Exactly solvable Schr\"odinger operators related to the hypergeometric equation
We study one-dimensional Schr\"odinger operators defined as closed operators that are exactly solvable in terms of the Gauss hypergeometric function. We allow the potentials to be complex. These operators fall into three groups. The first group can be reduced to the Gegenbauer equation, up to an affine transformation, a special case of the hypergeometric equation. The two other groups, which we call {\em hypergeometric of the first}, resp. {\em second kind}, can be reduced to the general Gauss hypergeometric equation. Each of the group is subdivided in three families, acting to on the Hilbert space $L^2]-1,1[,$ $L^2(\rr_+)$ resp. $L^2(\rr)$. Motivated by geometric applications of these families, we call them {\em spherical}, {\em hyperbolic}, resp. {\em deSitterian}. All these families are known from applications in Quantum Mechanics: e.g. spherical hypergeometric Schr\"odinger operators of the first kind are often called {\em trigonometric P\"oschl-Teller Hamiltonians}. For operators belonging to each family we compute their spectrum and determine their Green function (the integral kernel of their resolvent). We also describe transmutation identities that relate these Green functions. These identities interchange spectral parameters with coupling constants across different operator families. Finally, we describe how these operators arise from separation of variables of (pseudo-)Laplacians on symmetric manifolds. Our paper can be viewed as a sequel to \cite{DL}, where closed realizations of one-dimensional Schr\"odinger operators solvable in terms Kummer's confluent equation were studied.