It is known that, if a locally perturbed periodic self-adjoint operator on acombinatorial or quantum graph admits an eigenvalue embedded in the continuousspectrum, then the associated eigenfunction is compactly supported--that is, ifthe Fermi surface is irreducible, which occurs generically in dimension two orhigher. This article constructs a class of operators whose Fermi surface isreducible for all energies by coupling several periodic systems. The componentsof the Fermi surface correspond to decoupled spaces of hybrid states, and incertain frequency bands, some components contribute oscillatory hybrid states(corresponding to spectrum) and other components contribute only exponentialones. This separation allows a localized defect to suppress the oscillatory(radiation) modes and retain the evanescent ones, thereby leading to embeddedeigenvalues whose associated eigenfunctions decay exponentially but are notcompactly supported.
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