The almost periodic Gauge Transform: An abstract scheme with applications to Dirac operators

Jean Lagacé; Sergey Morozov; Leonid Parnovski; Bernhard Pfirsch; Roman Shterenberg

doi:10.5802/ahl.184

The almost periodic Gauge Transform: An abstract scheme with applications to Dirac operators

Jean Lagacé, Sergey Morozov, Leonid Parnovski, Bernhard Pfirsch, Roman Shterenberg

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Abstract

One of the main tools used to understand both qualitative and quantitative spectral behaviour of periodic and almost periodic Schrödinger operators is the gauge transform method. In this paper, we extend this method to an abstract setting, thus allowing for greater flexibility in its applications that include, among others, matrix-valued operators. In particular, we obtain asymptotic expansions for the density of states of certain almost periodic systems of elliptic operators, including systems of Dirac type. We also prove that a range of periodic systems including the two-dimensional Dirac operators satisfy the Bethe–Sommerfeld property, that the spectrum contains a semi-axis — or indeed two semi-axes in the case of operators that are not semi-bounded.

Original language	English
Pages (from-to)	1031-1113
Journal	Annales Henri Lebesgue
Volume	6
DOIs	https://doi.org/10.5802/ahl.184
Publication status	Published - Dec 2023

Keywords

Periodic and almost-periodic problems, Gauge transform, Density of states, Bethe–Sommerfeld property, Dirac operators

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title = "The almost periodic Gauge Transform: An abstract scheme with applications to Dirac operators",

abstract = "One of the main tools used to understand both qualitative and quantitative spectral behaviour of periodic and almost periodic Schr{\"o}dinger operators is the gauge transform method. In this paper, we extend this method to an abstract setting, thus allowing for greater flexibility in its applications that include, among others, matrix-valued operators. In particular, we obtain asymptotic expansions for the density of states of certain almost periodic systems of elliptic operators, including systems of Dirac type. We also prove that a range of periodic systems including the two-dimensional Dirac operators satisfy the Bethe–Sommerfeld property, that the spectrum contains a semi-axis — or indeed two semi-axes in the case of operators that are not semi-bounded.",

keywords = "Periodic and almost-periodic problems, Gauge transform, Density of states, Bethe–Sommerfeld property, Dirac operators",

author = "Jean Lagac{\'e} and Sergey Morozov and Leonid Parnovski and Bernhard Pfirsch and Roman Shterenberg",

year = "2023",

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T1 - The almost periodic Gauge Transform

T2 - An abstract scheme with applications to Dirac operators

AU - Lagacé, Jean

AU - Morozov, Sergey

AU - Parnovski, Leonid

AU - Pfirsch, Bernhard

AU - Shterenberg, Roman

PY - 2023/12

Y1 - 2023/12

N2 - One of the main tools used to understand both qualitative and quantitative spectral behaviour of periodic and almost periodic Schrödinger operators is the gauge transform method. In this paper, we extend this method to an abstract setting, thus allowing for greater flexibility in its applications that include, among others, matrix-valued operators. In particular, we obtain asymptotic expansions for the density of states of certain almost periodic systems of elliptic operators, including systems of Dirac type. We also prove that a range of periodic systems including the two-dimensional Dirac operators satisfy the Bethe–Sommerfeld property, that the spectrum contains a semi-axis — or indeed two semi-axes in the case of operators that are not semi-bounded.

AB - One of the main tools used to understand both qualitative and quantitative spectral behaviour of periodic and almost periodic Schrödinger operators is the gauge transform method. In this paper, we extend this method to an abstract setting, thus allowing for greater flexibility in its applications that include, among others, matrix-valued operators. In particular, we obtain asymptotic expansions for the density of states of certain almost periodic systems of elliptic operators, including systems of Dirac type. We also prove that a range of periodic systems including the two-dimensional Dirac operators satisfy the Bethe–Sommerfeld property, that the spectrum contains a semi-axis — or indeed two semi-axes in the case of operators that are not semi-bounded.

KW - Periodic and almost-periodic problems, Gauge transform, Density of states, Bethe–Sommerfeld property, Dirac operators

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SP - 1031

EP - 1113

JO - Annales Henri Lebesgue

JF - Annales Henri Lebesgue

ER -

The almost periodic Gauge Transform: An abstract scheme with applications to Dirac operators

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