A logarithmic improvement in the two-point Weyl Law for manifolds without conjugate points
Annales de l'Institut Fourier, Volume 74 (2024) no. 2, pp. 719-762.

In this paper, we study the two-point Weyl Law for the Laplace–Beltrami operator on a smooth, compact Riemannian manifold M with no conjugate points. That is, we find the asymptotic behavior of the Schwartz kernel, E λ (x,y), of the projection operator from L 2 (M) onto the direct sum of eigenspaces with eigenvalue smaller than λ 2 as λ. In the regime where x,y are restricted to a compact neighborhood of the diagonal in M×M, we obtain a uniform logarithmic improvement in the remainder of the asymptotic expansion for E λ and its derivatives of all orders, which generalizes a result of Bérard, who treated the on-diagonal case E λ (x,x). When x,y avoid a compact neighborhood of the diagonal, we obtain this same improvement in an upper bound for E λ . Our results imply that the rescaled covariance kernel of a monochromatic random wave locally converges in the C -topology to a universal scaling limit at an inverse logarithmic rate.

Dans cet article, nous étudions la loi de Weyl à deux points pour l’opérateur de Laplace–Beltrami sur une variété riemannienne lisse et compacte M sans points conjugués. Cela veut dire que nous donnons le comportement asymptotique du noyau de Schwartz, E λ (x,y), de l’opérateur de projection de L 2 (M) sur la somme directe des espaces propres pour des valeurs propres inférieures à λ 2 quand λ. Dans le régime où x,y sont restreints à un voisinage compact de la diagonale en M×M, on obtient une amélioration logarithmique uniforme dans le reste du développement asymptotique pour E λ et ses dérivées de tous ordres , ce qui généralise un résultat de Bérard, qui a traité le cas diagonal E λ (x,x). Lorsque x,y évitent un voisinage compact de la diagonale, on obtient cette même amélioration en une majoration de E λ . Nos résultats impliquent que le noyau de covariance redimensionné d’une onde aléatoire monochromatique converge localement dans la topologie C vers une limite d’échelle universelle à un taux logarithmique inverse.

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DOI: 10.5802/aif.3598
Classification: 35P20
Keywords: Weyl Laws, Spectral Theory, Riemannian Geometry, Random Waves.
Mot clés : Loi de Weyl, théorie spectrale, géométrie riemannienne, ondes aléatoires

Keeler, Blake 1

1 McGill University, 805 Sherbrooke St. W., Montreal QC (Canada)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Keeler, Blake. A logarithmic improvement in the two-point Weyl Law for manifolds without conjugate points. Annales de l'Institut Fourier, Volume 74 (2024) no. 2, pp. 719-762. doi : 10.5802/aif.3598. https://aif.centre-mersenne.org/articles/10.5802/aif.3598/

[1] Avakumović, V. G. Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten, Math. Z., Volume 65 (1956), pp. 327-344 | DOI | MR | Zbl

[2] Bérard, Pierre H. On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z., Volume 155 (1977) no. 3, pp. 249-276 | DOI | MR | Zbl

[3] Berry, M. Regular and irregular semiclassical wavefronts, J. Phys. A, Volume 10 (1977), pp. 2083-2091 | DOI | Zbl

[4] Blair, Matthew D. On logarithmic improvements of critical geodesic restriction bounds in the presence of nonpositive curvature, Israel J. Math., Volume 224 (2018) no. 1, pp. 407-436 | DOI | MR | Zbl

[5] Bonthonneau, Y. A lower bound for the Θ function on manifolds without conjugate points, Documenta Math., Volume 22 (2017), pp. 1279-1283 | MR | Zbl

[6] Canzani, Yaiza; Hanin, Boris Scaling limit for the kernel of the spectral projector and remainder estimates in the pointwise Weyl law, Anal. PDE, Volume 8 (2015) no. 7, pp. 1707-1731 | DOI | MR | Zbl

[7] Canzani, Yaiza; Hanin, Boris C scaling asymptotics for the spectral projector of the Laplacian, J. Geom. Anal., Volume 28 (2018) no. 1, pp. 111-122 | DOI | MR | Zbl

[8] do Carmo, Manfredo Perdigão Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992, xiv+300 pages (translated from the second Portuguese edition by Francis Flaherty) | DOI | MR | Zbl

[9] Dierickx, G.; Nourdin, I.; Peccati, G.; Rossi, M. Small Scale CLTs for the Nodal Length of Monochromatic Waves (2020) (https://arxiv.org/abs/2005.06577)

[10] Duistermaat, J. J.; Guillemin, V. W. The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math., Volume 29 (1975) no. 1, pp. 39-79 | DOI | MR | Zbl

[11] Fedoryuk, M. V. The multidimensional stationary phase method. The second term of the asymptotic expansions, U.S.S.R. Comput. Maths. Math. Phys., Volume 30 (1990) no. 3, pp. 104-107 | DOI | Zbl

[12] Hörmander, Lars The spectral function of an elliptic operator, Acta Math., Volume 121 (1968), pp. 193-218 | DOI | MR | Zbl

[13] Hörmander, Lars The Analysis of Linear Partial Differential Operators. III, Springer-Verlag, 1985 | DOI | MR

[14] Ingremeau, M.; Rivera, A How Lagranian states evolve into random waves (2020) (https://arxiv.org/abs/2011.02943)

[15] Ivriĭ, V. The second term of the spectral asymptotics for a Laplace–Beltrami operator on manifolds with boundary, Funktsional. Anal. i Prilozhen., Volume 14 (1980) no. 2, pp. 25-34 | MR

[16] Lee, John M. Introduction to Riemannian manifolds, Graduate Texts in Mathematics, 176, Springer, Cham, 2018, xiii+437 pages | DOI | MR | Zbl

[17] Levitan, B. M. On the asymptotic behavior of the spectral function of a self-adjoint differential equation of the second order and on expansion in eigenfunctions, Izvestiya Akad. Nauk SSSR. Ser. Mat., Volume 17 (1953), pp. 331-364 | MR

[18] Levitan, B. M. On the asymptotic behavior of a spectral function and on expansion in eigenfunctions of a self-adjoint differential equation of second order. II, Izv. Akad. Nauk SSSR. Ser. Mat., Volume 19 (1955), pp. 33-58 | MR

[19] Safarov, Yu G. Asymptotics of a spectral function of a positive elliptic operator without a nontrapping condition, Funktsional. Anal. i Prilozhen., Volume 22 (1988) no. 3, p. 53-65, 96 | DOI | MR | Zbl

[20] Safarov, Yu G.; Vassiliev, D. The asymptotic distribution of eigenvalues of partial differential operators, Translations of Mathematical Monographs, 155, American Mathematical Society, Providence, RI, 1997, xiv+354 pages (translated from the Russian manuscript by the authors) | DOI | MR

[21] Shubin, M. A. Spectral theory of elliptic operators on noncompact manifolds, Méthodes semi-classiques, Vol. 1 (Nantes, 1991) (Astérisque), Société mathématique de France, 1992 no. 207, pp. 35-108 | MR | Zbl

[22] Sogge, Christopher D. Hangzhou lectures on eigenfunctions of the Laplacian, Annals of Mathematics Studies, 188, Princeton University Press, Princeton, NJ, 2014, xii+193 pages | DOI | MR | Zbl

[23] Sogge, Christopher D. Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, 210, Cambridge University Press, Cambridge, 2017, xiv+334 pages | DOI | MR | Zbl

[24] Sogge, Christopher D.; Zelditch, Steve Riemannian manifolds with maximal eigenfunction growth, Duke Math. J., Volume 114 (2002) no. 3, pp. 387-437 | DOI | MR | Zbl

[25] Trèves, François Basic linear partial differential equations, Pure and Applied Mathematics, 62, Academic Press, 1975, xvii+470 pages | MR | Zbl

[26] Colin de Verdiere, Y. Spectre du laplacien et longueurs des géodésiques périodiques I., Compos. Math., Volume 27 (1973), pp. 83-106 | Numdam | Zbl

[27] Xu, B. Derivatives of the spectral function and Sobolev norms of eigenfunctions on a closed Riemannian manifold, Ann. Global Anal. Geom., Volume 26 (2004) no. 3, pp. 231-252 | DOI | MR | Zbl

[28] Zelditch, S. Real and complex zeros of Riemannian random waves, Spectral analysis in geometry and number theory (Contemporary Mathematics), Volume 484, American Mathematical Society, Providence, RI, 2009, pp. 321-342 | DOI | MR | Zbl

[29] Zworski, M. Semiclassical Analysis, Providence: AMS, 2012 | DOI

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