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

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

Annales Henri Lebesgue, Tome 6 (2023), pp. 1031-1113

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.

La méthode de la transformée de jauge est l’un des principaux outils utilisés pour étudier le comportement spectral des opérateurs de Schrödinger périodiques et presque périodiques, autant d’un point de vue qualitatif que quantitatif. Dans cet article, nous généralisons cette méthode dans un contexte abstrait, nous permettant une plus grande flexibilité dans les applications, entre autres aux matrices d’opérateurs. En particulier, nous obtenons une expansion asymptotique de la densité d’états de certain systèmes d’opérateurs presque périodiques elliptiques, dont des opérateurs de Dirac. Nous démontrons aussi que plusieurs systèmes périodiques, incluant l’opérateur de Dirac bidimensionnel, possèdent la propriété de Bethe–Sommerfeld, comme quoi leur spectre contient un demi-axe, ou même deux demi-axes lorsqu’ils ne sont pas semibornés.

Classification : 35P20, 35J46, 47A55, 81Q10
Keywords: Periodic and almost-periodic problems, Gauge transform, Density of states, Bethe–Sommerfeld property, Dirac operators

[BP09] Barbatis, Gerassimos; Parnovski, Leonid Bethe–Sommerfeld conjecture for pseudodifferential perturbation, Commun. Partial Differ. Equations, Volume 34 (2009) no. 4-6, pp. 383-418 | DOI | MR | Zbl

[BS87] Birman, Mikhail S.; Solomjak, Mikhaĭl Spectral Theory of Self-Adjoint Operators in Hilbert Space, Mathematics and its Applications. Soviet Series, Kluwer Academic Publishers, 1987 | Zbl | DOI

[CMS73] Coburn, Lewis A.; Moyer, R. D.; Singer, Isadore M. $C^{*}$-algebras of almost periodic pseudo-differential operators, Acta Math., Volume 130 (1973), pp. 279-307 | DOI | MR | Zbl

[CVN08] Charles, Laurent; Vũ Ngọc, San Spectral asymptotics via the semiclassical Birkhoff normal form, Duke Math. J., Volume 143 (2008) no. 3, pp. 463-511 | DOI | MR | Zbl

[Dix81] Dixmier, Jacques Von Neumann Algebras, North-Holland Mathematical Library, 27, North-Holland, 1981 (with a preface by E. C. Lance, Translated from the second French edition by F. Jellett) | MR | Zbl

[DT82] Dahlberg, Bjorn E. J.; Trubowitz, Eugene A remark on two-dimensional periodic potentials, Comment. Math. Helv., Volume 57 (1982) no. 1, pp. 130-134 | DOI | MR | Zbl

[GM91] Gilbert, John E.; Murray, Margaret A. M. Clifford algebras and Dirac operators in harmonic analysis, Cambridge Studies in Advanced Mathematics, 26, Cambridge University Press, 1991 | DOI | MR | Zbl

[Hör07] Hörmander, Lars The analysis of linear partial differential operators. III Pseudo-differential operators, Classics in Mathematics, Springer, 2007 (reprint of the 1994 edition) | DOI | MR | Zbl

[Ivr19] Ivrii, Victor Complete Semiclassical Spectral Asymptotics for Periodic and Almost Periodic Perturbations of Constant Operators, Springer (2019), pp. 583-606 | Zbl

[Kuc93] Kuchment, Peter Floquet theory for partial differential equations, Operator Theory: Advances and Applications, 60, Birkhäuser, 1993 | DOI | MR | Zbl

[MPS14] Morozov, Sergey; Parnovski, Leonid; Shterenberg, Roman Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic pseudo-differential operators, Ann. Henri Poincaré, Volume 15 (2014) no. 2, pp. 263-312 | DOI | MR | Zbl

[Naĭ72] Naĭmark, Mark A. Normed algebras, Wolters-Noordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics, Wolters-Noordhoff Publishing, Groningen, 1972 (translated from the second Russian edition by Leo F. Boron) | MR | Zbl

[Par08] Parnovski, Leonid Bethe–Sommerfeld conjecture, Ann. Henri Poincaré, Volume 9 (2008) no. 3, pp. 457-508 | DOI | MR | Zbl

[PS01] Parnovski, Leonid; Sobolev, Alexander V. On the Bethe–Sommerfeld conjecture for the polyharmonic operator, Duke Math. J., Volume 107 (2001) no. 2, pp. 209-238 | DOI | MR | Zbl

[PS10] Parnovski, Leonid; Sobolev, Alexander V. Bethe–Sommerfeld conjecture for periodic operators with strong perturbations, Invent. Math., Volume 181 (2010) no. 3, pp. 467-540 | DOI | MR | Zbl

[PS12] Parnovski, Leonid; Shterenberg, Roman Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic Schrödinger operators, Ann. Math., Volume 176 (2012) no. 2, pp. 1039-1096 | DOI | MR | Zbl

[PS16] Parnovski, Leonid; Shterenberg, Roman Complete asymptotic expansion of the spectral function of multidimensional almost-periodic Schrödinger operators, Duke Math. J., Volume 165 (2016) no. 3, pp. 509-561 | DOI | MR | Zbl

[PS19] Parnovski, Leonid; Shterenberg, Roman Perturbation theory for almost-periodic potentials I: one-dimensional case, Commun. Math. Phys., Volume 366 (2019) no. 3, pp. 1229-1257 | DOI | MR | Zbl

[Roz78] Rozenbljum, Grigorii V. Near-similarity of operators and the spectral asymptotic behavior of pseudodifferential operators on the circle, Tr. Mosk. Mat. O.-va, Volume 36 (1978), p. 59-84, 294 | MR | Zbl

[Shu78] Shubin, Mikhail A. Almost periodic functions and partial differential operators, Usp. Mat. Nauk, Volume 33 (1978) no. 2, p. 3-47, 247 | MR

[Shu79a] Shubin, Mikhail A. Pseudodifferential almost-periodic operators and von Neumann algebras, Trans. Mosc. Math. Soc., Volume 35 (1979), pp. 103-166 | Zbl

[Shu79b] Shubin, Mikhail A. Spectral theory and the index of elliptic operators with almost-periodic coefficients, Usp. Mat. Nauk, Volume 34 (1979) no. 2, pp. 95-135 | MR | Zbl

[Skr85] Skriganov, Maxim M. Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Tr. Mat. Inst. Steklova, Volume 171 (1985), pp. 3-122 | MR

[Sob05] Sobolev, Alexander V. Integrated density of states for the periodic Schrödinger operator in dimension two, Ann. Henri Poincaré, Volume 6 (2005) no. 1, pp. 31-84 | DOI | MR | Zbl

[Sob06] Sobolev, Alexander V. Asymptotics of the integrated density of states for periodic elliptic pseudo-differential operators in dimension one, Rev. Mat. Iberoam., Volume 22 (2006) no. 1, pp. 55-92 | DOI | MR | Zbl

[Tay11] Taylor, Michael E. Partial differential equations II. Qualitative studies of linear equations, Applied Mathematical Sciences, 116, Springer, 2011 | DOI | MR | Zbl

[Tha91] Thaller, Bernd The Dirac equation, Texts and Monographs in Physics, Springer, 1991 | DOI | MR | Zbl

[Upm02] Upmeier, Harald Dirac operator and real structure on Euclidean and Minkowski spacetime, Noncommutative geometry and the standard model of elementary particle physics (Hesselberg, 1999) (Scheck, Florian et al., eds.) (Lecture Notes in Physics), Volume 596, Springer, 2002, pp. 136-151 | DOI | MR | Zbl

[Wei77] Weinstein, Alan Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J., Volume 44 (1977) no. 4, pp. 883-892 | MR | Zbl