On the Structure of Eigenfunctions Corresponding to Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators

Abstract

The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z n . It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders whether this must always be the case. The paper answers this question affirmatively. What is more surprising, one can estimate that the eigenmode must be localized not far away from the perturbation (in a neighborhood of the perturbation's support, the width of the neighborhood determined by the unperturbed operator only).The validity of this result requires the condition of irreducibility of the Fermi (Floquet) surface of the periodic operator, which is expected to be satisfied for instance for periodic Schrödinger operators.

On the Spectra of Carbon Nano-Structures

“…This relation between quantum and combinatorial graph operators is well known and has been exploited many times (e.g., [1,5,8,34,35,39,50]). …”

On occurrence of spectral edges for periodic operators inside the Brillouin zone

“…The Hilbert space where the operator acts is Here, f ′ e (v) denotes the outgoing derivative of f at v along the edge e. It is well known (e.g., [16,40,43,49]) and easy to check that Floquet theory applies to the quantum graph case. In particular, the spectrum σ(H) coincides with the union over the Brillouin zone B of the spectra of Floquet Hamiltonians H(k) .16), and with the additional cyclic (Bloch, Floquet) condition (2.3):…”

Untitled

“…However, Theorem 6.25 ([259]). In the graph case, embedded eigenvalues might arise, but if the relevant Fermi surface is irreducible, the corresponding eigenfunction must be supported close to the support of the perturbation; (see details in [259]). …”