Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L2 (Rd )

Jean Michel Combes; Peter D. Hislop

doi:10.1007/978-3-030-68490-7_6

Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L² (R^d )

Jean Michel Combes, Peter D. Hislop

Mathematics

Producción científica: Conference contribution › revisión exhaustiva

Resumen

We prove some local estimates on the trace of spectral projectors for random Schrödinger operators restricted to cubes ⊂ R^d . We also present a new proof of the spectral averaging result based on analytic perturbation theory. Together, these provide another proof of the Wegner estimate with an explicit form of the constant and an alternate proof of the Birman-Solomyak formula. We also use these results to prove the Lipschitz continuity of the local density of states function for a restricted family of random Schrödinger operators on cubes ⊂ R^d, for d 1. The result holds for low energies without a localization assumption but is not strong enough to extend to the infinite-volume limit.

Idioma original	English
Título de la publicación alojada	Schrödinger Operators, Spectral Analysis and Number Theory - In Memory of Erik Balslev
Editores	Sergio Albeverio, Anindita Balslev, Ricardo Weder
Páginas	117-132
Número de páginas	16
DOI	https://doi.org/10.1007/978-3-030-68490-7_6
Estado	Published - 2021
Evento	Conference to celebrate Erik Balslev’s 75th birthday, 2010 - Aarhus, Denmark Duración: oct 1 2010 → oct 2 2010

Serie de la publicación

Nombre	Springer Proceedings in Mathematics and Statistics
Volumen	348
ISSN (versión impresa)	2194-1009
ISSN (versión digital)	2194-1017

Conference

Conference	Conference to celebrate Erik Balslev’s 75th birthday, 2010
País/Territorio	Denmark
Ciudad	Aarhus
Período	10/1/10 → 10/2/10

Nota bibliográfica

Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021.

ASJC Scopus subject areas

General Mathematics

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10.1007/978-3-030-68490-7_6

Otros archivos y enlaces

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Combes, J. M., & Hislop, P. D. (2021). Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L² (R^d ). En S. Albeverio, A. Balslev, & R. Weder (Eds.), Schrödinger Operators, Spectral Analysis and Number Theory - In Memory of Erik Balslev (pp. 117-132). (Springer Proceedings in Mathematics and Statistics; Vol. 348). https://doi.org/10.1007/978-3-030-68490-7_6

Combes, Jean Michel ; Hislop, Peter D. / Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L² (R^d ). Schrödinger Operators, Spectral Analysis and Number Theory - In Memory of Erik Balslev. editor / Sergio Albeverio ; Anindita Balslev ; Ricardo Weder. 2021. pp. 117-132 (Springer Proceedings in Mathematics and Statistics).

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abstract = "We prove some local estimates on the trace of spectral projectors for random Schr{\"o}dinger operators restricted to cubes ⊂ Rd . We also present a new proof of the spectral averaging result based on analytic perturbation theory. Together, these provide another proof of the Wegner estimate with an explicit form of the constant and an alternate proof of the Birman-Solomyak formula. We also use these results to prove the Lipschitz continuity of the local density of states function for a restricted family of random Schr{\"o}dinger operators on cubes ⊂ Rd, for d 1. The result holds for low energies without a localization assumption but is not strong enough to extend to the infinite-volume limit.",

keywords = "Density of states, Random schr{\"o}dinger operators, Spectral averaging",

author = "Combes, \{Jean Michel\} and Hislop, \{Peter D.\}",

note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive license to Springer Nature Switzerland AG 2021.; Conference to celebrate Erik Balslev{\textquoteright}s 75th birthday, 2010 ; Conference date: 01-10-2010 Through 02-10-2010",

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doi = "10.1007/978-3-030-68490-7\_6",

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series = "Springer Proceedings in Mathematics and Statistics",

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Combes, JM & Hislop, PD 2021, Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L² (R^d ). En S Albeverio, A Balslev & R Weder (eds.), Schrödinger Operators, Spectral Analysis and Number Theory - In Memory of Erik Balslev. Springer Proceedings in Mathematics and Statistics, vol. 348, pp. 117-132, Conference to celebrate Erik Balslev’s 75th birthday, 2010, Aarhus, Denmark, 10/1/10. https://doi.org/10.1007/978-3-030-68490-7_6

Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L² (R^d ). / Combes, Jean Michel; Hislop, Peter D.
Schrödinger Operators, Spectral Analysis and Number Theory - In Memory of Erik Balslev. ed. / Sergio Albeverio; Anindita Balslev; Ricardo Weder. 2021. p. 117-132 (Springer Proceedings in Mathematics and Statistics; Vol. 348).

Producción científica: Conference contribution › revisión exhaustiva

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T1 - Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L2 (Rd )

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AU - Hislop, Peter D.

N1 - Publisher Copyright: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021.

PY - 2021

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N2 - We prove some local estimates on the trace of spectral projectors for random Schrödinger operators restricted to cubes ⊂ Rd . We also present a new proof of the spectral averaging result based on analytic perturbation theory. Together, these provide another proof of the Wegner estimate with an explicit form of the constant and an alternate proof of the Birman-Solomyak formula. We also use these results to prove the Lipschitz continuity of the local density of states function for a restricted family of random Schrödinger operators on cubes ⊂ Rd, for d 1. The result holds for low energies without a localization assumption but is not strong enough to extend to the infinite-volume limit.

AB - We prove some local estimates on the trace of spectral projectors for random Schrödinger operators restricted to cubes ⊂ Rd . We also present a new proof of the spectral averaging result based on analytic perturbation theory. Together, these provide another proof of the Wegner estimate with an explicit form of the constant and an alternate proof of the Birman-Solomyak formula. We also use these results to prove the Lipschitz continuity of the local density of states function for a restricted family of random Schrödinger operators on cubes ⊂ Rd, for d 1. The result holds for low energies without a localization assumption but is not strong enough to extend to the infinite-volume limit.

KW - Density of states

KW - Random schrödinger operators

KW - Spectral averaging

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Combes JM, Hislop PD. Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L² (R^d ). En Albeverio S, Balslev A, Weder R, editores, Schrödinger Operators, Spectral Analysis and Number Theory - In Memory of Erik Balslev. 2021. p. 117-132. (Springer Proceedings in Mathematics and Statistics). doi: 10.1007/978-3-030-68490-7_6