Generalized Kepler problems. I. Without magnetic charges
For each simple euclidean Jordan algebra V of rank {rho} and degree {delta}, we introduce a family of classical dynamic problems. These dynamical problems all share the characteristic features of the Kepler problem for planetary motions, such as the existence of Laplace-Runge-Lenz vector and hidden symmetry. After suitable quantizations, a family of quantum dynamic problems, parametrized by the nontrivial Wallach parameter {nu}, is obtained. Here, {nu} Element-Of W(V):={l_brace}k({delta}/2) Double-Vertical-Line k=1,...,({rho}-1){r_brace} Union (({rho}-1)({delta}/2),{infinity}) and was introduced by N. Wallach to parametrize the set of nontrivial scalar-type unitary lowest weight representations of the conformal group of V. For the quantum dynamic problem labelled by {nu}, the bound state spectra is -(1/2/(I+{nu}({rho}/2)){sup 2}), I= 0, 1, Horizontal-Ellipsis and its Hilbert space of bound states gives a new realization for the afore-mentioned representation labelled by {nu}. A few results in the literature about these representations become more explicit and more refined. The Lagrangian for a classical Kepler-type dynamic problem introduced here is still of the simple form: (1/2) parallel x parallel {sup 2}+(1/r). Here, x is the velocity of a unit-mass particle moving on the space consisting of V's semi-positive elements of a fixed rank, and r is the inner product of x with the identity element of V.