Semiclassical estimates for eigenvalue means of Laplacians on spheres. (English) Zbl 1519.58007
The purpose of the authors is to study properties of the Laplacian operator on the hypersphere and on the hyperhemisphere, respectively, and deduce asymptotics and Weyl-sharp upper and lower bound for Riesz-means and counting functions. More precisely, concerning the first result, they look for further terms in the asymptotic expansions for counting functions. Also, they examine subsequent terms in the expansion of the Riesz-mean. Regarding the second main result, they state that Berezin-Li-Yau bounds for Dirichlet eigenvalues remain true for a hypersphere in dimension tree, four, and five.
The article is endowed with an appendice on a duality principle for Riesz-means.
The article is endowed with an appendice on a duality principle for Riesz-means.
Reviewer: Mohammed El Aïdi (Bogotá)
MSC:
| 58C40 | Spectral theory; eigenvalue problems on manifolds |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J40 | Boundary value problems for higher-order elliptic equations |
| 53A05 | Surfaces in Euclidean and related spaces |
Keywords:
eigenvalues; Pólya’s conjecture; spheres and hemispheres; Riesz-means; Berezin-Li-Yau inequality; Kröger inequality; averaged variational principle; semiclassical expansions; asymptotically sharp estimatesSoftware:
DLMFReferences:
| [1] | Bérard, P.; Besson, G., Spectres et groupes cristallographiques. II. Domaines sphériques, Ann. Inst. Fourier (Grenoble), 30, 3, 237-248 (1980) · Zbl 0426.35073 · doi:10.5802/aif.800 |
| [2] | Berezin, FA, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat., 36, 1134-1167 (1972) · Zbl 0247.47019 |
| [3] | Berger, M.; Gauduchon, P.; Mazet, E., Le spectre d’une variété riemannienne (The spectrum of a Riemannian manifold) Lect. Notes Math (1971), Cham: Springer, Cham · Zbl 0223.53034 · doi:10.1007/BFb0064643 |
| [4] | Buoso, D.; Luzzini, P.; Provenzano, L.; Stubbe, J., On the spectral asymptotics for the buckling problem, J. Math. Phys., 62, 12, 18 (2021) · Zbl 1490.35480 · doi:10.1063/5.0069529 |
| [5] | Buoso, D., Luzzini, P., Provenzano, L., Stubbe, J.: Semiclassical expansions and estimates for eigenvalue means of Laplacians on compact homogeneous spaces. In preparation (2023) · Zbl 1490.35480 |
| [6] | Buoso, D.; Provenzano, L.; Stubbe, J., Semiclassical bounds for spectra of biharmonic operators, Rend. Mat. Appl., 7, 43-4, 267-314 (2022) · Zbl 1495.35134 |
| [7] | Canzani, Y., Galkowsi, J.: Weyl remainders: an application of geodesic beams. Invent. Math. (2023) · Zbl 1514.35294 |
| [8] | Cheng, Q-M; Yang, H., Estimates for eigenvalues on Riemannian manifolds, J. Differ. Equ., 247, 8, 2270-2281 (2009) · Zbl 1180.35390 · doi:10.1016/j.jde.2009.07.015 |
| [9] | El Soufi, A.; Harrell, EM II; Ilias, S.; Stubbe, J., On sums of eigenvalues of elliptic operators on manifolds, J. Spectr. Theory, 7, 4, 985-1022 (2017) · Zbl 1382.58023 · doi:10.4171/JST/183 |
| [10] | Filonov, N., Levitin, M., Polterovich, I. et al.: Póya’s conjecture for Euclidean balls. Invent. math. (2023). doi:10.1007/s00222-023-01198-1 |
| [11] | Folland, GB, Introduction to Partial Differential Equations. Math Notes (Princeton) (1976), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 0325.35001 · doi:10.1515/9780691213033 |
| [12] | Frank, RL; Geisinger, L., Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain, Mathematical Results in Quantum Physics, 138-147 (2011), Hackensack, NJ: World Scientific Publications, Hackensack, NJ · Zbl 1253.47031 · doi:10.1142/9789814350365_0012 |
| [13] | Frank, RL; Geisinger, L., Semi-classical analysis of the Laplace operator with Robin boundary conditions, Bull. Math. Sci., 2, 2, 281-319 (2012) · Zbl 1253.35090 · doi:10.1007/s13373-012-0028-5 |
| [14] | Frank, RL; Larson, S., Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain, J. Reine Angew. Math., 766, 195-228 (2020) · Zbl 1467.35107 · doi:10.1515/crelle-2019-0019 |
| [15] | Freitas, P., Asymptotic behaviour of extremal averages of Laplacian eigenvalues, J. Stat. Phys., 167, 6, 1511-1518 (2017) · Zbl 1372.35198 · doi:10.1007/s10955-017-1789-8 |
| [16] | Freitas, P., A remark on Pólya’s conjecture at low frequencies, Arch. Math. (Basel), 112, 3, 305-311 (2019) · Zbl 1409.35157 · doi:10.1007/s00013-018-1258-x |
| [17] | Freitas, P., Mao, J., Salavessa, I.: Pólya-type inequalities on spheres and hemispheres. arXiv:2204.07277, (2022) |
| [18] | Freitas, P., Salavessa, I.: Families of non-tiling domains satisfying pólya’s conjecture. arXiv:2204.08902, (2022) |
| [19] | Gallot, S.: Inégalités isopérimétriques et analytiques sur les variétés riemanniennes. Astérisque, (163-164):5-6, 31-91, 281 (1989), 1988. On the geometry of differentiable manifolds (Rome, 1986) · Zbl 0674.53001 |
| [20] | Geisinger, L.; Laptev, A.; Weidl, T., Geometrical versions of improved Berezin-Li-Yau inequalities, J. Spectr. Theory, 1, 1, 87-109 (2011) · Zbl 1257.35138 · doi:10.4171/JST/4 |
| [21] | Giné, E.; The, M., addition formula for the eigenfunctions of the Laplacian, Adv. Math., 18, 1, 102-107 (1975) · Zbl 0309.33013 · doi:10.1016/0001-8708(75)90003-1 |
| [22] | Gradshteyn, I.S., Ryzhik I.M.: Table of integrals, series, and products. Elsevier/Academic Press, Amsterdam, eighth edition, (2015). Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Revised from the seventh edition [MR2360010] · Zbl 0918.65002 |
| [23] | Harrell, EM II; Hermi, L., On Riesz means of eigenvalues, Commun. Partial Differ. Equ., 36, 9, 1521-1543 (2011) · Zbl 1237.35123 · doi:10.1080/03605302.2011.595865 |
| [24] | Harrell, EM II; Provenzano, L.; Stubbe, J., Complementary asymptotically sharp estimates for eigenvalue means of Laplacians, Int. Math. Res. Not. IMRN, 11, 8405-8450 (2021) · Zbl 1473.35372 · doi:10.1093/imrn/rnz085 |
| [25] | Harrell, EM II; Stubbe, J., On sums of graph eigenvalues, Linear Algebra Appl., 455, 168-186 (2014) · Zbl 1305.05127 · doi:10.1016/j.laa.2014.05.001 |
| [26] | Harrell, EM II; Stubbe, J., Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian, J. Spectr. Theory, 8, 4, 1529-1550 (2018) · Zbl 1402.58020 · doi:10.4171/JST/234 |
| [27] | Helgason, S., The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds, Acta Math., 113, 153-180 (1965) · Zbl 0163.16602 · doi:10.1007/BF02391776 |
| [28] | Helgason, S.: Groups and geometric analysis, volume 83 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2000. Integral geometry, invariant differential operators, and spherical functions, Corrected reprint of the (1984) original · Zbl 0543.58001 |
| [29] | Ilyin, A.; Laptev, A., Berezin-Li-Yau inequalities on domains on the sphere, J. Math. Anal. Appl., 473, 2, 1253-1269 (2019) · Zbl 1439.35350 · doi:10.1016/j.jmaa.2019.01.020 |
| [30] | Ivriĭ, VJ, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary, Funktsional. Anal. Prilozhen., 14, 2, 25-34 (1980) |
| [31] | Kröger, P., Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space, J. Funct. Anal., 106, 2, 353-357 (1992) · Zbl 0777.35044 · doi:10.1016/0022-1236(92)90052-K |
| [32] | Laptev, A., Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal., 151, 2, 531-545 (1997) · Zbl 0892.35115 · doi:10.1006/jfan.1997.3155 |
| [33] | Li, P.; Yau, ST, On the Schrödinger equation and the eigenvalue problem, Commun. Math. Phys., 88, 3, 309-318 (1983) · Zbl 0554.35029 · doi:10.1007/BF01213210 |
| [34] | Melas, AD, A lower bound for sums of eigenvalues of the Laplacian, Proc. Am. Math. Soc., 131, 2, 631-636 (2003) · Zbl 1015.58011 · doi:10.1090/S0002-9939-02-06834-X |
| [35] | Melrose, RB, Weyl’s conjecture for manifolds with concave boundary. Geometry of the Laplace operator, Honolulu/Hawaii 1979, Proc. Symp. Pure Math., 36, 257-274 (1980) · Zbl 0436.58024 · doi:10.1090/pspum/036/573438 |
| [36] | Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds).: NIST handbook of mathematical functions. U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge (2010). With 1 CD-ROM (Windows, Macintosh and UNIX) · Zbl 1198.00002 |
| [37] | Pólya, G., On the eigenvalues of vibrating membranes, Proc. Lond. Math. Soc., 3, 11, 419-433 (1961) · Zbl 0107.41805 · doi:10.1112/plms/s3-11.1.419 |
| [38] | Pólya, G.: Mathematics and plausible reasoning. Vol. II. Princeton University Press, Princeton, NJ, second edition (1990). Patterns of plausible inference · Zbl 0034.15502 |
| [39] | Safarov, Y., Vassiliev, D.: The asymptotic distribution of eigenvalues of partial differential operators, volume 155 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI (1997). Translated from the Russian manuscript by the authors · Zbl 0870.35003 |
| [40] | Strichartz, RS, Estimates for sums of eigenvalues for domains in homogeneous spaces, J. Funct. Anal., 137, 1, 152-190 (1996) · Zbl 0848.58050 · doi:10.1006/jfan.1996.0043 |
| [41] | Weidl, T.: Improved Berezin-Li-Yau inequalities with a remainder term. In Spectral theory of differential operators, volume 225 of Amer. Math. Soc. Transl. Ser. 2, pp. 253-263. Amer. Math. Soc., Providence, RI (2008) · Zbl 1168.35404 |
| [42] | Weyl, H., Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., 71, 4, 441-479 (1912) · JFM 43.0436.01 · doi:10.1007/BF01456804 |
| [43] | Weyl, H., Über die Randwertaufgabe der Strahlungstheorie und asymptotische Spektralgesetze, J. Reine Angew. Math., 143, 177-202 (1913) · JFM 44.1053.02 · doi:10.1515/crll.1913.143.177 |
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.
