It is known that if a locally perturbed periodic self-adjoint operator on a combinatorial or quantum graph admits an eigenvalue embedded in the continuous spectrum, then the associated eigenfunction is compactly supported—that is, if the Fermi surface is irreducible, which occurs generically in dimension two or higher. This article constructs a class of operators whose Fermi surface is reducible for all energies by coupling several periodic systems. The components of the Fermi surface correspond to decoupled spaces of hybrid states, and in certain frequency bands, some components contribute oscillatory hybrid states (corresponding to spectrum) and other components contribute only exponential ones. This separation allows a localized defect to suppress the oscillatory (radiation) modes and retain the exponential ones, thereby leading to embedded eigenvalues whose associated eigenfunctions decay exponentially but are not compactly supported.
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Communicated by P. Deift
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Shipman, S.P. Eigenfunctions of Unbounded Support for Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators. Commun. Math. Phys. 332, 605–626 (2014). https://doi.org/10.1007/s00220-014-2113-y