A local test for global extrema in the dispersion relation of a periodic graph (Q2084563)

In this paper, the authors consider self-adjoint operators on a \(\mathbb{Z}^d\)-periodic graph. After a Floquet-Bloch transform, such operator can be viewed as the direct integral over the \(d\)-dimensional torus \(\mathbb{T}^d\) of self-adjoint matrices \(T(\alpha)\), that depends smoothly on \(\alpha\in\mathbb{T}^d\). Ordering the eigenvalues of \(T(\alpha)\) in increasing order, the authors consider critical points of the \(n\)th eigenvalue (for some integer \(n\), as function of \(\alpha\)) with the assumption that the nth eigenvalue is simple near such a critical point. The main aim of the paper is to provide a local test near such a critical point to check if it is a global extremum. On the periodic graph, the authors require a so called ``one crossing edge per generator'' property, which is not to restrictive. The latter implies some particular structure of the \(T(\alpha)\). Concerning the critical points \(\alpha^\circ\), it is also assume that some eigenvector \(f^\circ\) of \(T(\alpha^\circ)\) associated to the nth eigenvalue does not vanish ``on any crossing edge'' (some edge of the graph, that is related to the ``one crossing edge per generator'' property). This assumption plays a rôle in the behaviour of \(T(\alpha)\) near the critical point \(\alpha^\circ\). To formulate the first result, one needs a (a priori complex) matrix \(W(\alpha)\) that is explicitly defined in terms of \(T\), its derivatives up to order two, and the nth eigenvalue, all taken at \(\alpha\). Given a critical point \(\alpha^\circ\) satisfying the above assumptions and setting \(W:=W(\alpha^\circ)\), the first result states that if \(W\) is a self-adjoint, non-negative (resp. non-positive) matrix, then the nth eigenvalue achieves at \(\alpha^\circ\) its global minimal (resp. maximal) value. For the second result, the authors focus on real symmetric operators on the graph. In this setting, it turns out that so called ``corner points'' always are critical points for any eigenvalue. The second result states that a local extremum \(\alpha^\circ\) of the \(n\)th eigenvalue is a global extremum if the following cases: (1) \(\alpha^\circ\) is a corner point; (2) the dimension \(d\leq 2\); (3) the dimension \(d=3\) and the extremum \(\alpha^\circ\) is non-degenerate. We refer to the end of the Introduction for a heuristic of the proofs.