Under a mild condition on the branching mechanism, we provide an eigenvalue expansion for the pricing semigroup in a one-dimensional positive affine term structure model. This representation, which is based on results from Ogura [Publ. Res. Inst. Math. Sci., 6 (1970), pp. 307--321], recently improved by the authors in [J. Math. Anal. Appl., 459 (2018), pp. 619--660], allows us to get analytical expressions for the prices of interest rate sensitive European claims. As the pricing semigroups are non-self-adjoint linear operators, the computation of eigenfunctions and co-eigenmeasures is required in the expansions. We describe comprehensive methodologies to characterize these spectral objects from merely the knowledge of the branching and immigration mechanisms. To illustrate the computation power and advantages of our approach, we develop comparison analysis with Fourier--Laplace inversion techniques for some examples. Numerical experiments are provided and show that the spectral approach allows one to quickly price European vanilla options on bonds and yields for a whole range of strikes and maturities.