A General Dixmier Trace Formula for the Density of States on Open Manifolds

Eva-Maria Hekkelman ; Edward McDonald

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

We give an abstract formulation of the Dixmier trace formula for the density of states. This recovers prior versions and allows us to provide a Dixmier trace formula for the density of states of second order elliptic differential operators on manifolds of bounded geometry satisfying a certain geometric condition. This formula gives a new perspective on Roe's index on open manifolds.

Détail

Keywords: density of states, index theory, Dixmier trace, singular trace, bounded geometry, manifolds.

Bibliographie

[1] Adachi T., Sunada T., “Density of states in spectral geometry”, Comment. Math. Helv., 68 (1993), 480–493 | DOI | MR | Zbl

[2] Aizenman M., Warzel S., Random operators. Disorder effects on quantum spectra and dynamics, Grad. Stud. Math., 168, American Mathematical Society, Providence, RI, 2015 | DOI | MR | Zbl

[3] Alexander S., Orbach R., “Density of states on fractals: “fractons””, J. Physique Lett., 43 (1982), 625–631 | DOI

[4] Atiyah M.F., Singer I.M., “The index of elliptic operators on compact manifolds”, Bull. Amer. Math. Soc., 69 (1963), 422–433 | DOI | MR | Zbl

[5] Azamov N., Carey A., Dodds P., Sukochev F., “Operator integrals, spectral shift, and spectral flow”, Canad. J. Math., 61 (2009), 241–263, arXiv: math.OA/0703442 | DOI | MR | Zbl

[6] Azamov N., Hekkelman E., McDonald E., Sukochev F., Zanin D., “An application of singular traces to crystals and percolation”, J. Geom. Phys., 179 (2022), 104608, 22 pp., arXiv: 2202.03676 | DOI | MR | Zbl

[7] Azamov N., McDonald E., Sukochev F., Zanin D., “A Dixmier trace formula for the density of states”, Comm. Math. Phys., 377 (2020), 2597–2628, arXiv: 1910.12380 | DOI | MR | Zbl

[8] Bellissard J., van Elst A., Schulz-Baldes H., “The noncommutative geometry of the quantum Hall effect”, J. Math. Phys., 35 (1994), 5373–5451, arXiv: cond-mat/9411052 | DOI | MR | Zbl

[9] Berezin F.A., Shubin M.A., The Schrödinger equation, Math. Appl. (Soviet Ser.), 66, Kluwer Academic Publishers Group, Dordrecht, 1991 | DOI | MR | Zbl

[10] Birman M.Sh., Solomjak M.Z., “Estimates of singular numbers of integral operators. III Operators in unbounded domains”, Vestnik Leningrad. Univ., 24 (1969), 35–48 | MR | Zbl

[11] Bourgain J., Klein A., “Bounds on the density of states for Schrödinger operators”, Invent. Math., 194 (2013), 41–72, arXiv: 1112.1716 | DOI | MR | Zbl

[12] Bourne C., Prodan E., “Non-commutative Chern numbers for generic aperiodic discrete systems”, J. Phys. A, 51 (2018), 235202, 38 pp., arXiv: 1712.04136 | DOI | MR | Zbl

[13] Carmona R., Lacroix J., Spectral theory of random Schrödinger operators, Probab. Appl., Birkhäuser, Boston, MA, 1990 | DOI | MR | Zbl

[14] Chavel I., Riemannian geometry. A modern introduction, Cambridge Stud. Adv. Math., 98, 2nd ed., Cambridge University Press, Cambridge, 2006 | DOI | MR | Zbl

[15] Chayes J.T., Chayes L., Franz J.R., Sethna J.P., Trugman S.A., “On the density of states for the quantum percolation problem”, J. Phys. A, 19 (1986), L1173–L1177 | DOI | MR

[16] Connes A., “The action functional in noncommutative geometry”, Comm. Math. Phys., 117 (1988), 673–683 | DOI | MR | Zbl

[17] Conway J.B., A course in functional analysis, Grad. Texts in Math., 96, 2nd ed., Springer, New York, 1990 | DOI | MR | Zbl

[18] Doi S.-i., Iwatsuka A., Mine T., “The uniqueness of the integrated density of states for the Schrödinger operators with magnetic fields”, Math. Z., 237 (2001), 335–371 | DOI | MR | Zbl

[19] Eichhorn J., “The boundedness of connection coefficients and their derivatives”, Math. Nachr., 152 (1991), 145–158 | DOI | MR

[20] Eichhorn J., Global analysis on open manifolds, Nova Science Publishers, New York, 2007 | MR | Zbl

[21] Eichhorn J., Relative index theory, determinants and torsion for open manifolds, World Scientific Publishing, Hackensack, NJ, 2009 | DOI | MR | Zbl

[22] Greene R.E., “Complete metrics of bounded curvature on noncompact manifolds”, Arch. Math. (Basel), 31 (1978), 89–95 | DOI | MR | Zbl

[23] Grimaldi R., Pansu P., “Bounded geometry, growth and topology”, J. Math. Pures Appl., 95 (2011), 85–98, arXiv: 1008.4887 | DOI | MR | Zbl

[24] Große N., Schneider C., “Sobolev spaces on Riemannian manifolds with bounded geometry: general coordinates and traces”, Math. Nachr., 286 (2013), 1586–1613, arXiv: 1301.2539 | DOI | MR | Zbl

[25] Hof A., “Some remarks on discrete aperiodic Schrödinger operators”, J. Stat. Phys., 72 (1993), 1353–1374 | DOI | MR | Zbl

[26] Kirsch W., Müller P., “Spectral properties of the Laplacian on bond-percolation graphs”, Math. Z., 252 (2006), 899–916, arXiv: math-ph/0407047 | DOI | MR | Zbl

[27] Kodani S., “An estimate on the volume of metric balls”, Kodai Math. J., 11 (1988), 300–305 | DOI | MR | Zbl

[28] Kordyukov Yu.A., “$L^p$-theory of elliptic differential operators on manifolds of bounded geometry”, Acta Appl. Math., 23 (1991), 223–260 | DOI | MR | Zbl

[29] Kosaki H., “An inequality of Araki–Lieb–Thirring (von Neumann algebra case)”, Proc. Amer. Math. Soc., 114 (1992), 477–481 | DOI | MR | Zbl

[30] Lang R., Spectral theory of random Schrödinger operators. A genetic introduction, Lecture Notes in Math., 1498, Springer, Berlin, 1991 | DOI | MR | Zbl

[31] Lenz D., Peyerimhoff N., Post O., Veselić I., “Continuity properties of the integrated density of states on manifolds”, Jpn. J. Math., 3 (2008), 121–161, arXiv: 0705.1079 | DOI | MR | Zbl

[32] Lenz D., Peyerimhoff N., Veselić I., “Integrated density of states for random metrics on manifolds”, Proc. Lond. Math. Soc., 88 (2004), 733–752, arXiv: math-ph/0212058 | DOI | MR | Zbl

[33] Lenz D., Peyerimhoff N., Veselić I., “Groupoids, von Neumann algebras and the integrated density of states”, Math. Phys. Anal. Geom., 10 (2007), 1–41, arXiv: math-ph/0203026 | DOI | MR | Zbl

[34] Levitina G., Sukochev F., Zanin D., “Cwikel estimates revisited”, Proc. Lond. Math. Soc., 120 (2020), 265–304, arXiv: 1703.04254 | DOI | MR | Zbl

[35] Lord S., Sukochev F., Zanin D., Singular traces, v. 1, De Gruyter Stud. Math., 46/1, Theory, De Gruyter, Berlin, 2021 | DOI | MR | Zbl

[36] Lord S., McDonald E., Sukochev F., Zanin D., Singular traces, v. 2, De Gruyter Stud. Math, 46/2, Trace formulas, De Gruyter, Berlin, 2023 | DOI | MR | Zbl

[37] Pastur L., Figotin A., Spectra of random and almost-periodic operators, Grundlehren Math. Wiss., 297, Springer, Berlin, 1992 | DOI | MR | Zbl

[38] Petersen P., Riemannian geometry, Grad. Texts in Math., 171, 2nd ed., Springer, New York, 2006 | DOI | MR | Zbl

[39] Peyerimhoff N., Veselić I., “Integrated density of states for ergodic random Schrödinger operators on manifolds”, Geom. Dedicata, 91 (2002), 117–135, arXiv: math-ph/0210047 | DOI | MR | Zbl

[40] Roe J., “An index theorem on open manifolds. I”, J. Differential Geom., 27 (1988), 87–113 | DOI | MR | Zbl

[41] Roe J., Elliptic operators, topology and asymptotic methods, Pitman Res. Notes Math. Ser., 395, 2nd ed., Longman, Harlow, 1998 | DOI | MR | Zbl

[42] Rudin W., Functional analysis, Internat. Ser. Pure Appl. Math., 2nd ed., McGraw-Hill, New York, 1991 | MR | Zbl

[43] Sakai T., Riemannian geometry, Transl. Math. Monogr., 149, American Mathematical Society, Providence, RI, 1996 | DOI | MR | Zbl

[44] Shubin M.A., “The spectral theory and the index of elliptic operators with almost periodic coefficients”, Russian Math. Surveys, 34 (1979), 109–157 | DOI | MR | Zbl

[45] Shubin M.A., “Spectral theory of elliptic operators on noncompact manifolds”, Astérisque, 207, 1992, 35–108 | MR | Zbl

[46] Simon B., “Schrödinger semigroups”, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447–526 | DOI | MR | Zbl

[47] Simon B., Trace ideals and their applications, Math. Surveys Monogr., 120, 2nd ed., American Mathematical Society, Providence, RI, 2005 | DOI | MR | Zbl

[48] Simon B., Operator theory: A comprehensive course in analysis, v. 4, American Mathematical Society, Providence, RI, 2015 | DOI | MR | Zbl

[49] Strichartz R.S., “Spectral asymptotics revisited”, J. Fourier Anal. Appl., 18 (2012), 626–659, arXiv: 1012.0272 | DOI | MR | Zbl

[50] Sukochev F., Zanin D., The Connes character formula for locally compact spectral triples, arXiv: 1803.01551

[51] Veselić I., “Spectral analysis of percolation Hamiltonians”, Math. Ann., 331 (2005), 841–865, arXiv: math-ph/0405006 | DOI | MR | Zbl

[52] Veselić I., Existence and regularity properties of the integrated density of states of random Schrödinger operators, Lecture Notes in Math., 1917, Springer, Berlin, 2008 | DOI | MR | Zbl

[53] Wegge-Olsen N.E., $K$-theory and $C^*$-algebras. A friendly approach, Oxford Sci. Publ., The Clarendon Press, Oxford University Press, New York, 1993 | MR

[54] Wegner F., “Bounds on the density of states in disordered systems”, Z. Phys. B, 44 (1981), 9–15 | DOI | MR