Mathematical Physics and Harmonic Analysis Seminar

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Title: Spectrum, Lifshitz tails and localization for the Anderson model on the Sierpinski gasket graph

Abstract: We study the Anderson model on the Sierpinski gasket graph, a canonical self-similar fractal. Assuming that the single-site distribution has connected support, we identify the almost-sure spectrum. We establish the existence of the integrated density of states and prove Lifshitz-tail behavior at the bottom of the spectrum, with Lifshitz exponent equal to the ratio of the volume-growth exponent to the random-walk dimension of the Sierpinski gasket. Moreover, we show that this Lifshitz-tail upper bound yields Anderson localization near the spectral bottom via the fractional-moment method, under a regularity assumption on the single-site distribution. The Lifshitz-tail and Anderson localization results extend to “Ahlfors α-regular” graphs, under suitable power-law assumptions on the low-lying eigenvalues of the Dirichlet and Neumann Laplacians on graph balls. If time permits, we will also discuss open problems, including full localization for the Anderson model on the gasket graph and the dependence of the metal–insulator transition on the dimension of general fractal graphs. 

Title: The Fibered Rotation Number

Abstract: In the talk we will describe the properties of the fibered rotation numbers of one-parameter families of smooth circle cocycles over an ergodic transformation in the base. In the case of projective cocycles we will discuss an analog of Johnson's Theorem connecting intervals of constancy of the rotation number with uniform hyperbolicty of the cocycle. Also, it turns out that the rotation numbers must be log-Holder regular with respect to the parameter. As an immediate application, one gets a dynamical proof of 1D version of the Craig-Simon’s theorem from spectral theory that claims that the integrated density of states of an ergodic Schrodinger operator must be log-Holder. Finally, we will present a new explicit formula for the rotation number of a cocycle and discuss its application to spectral theory of ergodic discrete Schrodinger operators.  The talk is based on a series of results obtained jointly with Victor Kleptsyn and Pedro Duarte.  

Title: Infinite volume topological index via finite volume Spectral Localizer

Abstract: In the study of topological insulators, quantum Hamiltonians are labelled by topological indices. Mathematically, these indices can be defined by a variety of approaches, all of which give trivial

(zero) answers as soon as the Hamiltonian is restricted to a finite volume. This presents a challenge to computing the indices numerically.

The "spectral localizer", introduced and developed by Loring and Schulz-Baldes in a series of papers starting from 2015, gives a practical computational procedure based on combining a space cut-off

with a particular unbounded perturbation of the Hamiltonian. We propose an alternative proof of the Loring--Schulz-Blades theorem, linking the signature of their localizer matrix with a spectral flow inspired by the Integer Quantum Hall Effect. In the talk, we will discuss the physical intuition behind the steps of the proof and illustrate them by numerical experiments.

Based on the joint work with Jacob Shapiro (Princeton) and Beyer White (IST Austria), arXiv:2512.21843. 

Title: Discrepancy Estimates and Quantum Dynamics

Abstract:  In this talk, we present new quantum dynamical upper bounds for one-dimensional analytic long-range operators with ergodic potentials, which improve over several previously known results. Our approach is based on novel discrepancy estimates for semi-algebraic sets. Specifically, for shift dynamics, we establish an asymptotically sharp discrepancy bound by constructing a matching lower bound. For skew-shift dynamics, we reduce the exponential upper bound to a polynomial one by Vinogradov method. 

Title: An Overview of Wave Propagation in Periodic Composites

Abstract:  In this talk, we shall discuss the problem of homogenization of the wave equation, which is the study of approximating the effective transport properties of a highly heterogeneous medium by a homogeneous one. While one measures the goodness of an approximation in the elliptic setting solely by its accuracy (ε), the error for the wave equation is measured in accuracy (ε) and time (t) simultaneously. However, achieving high accuracy and long times are competing objectives in the hyperbolic setting, and it turns out that the classical two-scale ansatz is only good to an O(ε^{-2+δ}) timescale. We shall explain the reason behind this fundamental limitation, and several approaches to overcome this, namely, a "criminal ansatz" (Conca-Orive-Vanninathan 2022) and a "spectral ansatz" (Benoit-Duerinckx-Gloria-Ruf 2023). 

Title: Mixed Quantization and Quantum Ergodicity on Graph Vector Bundles

Abstract:  Quantum Ergodicity (QE) is a classical topic in quantum chaos. It asserts that on a compact Riemannian manifold whose geodesic flow is ergodic, the Laplacian admits a density-one subsequence of eigenfunctions that become equidistributed. In this talk, we present a discrete version of QE for vector bundles over regular graphs. The main technique is mixed quantization, which brings together semiclassical and geometric quantizations.

Title: Quantum Dynamics for Periodic Schr\"odinger Operators

Abstract:  Periodic Schr\"odinger operators exhibit ballistic motion, that is, a linear scaling of the position observable in time. I will discuss a pair of recent results that give improved understanding of these phenomena in the setting of discrete operators by leveraging tools from complex analysis: (1) a dispersive estimate for arbitrary one-dimensional Schr\"odinger operators and (2) sharp dynamical estimates for generic multi-dimensional potentials in the large-coupling regime.

Title: Almost Periodicity, the Kotani-Last and Deift's Conjecture

Abstract:  In this talk I will talk about two conjectures about almost periodicity in the study of spectral theory. Then we will take a second look at the known results from the perspective of different notions of almost periodicity.

Title: Oscillation of graph eigenfunctions and its applications  

Abstract:  We discuss a formula that expresses the number of sign changes of the $k$-th eigenvector of a graph operator as the sum of the "Sturm contribution" $k-1$ and a "cycle contribution" which takes the form of the Morse index (number of negative eigenvalues) of a weighted cycle intersection form associated with the graph. 

This result has many interesting connections.  First, it allows one to derive a simple formula for the Hessian of the dispersion relation of a particular class of crystals (periodic lattices), namely maximal abelian covers of finite graphs.  Second, it can be used to efficiently determine stability of a stationary solution on a coupled oscillator network, such as the non-uniform Kuramoto model for the synchronization of a network of electrical oscillators.  Finally, the determinant of the weighted cycle intersection form is the first Symanzik polynomial of the graph (closely related to the Kirchhoff polynomial), hinting at connections to both Feynman amplitudes and matroids. 

Based on a joint work with Jared Bronski and Mark Goresky.

Title: Families of graphs whose Bloch varieties are perfect Morse functions

Abstract:  A Bloch variety of a Z^d periodic graph with m orbits of vertices and isolated critical points has at least 2^d m critical points. (These occur over the corner points, those points satisfying z^2=1.) 

I will describe two families of Z^d periodic graphs whose Bloch varieties have only these critical points.  Such graphs satisfy Kuchment's Spectral Edges nondegeneracy conjecture.  The first family are graphs whose Newton polytope has base a cross polytope, while the second are given in terms of the graph. 

This is joint work with Matthew Faust.