arXiv subjects
Publications and source records attributed to Ismail Abouamal.
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 $III_1$, with this transition occurring precisely at the inverse temperature $β= 4$.
We construct a quantum satisitical mechanical system which generalizes the Connes-Marcolli $GL_2$ system. In particular we introduce the Connes-Marcolli system associated to the Siegel modular variety of degree $2$. We classify its $\text{KMS}_β$-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 $\text{KMS}_β$ state for $β< 3$ with $β\neq 1$, construct the explicit extremal Gibbs states for $β>4$ and show that a unique $\text{KMS}_β$ state exists for every $β>0$ with $3 < β\leq 4$.
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 ODE) 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.