A Note on the Spectrum of Magnetic Dirac Operators
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this article, we study the spectrum of the magnetic Dirac operator, and the magnetic Dirac operator with potential over complete Riemannian manifolds. We find sufficient conditions on the potentials as well as the manifold so that the spectrum is either maximal, or discrete. We also show that magnetic Dirac operators can have a dense set of eigenvalues.
Keywords: Dirac operator, potentials, spectrum.
@article{SIGMA_2023_19_a101,
     author = {Nelia Charalambous and Nadine Grosse},
     title = {A {Note} on the {Spectrum} of {Magnetic} {Dirac} {Operators}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2023},
     volume = {19},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a101/}
}
TY  - JOUR
AU  - Nelia Charalambous
AU  - Nadine Grosse
TI  - A Note on the Spectrum of Magnetic Dirac Operators
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2023
VL  - 19
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a101/
LA  - en
ID  - SIGMA_2023_19_a101
ER  - 
%0 Journal Article
%A Nelia Charalambous
%A Nadine Grosse
%T A Note on the Spectrum of Magnetic Dirac Operators
%J Symmetry, integrability and geometry: methods and applications
%D 2023
%V 19
%U http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a101/
%G en
%F SIGMA_2023_19_a101
Nelia Charalambous; Nadine Grosse. A Note on the Spectrum of Magnetic Dirac Operators. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a101/

[1] Agmon S., Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of $N$-body Schrödinger operators, Math. Notes, 29, Princeton University Press, Princeton, NJ, 1982 | MR | Zbl

[2] Ammann B., Große N., “$L^p$-spectrum of the Dirac operator on products with hyperbolic spaces”, Calc. Var. Partial Differential Equations, 55 (2016), 127, 36 pp., arXiv: 1405.2830 | DOI | MR | Zbl

[3] Cecchini S., Zeidler R., “Scalar curvature and generalized Callias operators”, Perspectives in Scalar Curvature, v. 1, World Scientific Publishing, Hackensack, NJ, 2023, 515–542 | DOI | MR

[4] Charalambous N., Lu Z., “On the spectrum of the Laplacian”, Math. Ann., 359 (2014), 211–238, arXiv: 1211.3225 | DOI | MR | Zbl

[5] Chernoff P.R., “Essential self-adjointness of powers of generators of hyperbolic equations”, J. Funct. Anal., 12 (1973), 401–414 | DOI | MR | Zbl

[6] Chernoff P.R., “Schrödinger and Dirac operators with singular potentials and hyperbolic equations”, Pacific J. Math., 72 (1977), 361–382 | DOI | MR | Zbl

[7] Cycon H.L., Froese R.G., Kirsch W., Simon B., Schrödinger operators, with application to quantum mechanics and global geometry, Texts Monog. Phys., Springer, Berlin, 1987 | DOI | MR | Zbl

[8] Friedrich T., Dirac-Operatoren in der Riemannschen Geometrie. Mit einem Ausblick auf die Seiberg–Witten-Theorie, Adv. Lect. Math., Friedr. Vieweg Sohn, Braunschweig, 1997 | DOI | MR

[9] Ginoux N., The Dirac spectrum, Lecture Notes in Math., 1976, Springer, Berlin, 2009 | DOI | MR | Zbl

[10] Helffer B., Semi-classical analysis for the Schrödinger operator and applications, Lecture Notes in Math., 1336, Springer, Berlin, 1988 | DOI | MR | Zbl

[11] Helffer B., Nourrigat J., Wang X.P., “Sur le spectre de l'équation de Dirac (dans ${\mathbb R}^3$ ou ${\mathbb R}^2$) avec champ magnétique”, Ann. Sci. École Norm. Sup., 22 (1989), 515–533 | DOI | MR | Zbl

[12] Lawson Jr. H.B., Michelsohn M.L., Spin geometry, Princeton Math. Ser., 38, Princeton University Press, Princeton, NJ, 1989 | MR | Zbl

[13] Miller K., Simon B., “Quantum magnetic Hamiltonians with remarkable spectral properties”, Phys. Rev. Lett., 44 (1980), 1706–1707 | DOI | MR | Zbl

[14] Paeng S.-H., “Relative volume comparison for asymptotically flat manifolds and rigidity of total mass”, Ann. Global Anal. Geom., 56 (2019), 567–580 | DOI | MR | Zbl

[15] Reed M., Simon B., Methods of modern mathematical physics. IV Analysis of operators, Academic Press, New York, 1978 | MR | Zbl

[16] Roe J., Elliptic operators, topology and asymptotic methods, Pitman Research Notes in Math. Ser., 395, Longman, Harlow, 1998 | DOI | MR | Zbl

[17] Savale N., “Koszul complexes, Birkhoff normal form and the magnetic Dirac operator”, Anal. PDE, 10 (2017), 1793–1844, arXiv: 1511.08545 | DOI | MR | Zbl

[18] Savale N., “A Gutzwiller type trace formula for the magnetic Dirac operator”, Geom. Funct. Anal., 28 (2018), 1420–1486, arXiv: 1806.10956 | DOI | MR | Zbl

[19] Shigekawa I., “Eigenvalue problems for the Schrödinger operator with the magnetic field on a compact Riemannian manifold”, J. Funct. Anal., 75 (1987), 92–127 | DOI | MR | Zbl

[20] Shigekawa I., “Spectral properties of Schrödinger operators with magnetic fields for a spin $\frac12$ particle”, J. Funct. Anal., 101 (1991), 255–285 | DOI | MR | Zbl

[21] Shubin M., “Essential self-adjointness for semi-bounded magnetic Schrödinger operators on non-compact manifolds”, J. Funct. Anal., 186 (2001), 92–116, arXiv: math.SP/0007019 | DOI | MR | Zbl

[22] Simon B., “Spectrum and continuum eigenfunctions of Schrödinger operators”, J. Funct. Anal., 42 (1981), 347–355 | DOI | MR | Zbl

[23] Thaller B., The Dirac equation, Texts Monog. Phys., Springer, Berlin, 1992 | DOI | MR

[24] Yamada O., “On the spectrum of Dirac operators with the unbounded potential at infinity”, Hokkaido Math. J., 26 (1997), 439–449 | DOI | MR | Zbl