Exact and approximate energy sums in potential wells (bibtex)

by M V Berry, Kieron Burke

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.

Reference:

Exact and approximate energy sums in potential wells M V Berry, Kieron Burke, Journal of Physics A: Mathematical and Theoretical 53, 095203 (2020).

Bibtex Entry:

@article{BB19b, Pub-num = {196}, Title = {Exact and approximate energy sums in potential wells}, Author = {M V Berry, Kieron Burke}, 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.}, Doi = {10.1088/1751-8121/ab69a6}, %% Issn = {}, Year = {2020}, Month = {Feb}, Journal = {Journal of Physics A: Mathematical and Theoretical}, Volume = {53}, %% Issue = {}, Number = {9}, Pages = {095203}, Publisher = {{IOP} Publishing}, Url = {https://doi.org/10.1088%2F1751-8121%2Fab69a6}, %% arXiv = {}, %% keywords = {} %%}

Powered by bibtexbrowser