Embedded eigenvalues for asymptotically periodic ODE systems

We investigate the persistance of embedded eigenvalues under perturbations of a certain self-adjoint Schrödinger-type differential operator in $L^2 (\mathbb{R} ; \mathbb{R}^n)$, with an asymptotically periodic potential. The studied perturbations are small and belong to a certain Banach space with a specified decay rate, in particular, a weighted space of continuous matrix valued functions. Our main result is that the set of perturbations for which the embedded eigenvalue persists forms a smooth manifold with a specified co-dimension. This is done using tools from Floquet theory, basic Banach space calculus, exponential dichotomies and their roughness properties, and Lyapunov- Schmidt reduction. A second result is provided, where under an extra assumption, it can be proved that the first result holds even when the space of perturbations is replaced by a much smaller space, as long as it contains a minimal subspace. In the end, as a way of showing that the investigated setting exists, a concrete example is presented. The example itself relates to a problem from quantum mechanics and represents a system of electrons in an infinite one-dimensional crystal.

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Received 18 April 2023

Received revised 22 November 2023

Accepted 26 December 2023

Published 1 June 2024