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 	   = {}
%%}
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