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A Poisson realization of the simple real Lie algebra $mathfrak {so}^*(4n)$ on the phase space of each $mathrm {Sp}(1)$-Kepler problem is exhibited. As a consequence one obtains the Laplace-Runge-Lenz vector for each classical $mathrm{Sp}(1)$-Kepler problem. The verification of these Poisson realizations is greatly simplified via an idea due to A. Weinstein. The totality of these Poisson realizations is shown to be equivalent to the canonical Poisson realization of $mathfrak {so}^*(4n)$ on the Poisson manifold $T^*mathbb H_*^n/mathrm{Sp}(1)$. (Here $mathbb H_*^n:=mathbb H^nbackslash {0}$ and the Hamiltonian action of $mathrm{Sp}(1)$ on $T^*mathbb H_*^n$ is induced from the natural right action of $mathrm{Sp}(1)$ on $mathbb H_*^n$. )
We prove the stability (instability) of the outer (inner) catenoid connecting two concentric circular rings, and explicitly construct the unstable mode of the inner catenoid, by studying the spectrum of an exactly solvable one-dimensional Schrodinger
We solve the group classification problem for the $2+1$ generalized quantum Zakharov-Kuznetsov equation. Particularly we consider the generalized equation $u_{t}+fleft( uright) u_{z}+u_{zzz}+u_{xxz}=0$, and the time-dependent Zakharov-Kuznetsov equat
In this paper we study the convex cone of not necessarily smooth measures satisfying the classical KMS condition within the context of Poisson geometry. We discuss the general properties of KMS measures and its relation with the underlying Poisson ge
We extend duality between the quantum integrable Gaudin models with boundary and the classical Calogero-Moser systems associated with root systems of classical Lie algebras $B_N$, $C_N$, $D_N$ to the case of supersymmetric ${rm gl}(m|n)$ Gaudin model
The concept of duality reflects a link between two seemingly different physical objects. An example in quantum mechanics is a situation where the spectra (or their parts) of two Hamiltonians go into each other under a certain transformation. We term