Wilhelm Treschow | Searcharxiv

arXiv2025

Perturbations of embedded eigenvalues of asymptotically periodic magnetic Schr\"odinger operators on a cylinder

We investigate the persistence of embedded eigenvalues for a class of magnetic Laplacians on an infinite cylindrical domain. The magnetic potential is assumed to be $C^2$ and asymptotically periodic along the unbounded direction, with an algebraic decay rate towards a periodic background potential. Under the condition that the embedded eigenvalue of the unperturbed operator lies away from the thresholds of the continuous spectrum, we show that the set of nearby potentials for which the embedded eigenvalue persists forms a smooth manifold of finite and even codimension. The proof employs tools from Floquet theory, exponential dichotomies, and Lyapunov--Schmidt reduction. Additionally, we give an example of a potential which satisfies the assumptions of our main theorem.

math.FA↗

arXiv2024

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.

math.FA↗