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A mathematical framework is constructed for the sum of the lowest Neigenvalues of a potential. Exactness is illustrated on several model systems(harmonic oscillator, particle in a box, and Poschl-Teller well). Itsorder-by-order semiclassical expansion reduces to the gradient expansion forslowly-varying densities, but also yields a correction when the system isfinite and the spectrum discrete. Some singularities can be avoided whenevaluating the correction to the leading term. Explicit corrections to thegradient expansion to the kinetic energy in one dimension are found which, insimple cases, greatly improve accuracy. We discuss the relevance to practicaldensity functional calculations.

Explicit corrections to the gradient expansion for the kinetic energy in one dimension