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**Spectral properties of the equation (' + ieA) ' u = '?** / Elton, Daniel M.

Research output: Contribution to journal › Journal article › peer-review

Elton, DM 2001, 'Spectral properties of the equation (' + ieA) ' u = '?', *Proceedings of the Royal Society of Edinburgh: Section A Mathematics*, vol. 131, no. 5, pp. 1065-1089. https://doi.org/10.1017/S030821050000127X

Elton, D. M. (2001). Spectral properties of the equation (' + ieA) ' u = '? *Proceedings of the Royal Society of Edinburgh: Section A Mathematics*, *131*(5), 1065-1089. https://doi.org/10.1017/S030821050000127X

Elton DM. Spectral properties of the equation (' + ieA) ' u = '? Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 2001 Oct 1;131(5):1065-1089. https://doi.org/10.1017/S030821050000127X

@article{a6cb190069a64381839536cbe40b2682,

title = "Spectral properties of the equation (' + ieA) ' u = '?",

abstract = "We develop a spectral theory for the equation ( + ieA) × u = ±mu on Minkowski 3-space (one time variable and two space variables); here, A is a real vector potential and the vector product is defined with respect to the Minkowski metric. This equation was formulated by Elton and Vassiliev, who conjectured that it should have properties similar to those of the two-dimensional Dirac equation. Our equation contains a large parameter c (speed of light), and this motivates the study of the asymptotic behaviour of its spectrum as c → +∞. We show that the essential spectrum of our equation is the same as that of Dirac (theorem 3.1), whereas the discrete spectrum agrees with Dirac to a relative accuracy δλ/mc2 ~ O(c−4) (theorem 3.3). In other words, we show that our equation has the same accuracy as the two-dimensional Pauli equation, its advantage over Pauli being relativistic invariance.",

author = "Elton, {Daniel M.}",

note = "The final, definitive version of this article has been published in the Journal, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 131 (5), pp 1065-1089 2001, {\textcopyright} 2001 Cambridge University Press. RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics",

year = "2001",

month = oct,

day = "1",

doi = "10.1017/S030821050000127X",

language = "English",

volume = "131",

pages = "1065--1089",

journal = "Proceedings of the Royal Society of Edinburgh: Section A Mathematics",

issn = "0308-2105",

publisher = "Cambridge University Press",

number = "5",

}

TY - JOUR

T1 - Spectral properties of the equation (' + ieA) ' u = '?

AU - Elton, Daniel M.

N1 - The final, definitive version of this article has been published in the Journal, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 131 (5), pp 1065-1089 2001, © 2001 Cambridge University Press. RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics

PY - 2001/10/1

Y1 - 2001/10/1

N2 - We develop a spectral theory for the equation ( + ieA) × u = ±mu on Minkowski 3-space (one time variable and two space variables); here, A is a real vector potential and the vector product is defined with respect to the Minkowski metric. This equation was formulated by Elton and Vassiliev, who conjectured that it should have properties similar to those of the two-dimensional Dirac equation. Our equation contains a large parameter c (speed of light), and this motivates the study of the asymptotic behaviour of its spectrum as c → +∞. We show that the essential spectrum of our equation is the same as that of Dirac (theorem 3.1), whereas the discrete spectrum agrees with Dirac to a relative accuracy δλ/mc2 ~ O(c−4) (theorem 3.3). In other words, we show that our equation has the same accuracy as the two-dimensional Pauli equation, its advantage over Pauli being relativistic invariance.

AB - We develop a spectral theory for the equation ( + ieA) × u = ±mu on Minkowski 3-space (one time variable and two space variables); here, A is a real vector potential and the vector product is defined with respect to the Minkowski metric. This equation was formulated by Elton and Vassiliev, who conjectured that it should have properties similar to those of the two-dimensional Dirac equation. Our equation contains a large parameter c (speed of light), and this motivates the study of the asymptotic behaviour of its spectrum as c → +∞. We show that the essential spectrum of our equation is the same as that of Dirac (theorem 3.1), whereas the discrete spectrum agrees with Dirac to a relative accuracy δλ/mc2 ~ O(c−4) (theorem 3.3). In other words, we show that our equation has the same accuracy as the two-dimensional Pauli equation, its advantage over Pauli being relativistic invariance.

U2 - 10.1017/S030821050000127X

DO - 10.1017/S030821050000127X

M3 - Journal article

VL - 131

SP - 1065

EP - 1089

JO - Proceedings of the Royal Society of Edinburgh: Section A Mathematics

JF - Proceedings of the Royal Society of Edinburgh: Section A Mathematics

SN - 0308-2105

IS - 5

ER -