Abstract. We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L 1 estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.

For the BCS equation with local two-body interaction λV (x), we give a rigorous analysis of the asymptotic behavior of the critical temperature as λ → 0. We derive necessary and sufficient conditions on V (x) for the existence of a non-trivial solution for all values of λ > 0.

Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses, we consider the Weyl M‐function of extensions of the operators. The extensions are determined by abstract boundary conditions and we establish results on the relationship between the M‐function as an analytic function of a spectral parameter and the spectrum of the extension. We also give an example where the M‐function does not contain the whole spectral information of the resolvent, and show that the results can be applied to elliptic PDEs where the M‐function corresponds to the Dirichlet to Neumann map.

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