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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)

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
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[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)
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