We develop the rational dynamics for the long-term investor among boundedly rational speculators in the Carfì–Musolino speculative and hedging model. Numerical evidence is given that indicates there are various phases determined by the degree of nonrational behavior of speculators. The dynamics are shown to be influenced by speculator “noise”. This model has two types of operators: a real economic subject (Air, a long-term trader) and one or more investment banks (Bank, short-term speculators). It also has two markets: oil spot market and U.S. dollar futures. Bank agents react to Air and equilibrate much more quickly than Air, thus we consider rational, best-local-response dynamics for Air based on averaged values of equilibrated Bank variables. The averaged Bank variables are effectively parameters for Air dynamics that depend on deviations-from-rationality (temperature) and Air investment (external field). At zero field, below a critical temperature, there is a phase transition in the speculator system which creates two equilibriums for bank variables, hence in this regime the parameters for the dynamics of the long-term investor Air can undergo a rapid change, which is exactly what happens in the study of quenched dynamics for physical systems. It is also shown that large changes in strategy by the long-term Air investor are always preceded by diverging spatial volatility of Bank speculators. The phases resemble those for unemployment in the “Mark 0” macroeconomic model.
We present a new approach to the eigensystem multiscale analysis (EMSA) for the Anderson model that relies on the Wegner estimate. The EMSA treats all energies of the finite volume operator in an energy interval at the same time, simultaneously establishing localization of all eigenfunctions with eigenvalues in the energy interval with high probability. It implies all the usual manifestations of localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization) for the Anderson model. The new method removes the restrictive level spacing hypothesis used in the previous versions of the EMSA, allowing for single site probability distributions that are H\"older continuous of order $\alpha \in (0,1]$. (Joint work with Alex Elgart.)
In this talk, we first consider quasi-periodic Schr\"odinger operators with finitely differentiable potentials. If the potential is analytic, there are numerous results. But not every result holds if one replaces the analyticity with a smoothness condition. We will give some positive results in this aspect, generalizing some interesting results in the analytic case to the finitely smooth case. This includes the global reducibility results, generalized Chamber's formula and their applications to the study of continuity of the spectra. Finally we will give a recent result on the continuity of spectral measure of multi frequency quasi-periodic Schr\"odinger operators with small analytic quasi-periodic potentials.
In 2012, Filoche and Mayboroda introduced the concept of the landscape function u, for an elliptic operator L, which solves the inhomogeneous equation Lu=1. This landscape function has remarkable power to predict the shape and location of localized low energy eigenfunction. These ideas led to beautiful results in mathematics, as well as theoretical and experimental physics. In this talk, we first briefly review these results of landscape theory for differential operators on R^d. We will then discuss some recent progress of extending landscape theory to tight-binding Hamiltonians on discrete lattice Z^d. In particular, we show that the effective potential 1/u creates barrier for appropriate exponential decay eigenfunctions of Agmon type for some discrete Schrodinger operators. We also show that the minimum of 1/u leads to a new counting function, which gives non-asymptotic estimates on the integrated density of states of the Schrodinger operators. This talk contains joint work in progress with S. Mayboroda and some numerical experiments with W. Wang.
We established an axiomatic version of Christ-Kiselev's multi-linear operator techniques.
As applications, several spectral results of perturbed periodic Schrodinger operators are obtained, including WKB solutions, sharp transitions of preservation of absolutely continuous spectra, criteria of absence of singular spectra and sharp bounds of Hausdorff dimensions of singular spectra.
We develop the basic spectral theory of ergodic Schrodinger operators when the underlying dynamics are given by a conservative
ergodic transformation of a \sigma-finite measure space. Some fundamental results, such as the Ishii--Pastur theorem carry over to the
infinite-measure setting. We also discuss some examples in which straightforward analogs of results from the probability-measure case do not hold. We will discuss some examples and some interesting open problems.
The talk is based on a joint work with M. Boshernitzan, D. Damanik, and M. Lukic.
Understanding the statistical properties of Laplacian eigenfunctions in general and their nodal sets in particular, have an important role in the field of spectral geometry, and interest both mathematicians and physicists. A quantum graph is a system of a metric graph with a self-adjoint Schrodinger operator. It was proven for quantum graphs that the number of points on
which each eigenfunction vanish (also known as the nodal count) is
bounded away from the spectral position of the eigenvalue by the first Betti number of the graph. A remarkable result by Berkolaiko and Weyand showed that the nodal surplus is equal to a magnetic stability index of the corresponding eigenvalue. A similar result for discrete graphs holds as well proved first by Berkoliako and later by Colin deVerdiere.
Both from the nodal count point of view and the magnetic point of view, it is interesting to consider the distribution of these indices over the spectrum. In our work, we show that such a density exists and defines a nodal count distribution. Moreover, this distribution is symmetric, which allows deducing the topology of a graph from its nodal count. Although for general graphs we can not a priori calculate the nodal count distribution, we proved that a certain family of graphs will have a binomial distribution. As a corollary, given any sequence of graphs from that family with an increasing number of cycles, the sequence of nodal count distributions, properly normalized, will converge to a normal distribution.
A numerical study indicates that this property might be universal and led us to state the following conjecture. For every sequence of graphs with an increasing number of cycles, the corresponding sequence of properly normalized nodal count distributions will converge to a normal distribution.
In my talk, I will present our latest results extending the number
of families of graphs for which we can prove the conjecture.
This talk is based on joint works with Ram Band (Technion) and Gregory Berkolaiko (Texas A&M)
By judiciously constructing local defects in graph models of multi-layer graphene, bound states can be constructed at energies that lie within the continuous spectrum of the associated Schrödinger operator. The layers may be stacked in AA or AB fashion. A necessary condition for this construction is the reducibility of the Fermi surface for the multi-layer structure. This is achieved due to a special reduction of the complex dispersion relation to a function of a single polynomial "composite" function of the quasimomenta.
This is joint work with Wei Li and Stephen Shipman at LSU.