Friday, July 26, 2013

1307.6838 (Stephen P. Shipman)

Eigenfunctions of Unbounded Support for Embedded Eigenvalues of Locally
Perturbed Periodic Graph Operators

Stephen P. Shipman
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 of the hybrid states are oscillatory while the others are exponential. This separation allows a localized defect to suppress the oscillatory (radiation) modes and retain the evanescent ones, thereby leading to embedded eigenvalues whose associated eigenfunctions decay exponentially but are not compactly supported.
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