Our paper investigates one-dimensional Schrödinger operators defined as closed operators on \(L^2({{\mathbb {R}}})\) or \(L^2({{\mathbb {R}}}_+)\) that are exactly solvable in terms of confluent functions (or, equivalently, Whittaker functions). We allow the potentials to be complex. They fall into three families: Whittaker operators (or radial Coulomb Hamiltonians), Schrödinger operators with Morse potentials and isotonic oscillators. For each of them, we discuss the corresponding basic holomorphic family of closed operators and the integral kernel of their resolvents. We also describe transmutation identities that relate these resolvents. These identities interchange spectral parameters with coupling constants across different operator families. A similar analysis is performed for one-dimensional Schrödinger operators solvable in terms of Bessel functions (which are reducible to special cases of Whittaker functions). They fall into two families: Bessel operators and Schrödinger operators with exponential potentials. To make our presentation self-contained, we include a short summary of the theory of closed one-dimensional Schrödinger operators with singular boundary conditions. We also provide a concise review of special functions that we use.
