Bound eigenstates and generalized eigenstates (scattering eigenstates) are two kinds of eigenstates in quantum mechanics. In this work, we first introduce a systematic way to regularize a generalized eigenstates by using the Wick rotation. The states that can be regularized are, in fact, Gamow states since they form poles in the scattering matrix but not localized before the Wick rotation. We then demonstrate an example where a bosonic field interacting with an array of two level systems can have Gamow states with positive real eigenenergies, and the scattering spectrum diverges at the eigenenergy. Since the eigenenergies of this kind locate in a real continuous scattering spectrum, from the scattering matrix point of view, these states resemble the bound states in the continuum (BIC). Unlike BIC, however, these states are non-localized in space and possess the frequency-filtering nature which may lead to potential applications in tunable quantum frequency filters for scattering states.
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