We derive an adiabatic-type theorem that expresses the section determinants of spectral projections of a self-adjoint operator through the solution to a Wiener-Hopf equation. The solution theory of this equation is developed and for a special case a concrete criterion that ensures uniqueness of the solution is presented. Furthermore, for a special class of operators a dichotomy criterion, which is used in the proof of the adiabatic theorem, is proved.

Section determinant, spectral projection, operator-valued Wiener-Hopf equation, dichotomy

47B25 (47A62, 47A10, 15A15)