Institute for Computational and Experimental Research in Mathematics (ICERM)

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Abstract

We will discuss Bloch and Fermi varieties associated with Schr\"odinger operators and some works concerning their irreducibility.

Abstract

Let G be a finite graph of N edges. A finite standard quantum graph (G,L), with L=(L_1,…,L_N), is a collection of N intervals e_j=[0,L_j] glued at their endpoints, corresponding to the edges of G, equipped with the one-dimensional Laplacian (2ed derivative edgewise) that acts on functions that satisfy standard vertex conditions. We will be interested in this operator's spectral properties (eigenvalues and eigenfunctions) and their dependence on G and L. For a fixed G, its ``Secular Manifold’’ S_G is the set of points (e^{ikL_1},…, e^{ikL_N}) in the N-dimensional torus, such that k^2 is an eigenvalue of (G,L). The eigenvalues and eigenfunctions of (G,L), for any L, are determined by intersections of a curve depending on L with S_G. This allows to decouple spectral properties into G and L dependence and provides an algebraic toolkit for spectral geometry on quantum graphs. After providing the background and defining the secular manifold, I will review previous results: spectral gap distribution, nodal count distribution, and the arithmetic structure of the spectrum. I will discuss the role of the secular manifold in those results, and some open conjectures that may benefit from investigating the algebraic structure and Morse structure of this variety.

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Abstract

In recent work, Kravaris uses computes the Density of States for discrete Schrodinger operators as the (normalized) rank of a certain module over the ring of Laurent polynomials. I will discuss some of the computational and homological tools used in his result.

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Abstract

I will discuss analytic methods of showing that the level sets of spectral band functions at the edges of spectral bands have dimension at most $d-2$, where $d$ is the dimension of the lattice of periods. The approach works in the continuum for $d=2$ and for a large class of discrete Schrodinger operators in all dimensions. I will also summarize some current open questions in the area. The results in the talk are based on joint works with N. Filonov.

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