7)

• Exceptional Points of Non-hermitian Operators
• No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model
• Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart
• Ghost-free higher-derivative theory
• Comments on the Dynamics of the Pais-Uhlenbeck Oscillator
• Maximal couplings in PT-symmetric chain-models with the real spectrum of energies
• Three solvable matrix models of a quantum catastrophe

Cited by in corpus (5)

• Classical and Quantum Dynamics of Higher-Derivative Systems
• Multiply Degenerate Exceptional Points and Quantum Phase Transitions
• Level crossing in random matrices. II Random perturbation of a random matrix
• Non-standard quantum algebras and infinite-dimensional PT-symmetric systems
• Swanson Hamiltonian: non-PT-symmetry phase