Publications

Transition of type in the von Neumann algebras associated to the Connes-Marcolli $GSp_4$-system

Author: Ismail Abouamal
Published on: 2024-07-19
arXiv Link

Abstract:
We study different types of von Neumann algebras arising from the Connes-Marcolli $GSp_4$-system and show that a phase transition occurs at the level of these algebras. More precisely, we show that the type of these algebras transitions from type $I_{\infty}$ to type $\textrm{III}_1$, with this transition occurring precisely at the inverse temperature $β=4$.

Bost-Connes-Marcolli system for the Siegel modular variety

Author: Ismail Abouamal
Published on: 2022-10-14
arXiv Link

Abstract:
We construct a quantum satisitical mechanical system which generalizes the Connes-Marcolli GL2 system. In particular we introduce the Connes-Marcolli system associated to the Siegel modular variety of degree 2. We classify its KMSβ-states for inverse temperatures β>0 and show that a spontaneous phase transition occurs at $β=3$. More precisely, we prove that the system does not admit a KMSβ state for $β<3$ with $β≠1$, construct the explicit extremal Gibbs states for $β>4$ and show that a unique KMSβ state exists for every $β>0$ with $3<β≤4$.

Fifth-order superintergrable quantum system separating in Cartesian coordinates. Doubly exotic potentials

Author: Ismail Abouamal; Pavel Winternitz
Published on: 2018-02-06
Link

Abstract:
We consider a two-dimensional quantum Hamiltonian separable in Cartesian coordinates and allowing a fifth-order integral of motion. We impose the superintegrablity condition and find all doubly exotic superintegrable potentials (i.e., potentials $V(x, y) = V_1(x) + V_2(y)$, where neither $V_1(x)$ nor $V_2(y)$ satisfy a linear ordinary differential equation), allowing the existence of such an integral. All of these potentials are found to have the Painlevé property. Most of them are expressed in terms of known Painlevé transcendents or elliptic functions but some may represent new higher order Painlevé transcendents.