Exact and approximate energy sums in potential wells

Abstract

Sums of the N lowest energy levels for quantum particles bound by potentials are calculated, emphasising the semiclassical regime N ≫ 1. Euler-Maclaurin summation, together with a regularisation, gives a formula for these energy sums, involving only the levels N + 1, N + 2…. For the harmonic oscillator and the particle in a box, the formula is exact. For wells where the levels are known approximately (e.g. as a WKB series), with the higher levels being more accurate, the formula improves accuracy by avoiding the lower levels. For a linear potential, the formula gives the first Airy zero with an error of order 10−7. For the Pöschl–Teller potential, regularisation is not immediately applicable but the energy sum can be calculated exactly; its semiclassical approximation depends on how N and the well depth are linked. In more dimensions, the Euler–Maclaurin technique is applied to give an analytical formula for the energy sum for a free particle on a torus, using levels determined by the smoothed spectral staircase plus some oscillatory corrections from short periodic orbits.