Continuum of bound states in a non-hermitian model

In a Hermitian system, bound states must have quantized energies, whereas free states can form a continuum. We demonstrate how this principle fails for non-Hermitian systems, by analyzing non-Hermitian continuous Hamiltonians with an imaginary momentum and Landau-type vector potential. The eigenstates, which we call "continuum Landau modes" (CLMs), have Gaussian spatial envelopes and form a continuum filling the complex energy plane. We present experimentally realizable 1D and 2D lattice models that host CLMs; the lattice eigenstates are localized and have other features matching the continuous model. One of these lattices can serve as a rainbow trap, whereby the response to an excitation is concentrated at a position proportional to the frequency. Another lattice can act a wave funnel, concentrating an input excitation onto a boundary over a wide frequency bandwidth. Unlike recent funneling schemes based on the non-Hermitian skin effect, this requires a simple lattice design with reciprocal couplings.Ministry of Education (MOE)National Research Foundation (NRF)Published versionThis work was supported by the Singapore MOE Academic Research Fund Tier 3 Grant No. MOE2016-T3- 1-006 and Tier 1 Grant No. RG148/20, and by the National Research Foundation Competitive Research Programs NRFCRP23-2019-0005 and NRF-CRP23-2019-0007