Results are obtained on resolvent expansions around zero energy for Schrödinger operators H=-Δ+V(x) on L2(Rm), where V(x) is a sufficiently rapidly decaying real potential. The emphasis is on a unified approach, valid in all dimensions, which does not require one to distinguish between ∫V(x)dx=0 and ∫V(x)dx≠0 in dimensions m=1,2. It is based on a factorization technique and repeated decomposition of the Lippmann–Schwinger operator. Complete results are given in dimensions m=1 and m=2.

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A general adiabatic expansion is written down. The basic result is that, under appropriate smoothness conditions, the adiabatic estimations can be pushed up to exponential order. The results imply exponential estimates not only for hamiltonians analytic in a neighbourhood of the real axis but also for hamiltonians which (in an appropriate sense) beong to Gevrey classes.

Some general results about perturbations of not-semibounded selfadjoint operators by quadratic forms are obtained. These are applied to obtain the distinguished self-adjoint extension for Dirac operators with singular potentials (including potentials dominated by the Coulomb potential with Z<137). The distinguished self-adjoint extension, is the unique selfadjoint extension, for which the wave functions in its domain possess finite mean kinetic energy. It is shown moreover that the essential spectrum of the distinguished extension is contained in the spectrum of the free Hamiltonian.

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