Spectral Moments and Continuum Perturbation Theory

Abstract

The $p\mathrm{th}$ spectral moment is shown to be given exactly by the perturbation expansion of the resolvent operator carried to order $p$. Any individual term of order $n$ in the perturbation expansion makes an identically zero contribution to the $p\mathrm{th}$ moment for $n$ greater than $p$. The above is true even though perturbation theory is nonconvergent. The result may be useful in cases where perturbation theory converges for high energy but not for low energy. An application to the problem of band structure "tails" in impure crystals is suggested.