Spectral estimates for Schrödinger and Dirac-type operators on Riemannian manifolds. (English) Zbl 0791.58094
From a Hilbertian theorem (which generalizes a Theorem of S. Gallot and D. Meyer), we deduce a general criterion to compare the spectra of two operators \(T\) and \(T'\) acting on Hilbert spaces \(H\) and \(H'\), under the assumption that they verify Kato’s inequality with respect to a map \(\omega: H' \to H\). We apply this principle in several contexts. We obtain estimates for the spectra of Schrödinger and Dirac-type operators by the spectrum of the Riemannian base manifold. Our results contain, as particular cases, previous estimates given by other authors (S. Gallot and D. Meyer for the Hodge-de Rham Laplacian acting on differential forms, T. Friedrich for Dirac operator acting on spinors). The criterion, applied to Riemannian submersions, or coverings, or local quasi-isometries, gives estimates of the spectrum of the total space in terms of the spectrum of the base manifold.
Reviewer: M.Bordoni (Roma)
MSC:
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 53C20 | Global Riemannian geometry, including pinching |
| 53C27 | Spin and Spin\({}^c\) geometry |
Keywords:
comparison of spectra; Schrödinger; Hodge-de Rham; Dirac operator; Riemannian submersions; coverings; local quasi-isometriesReferences:
| [1] | B?rard, P.: Spectral geometry: Direct and inverse problem, with an appendix of G. Besson. (Lect. Notes Math., vol. 1207) Berlin Heidelberg New York: Springer 1986 |
| [2] | B?rard, P., Gallot, S.: In?galit?s isop?rimetriques pour l’?quation de la chaleur et applications ? l’estimation de quelques invariants g?om?triques. In: S?minaire Goulaouic-Meyer-Schwarz 1983-84. Palaiseau: ?cole Polytechnique 1984 |
| [3] | B?rard Bergery, L., Bourguignon, J.P.: Laplacians and Riemannian submersions with totally geodesic fibers. III. J. Math.26 (n. 2), 181-200 (1982) · Zbl 0483.58021 |
| [4] | Berger, M., Gauduchon, P., Mazet, E.: Le spectre d’une vari?t? riemannienne. (Lect. Notes Math., vol. 194) Berlin Heidelberg New York: Springer 1971 · Zbl 0223.53034 |
| [5] | Besson, G., Bordoni, M.: On the spectrum of Riemannian submersions with totally geodesic fibers. Rend. Mat. Accad. Naz. Lincei, Ser. IX1, 335-340 (1990) · Zbl 0716.53031 |
| [6] | Bochner, S., Yano, K.: Curvature and Betti numbers. (Ann. Math. Stud., vol. 32) Princeton: Princeton University Press 1953 · Zbl 0051.39402 |
| [7] | Bordoni, M.: An estimate for finite sums of eigenvalues of fiber spaces. C.R. Acad. Sci. Paris, Ser. I315, 1079-1083 (1992) · Zbl 0761.53019 |
| [8] | Bourguignon, J.P.: Formules de Weitzenb?ck en dimension 4. In: B?rard Bergery et al. (eds.) S?minaire Arthur Besse: G?om?trie riemannienne en dimension 4. (Textes Math. vol. 3, pp. 308-333) Paris: Cedic Fernard Nathan 1981 |
| [9] | Burago, Yu.D., Zalgaller, V.A.: Geometric inequalities (Grundlehren Math. Wiss., vol. 285) Berlin Heidelberg New York: Springer 1988 · Zbl 0633.53002 |
| [10] | Cheeger, J., Ebin, D.G.: Comparison theorems in Riemannian geometry. Amsterdam Oxford: North-Holland 1975 · Zbl 0309.53035 |
| [11] | Cheng, S.Y.: Eigenvalue comparison theorems and its geometric applications. Math. Z.143, 289-297 (1975) · Zbl 0329.53035 · doi:10.1007/BF01214381 |
| [12] | Colin de Verdi?re, Y.: Spectre de vari?t?s riemanniennes et spectre de graphes. In: Gleason, A.M. (ed.) Proc. Int. Cong. of Math. Berkeley 1986, pp. 522-530. Providence, RI: Am. Math. Soc. 1987 |
| [13] | Colin de Verdi?re, Y.: Construction de laplaciens dont une partie finie du spectre est donn?e. Ann. Sci. ?c. Norm. Sup?r., IV. S?r.20, 599-615 (1987) · Zbl 0636.58036 |
| [14] | Courtois, G.: Comportement du spectre d’une vari?t? riemannienne compacte sous perturbation topologique par excision d’un domaine. Th?se de doctorat. Grenoble: Institut Fourier 1987 |
| [15] | Dvoretzky, A.: A theorem on convex bodies and applications to Banach spaces. Proc. Natl. Acad. Sci. USA45, 223-226 (1959) · Zbl 0088.31802 · doi:10.1073/pnas.45.2.223 |
| [16] | Dvoretzky, A.: Some results on convex bodies and Banach spaces. In: Proc. Symp. Linear Spaces. Jerusalem 1960, pp. 123-160. Jerusalem: Academic Press & Oxford: Pergamon 1961 |
| [17] | Gallot, S.: Sur les vari?t?s riemanniennes ? op?rateur de courbure positif v?rifiant certaines conditions sur leurp-spectre. C.R. Acad. Sci. Paris277, 457-459 (1973) · Zbl 0264.53020 |
| [18] | Gallot, S.: Isoperimetric inequalities based on integral norms of Ricci curvature. Ast?risque157-158, 191-216 (1988) |
| [19] | Gallot, S., Meyer, D.: Op?ratuer de courbure et laplaciens des formes diff?rentielles d’une vari?t? riemannienne. J. Math. Pures Appl.54, 259-284 (1975) · Zbl 0316.53036 |
| [20] | Gallot, S., Meyer, D.: D’un r?sultat hilbertien ? un principe de comparaison entre spectres. Applications. Ann. Sci. ?c. Norm. Sup?r., IV. S?r.21, 561-591 (1988) · Zbl 0722.53037 |
| [21] | Gallot, S., Meyer, D.: Sur la premi?re valeur propre dup-spectre des vari?t?s ? op?rateur de courbure positif. C.R. Acad. Sci. Paris276, 1619-1621 (1973) · Zbl 0256.53035 |
| [22] | Friedrich, T.: Der erste Eigenwert des Dirac-operators einer kompakten Riemannschen Mannigfaltgkeit nichtnegativer Skalar-Kr?mmung. Math. Nachr.97, 117-146 (1980) · Zbl 0462.53027 · doi:10.1002/mana.19800970111 |
| [23] | Hess, H., Schrader, R., Uhlenbrock, D.A.: Kato’s inequality and the spectral distribution of Laplacians on compact Riemannian manifolds. J. Differ. Geom.15, 27-38 (1980) · Zbl 0442.58032 |
| [24] | Hijazi, O.: A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors. Commun. Math. Phys.104, 151-162 (1986) · Zbl 0593.58040 · doi:10.1007/BF01210797 |
| [25] | Hijazi, O.: Lower bounds for eigenvalues of the Dirac operator through modified connections. (Preliminary version, 1992) |
| [26] | Lichnerowicz, A.: Spineurs harmoniques. C.R. Acad. Sci. Paris, S?r. I257, 7-9 (1963) |
| [27] | Meyer, D.: Sur les vari?t?s riemanniennes ? op?rateur de courbure positif ou nul. C.R. Acad. Sci. Paris, S?r. A272, 482-485 (1971) · Zbl 0209.25301 |
| [28] | Meyer, D.: Un lemme de g?om?trie hilbertienne et des applications ? la g?om?trie riemannienne. C.R. Acad. Sci. Paris, S?r. I295, 467-469 (1982) · Zbl 0515.53039 |
| [29] | Montiel, S., Ros, A.: Schr?dinger operators associated to a holomorphic map. In: Ferus, D. et al. (eds.) Global differ. geom. and global analysis. (Lect. Notes Math. vol. 1481, pp. 147-174) Berlin Heidelberg New York: Springer 1986 |
| [30] | Osserman, R.: A survey on minimal surfaces. New York: Dover 1986 · Zbl 0209.52901 |
| [31] | Tisk, J.: Eigenvalues estimates with applications to minimal surfaces. Pac. J. Math.128, 361-366 (1987) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
