Let \(X=X_{int}\cup X_{\infty}\) be a non-compact, complete Riemannian manifold, where \(X_{int}\) is compact and \(X_{\infty}\) is the union of a finite number of either cylindrical or pointed ends. The qualitative spectral theory of operators, such as Laplacians or Dirac operators, defined on the space of sections of certain geometric fibrations over X is studied. This study is based on the following model, consisting of the system given by a discrete part \({\mathcal M}\) of \({\mathbb{R}}^+\), a multiplicity mapping m: \({\mathcal M}\to {\mathbb{Z}}\), a Hilbert space \({\mathcal H}\) and two Hamiltonians \(H_ 0\) and H operating in \({\mathcal H}\). The Hilbertian spectral analysis of H is performed. For example, it is shown that the singularly continuous spectrum of H is empty. Within the framework of the \(L^ 2\)-spectral theory, let \({\mathcal H}_{comp}\) be the space of elements of \({\mathcal H}\) with compact support, \({\mathcal H}_{loc}\) the space of elements which are locally in \({\mathcal H}\) and \({\mathcal L}_ s({\mathcal H}_{comp},{\mathcal H}_{loc})\) the space of operators of \({\mathcal H}_{comp}\) into \({\mathcal H}_{loc}\) endowed with the topology of simple convergence. The central result of this paper is that the resolvent \((H- \lambda)^{-1}\) \((\lambda \in {\mathbb{C}}\setminus [\mu_ 1,+\infty)),\) where \(\mu_ 1=\inf_{\mu \in {\mathcal M}}(\mu)\), considered as a function with values into \({\mathcal L}_ s({\mathcal H}_{comp},{\mathcal H}_{loc})\), admits a meromorphic continuation to the Riemann surface which is associated to the square roots (\(\sqrt{\lambda -\mu}\), \(\mu\in {\mathcal M})\). The study of the resolvent allows one to define a system of generalized eigenfunctions for H connected with the continuous spectrum. In the locally symmetric case such a system of generalized eigenfunctions is given by the Eisenstein series. The nature of the absolutely continuous part using these functions, which satisfy a family of functional equations, is pointed out. Certain spectral invariants of the considered systems, called transfer coefficients, are defined and their general properties are established. Additional properties of these coefficients in the particular case when X is a non-compact \({\mathbb{Q}}\)- rank one locally symmetric space of finite volume or H is a Hamiltonian with super-symmetry, are given.