This is an ongoing project.  Let $H_0$ be a discrete periodic  Schr\"odinger operator on $\Z^d$:

$$H_0=-\Delta+v_0,$$

where $-\Delta$ is the discrete Laplacian and $v_0$ is periodic in the sense that it is well defined on  $\Z^d/q_1\Z\oplus q_2 \Z\oplus\cdots\oplus q_d\Z$. For $d=2$, we tentatively proved that the Fermi variety $F_{\lambda}(v_0)/\Z^2$ is irreducible except for one value of  $\lambda$. We also construct a non-constant periodic function $v_0$ such that its Fermi variety is reducible for  some $\lambda$, which disproves a conjecture by  Gieseker, Kn\"orrer and Trubowitz.

Under some assumptions of irreducibility of Fermi variety $F_{\lambda}(v_0)/\Z^d$, we show that $H=-\Delta +v_0+v$ does not have any embedded eigenvalues provided that $v$ decays  exponentially. The assumptions are conjectured to be true for any periodic function $v_0$. As an application, we show that when $d=2$, $H=-\Delta +v_0+v$ does not have any embedded eigenvalues provided that $v$ decays exponentially. 

We study the free XXZ quantum spin model defined on a ring of size L and show that the bipartite entanglement entropy of eigenstates belonging to the first energy band above the vacuum ground state satisfy a logarithmically corrected area law. 

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.