Legendre Transform, Hessian Conjecture and Tree Formula

Let $\phi$ be a polynomial over $K$ (a field of characteristic 0) such that the Hessian of $\phi$ is a nonzero constant. Let $\bar\phi$ be the formal Legendre Transform of $\phi$. Then $\bar\phi$ is well-defined as a formal power series over $K$. The Hessian Conjecture introduced here claims that $\bar\phi$ is actually a polynomial. This conjecture is shown to be true when K=\bb{R} and the Hessian matrix of $\phi$ is either positive or negative definite somewhere. It is also shown to be equivalent to the famous Jacobian Conjecture. Finally, a tree formula for $\bar\phi$ is derived; as a consequence, the tree inversion formula of Gurja and Abyankar is obtained.Comment: 9 pages, references are update

Lorentz Group and Oriented MICZ-Kepler Orbits

The MICZ-Kepler orbits are the non-colliding orbits of the MICZ Kepler problems (the magnetized versions of the Kepler problem). The oriented MICZ-Kepler orbits can be parametrized by the canonical angular momentum $\mathbf L$ and the Lenz vector $\mathbf A$, with the parameter space consisting of the pairs of 3D vectors $(\mathbf A, \mathbf L)$ with ${\mathbf L}\cdot {\mathbf L} > (\mathbf L\cdot \mathbf A)^2$. The recent 4D perspective of the Kepler problem yields a new parametrization, with the parameter space consisting of the pairs of Minkowski vectors $(a,l)$ with $l\cdot l =-1$, $a\cdot l =0$, $a_0>0$. This new parametrization of orbits implies that the MICZ-Kepler orbits of different magnetic charges are related to each other by symmetries: \emph{${\mathrm {SO}}^+(1,3)\times {\mathbb R}_+$ acts transitively on both the set of oriented elliptic MICZ-Kepler orbits and the set of oriented parabolic MICZ-Kepler orbits}. This action extends to ${\mathrm {O}}^+(1,3)\times {\mathbb R}_+$, the \emph{structure group} for the rank-two Euclidean Jordan algebra whose underlying Lorentz space is the Minkowski space.Comment: 7 page

The Representation Aspect of the Generalized Hydrogen Atoms

Let $D\ge 1$ be an integer. In the Enright-Howe-Wallach classification list of the unitary highest weight modules of \widetilde{\mr{Spin}}(2, D+1), the (nontrivial) Wallach representations in Case II, Case III, and the mirror of Case III are special in the sense that they are precisely the ones that can be realized by the Hilbert space of bound states for a generalized hydrogen atom in dimension D. It has been shown recently that each of these special Wallach representations can be realized as the space of L^2-sections of a canonical hermitian bundle over the punctured {\bb R}^D. Here a simple algebraic characterization of these special Wallach representations is found.Comment: 18 pages, simplified proo

The O(1)-Kepler Problems

Let $n\ge 2$ be an integer. To each irreducible representation $\sigma$ of $\mathrm O(1)$, an $\mathrm {O}(1)$-Kepler problem in dimension $n$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to a generalized MICZ-Kepler problem in dimension two. The dynamical symmetry group of this system is $\widetilde {\mathrm{Sp}}_{2n}(\mathbb R)$ with the Hilbert space of bound states ${\mathscr H}(\sigma)$ being the unitary highest weight representation of $\widetilde {\mathrm{Sp}}_{2n}(\mathbb R)$ with highest weight $$(\underbrace{-1/2, ..., -1/2}_{n-1}, -(1/2+|\sigma|)),$$ which occurs at the right-most nontrivial reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. (Here $|\sigma|=0$ or 1 depending on whether $\sigma$ is trivial or not.) Furthermore, it is shown that the correspondence $\sigma\leftrightarrow \mathscr H(\sigma)$ is the theta-correspondence for dual pair $(\mathrm{O}(1), \mathrm{Sp}_{2n}(\mathbb R))\subseteq \mathrm{Sp}_{2n}(\mathbb R)$.Comment: Final published for