A microlocal approach to eigenfunction concentration

Galkowski, Jeffrey

Journées équations aux dérivées partielles (2018), Exposé no. 3, 14 p.

We describe a new approach to understanding averages of high energy Laplace eigenfunctions, $u_h$, over submanifolds,

$$\left|{\int }_H{u}_h\mathrm{d}{\sigma }_H\right|$$

where $H\subset M$ is a submanifold and ${\sigma }_H$ the induced by the Riemannian metric on $M$. This approach can be applied uniformly to submanifolds of codimension $1\le k\le n$ and in particular, gives a new approach to understanding $\parallel u_h{\parallel }_{L^{\infty }\left(M\right)}$. The method, developed in [4, 5, 6, 7, 12, 13], relies on estimating averages by the behavior of $u_h$ microlocally near the conormal bundle to $H$. By doing this, we are able to obtain quantitative improvements on eigenfunction averages under certain uniform non-recurrent conditions on the conormal directions to $H$. In particular, we do not require any global assumptions on the manifold $(M,g)$.

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

[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

[3] Bonthonneau, Yannick A lower bound for the $\Theta$ function on manifolds without conjugate points (2016) (https://arxiv.org/abs/1603.05697)

[4] Canzani, Yaiza; Galkowski, Jeffrey On the growth of eigenfunction averages: microlocalization and geometry (2017) (https://arxiv.org/abs/1710.07972, to appear in Duke Math. J.)

[5] Canzani, Yaiza; Galkowski, Jeffrey Eigenfunction concentration via geodesic beams (2019) (https://arxiv.org/abs/1903.08461)

[6] Canzani, Yaiza; Galkowski, Jeffrey Improvements for eigenfunction averages: An application of geodesic beams (2019) (https://arxiv.org/abs/1809.06296)

[7] Canzani, Yaiza; Galkowski, Jeffrey; Toth, John A. Averages of eigenfunctions over hypersurfaces, Commun. Math. Phys., Volume 360 (2018) no. 2, pp. 619-637 | Zbl

[8] Chen, Xuehua; Sogge, Christopher D. On integrals of eigenfunctions over geodesics, Proc. Am. Math. Soc., Volume 143 (2015) no. 1, pp. 151-161 | DOI | MR

[9] Dyatlov, Semyon; Guillarmou, Colin Microlocal limits of plane waves and Eisenstein functions, Ann. Sci. Éc. Norm. Supér., Volume 47 (2014) no. 2, pp. 371-448 | DOI | MR

[10] Dyatlov, Semyon; Zworski, Maciej Mathematical theory of scattering resonances (2016) (Book in progress, http://math.mit.edu/dyatlov/res/)

[11] Eberlein, Patrick When is a geodesic flow of ANOSOV type? I, J. Differ. Geom., Volume 8 (1973), pp. 437-463

[12] Galkowski, Jeffrey Defect measures of eigenfunctions with maximal $L^{\infty }$ growth (2017) (https://arxiv.org/abs/1704.01452 to appear in Ann. Inst. Fourier)

[13] Galkowski, Jeffrey; Toth, John A. Eigenfunction scarring and improvements in $L^{\infty }$ bounds, Anal. PDE, Volume 11 (2017) no. 3, pp. 801-812

[14] Good, Anton Local analysis of Selberg’s trace formula, Lecture Notes in Mathematics, 1040, Springer, 1983, i+128 pages | DOI | MR

[15] Hejhal, Dennis A. Sur certaines séries de Dirichlet associées aux géodésiques fermées d’une surface de Riemann compacte, C. R. Math. Acad. Sci. Paris, Volume 294 (1982) no. 8, pp. 273-276 | MR

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

[17] Iwaniec, Henryk; Sarnak, Peter $L^{\infty }$ norms of eigenfunctions of arithmetic surfaces, Ann. Math., Volume 141 (1995) no. 2, pp. 301-320 | DOI | MR | Zbl

[18] Lee, John M. Riemannian manifolds: an introduction to curvature, Graduate Texts in Mathematics, 176, Springer, 2006 | Zbl

[19] Levitan, Boris M. On the asymptotic behavior of the spectral function of a self-adjoint differential equation of the second order, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 16 (1952), pp. 325-352 | MR

[20] Sogge, Christopher D.; Toth, John A.; Zelditch, Steve About the blowup of quasimodes on Riemannian manifolds, J. Geom. Anal., Volume 21 (2011) no. 1, pp. 150-173 | DOI | MR

[21] Sogge, Christopher D.; Xi, Yakun; Zhang, Cheng Geodesic period integrals of eigenfunctions on Riemann surfaces and the Gauss–Bonnet Theorem (2016) (https://arxiv.org/abs/1604.03189)

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

[23] Sogge, Christopher D.; Zelditch, Steve Focal points and sup-norms of eigenfunctions, Rev. Mat. Iberoam., Volume 32 (2016) no. 3, pp. 971-994 | DOI | MR | Zbl

[24] Sogge, Christopher D.; Zelditch, Steve Focal points and sup-norms of eigenfunctions II: the two-dimensional case, Rev. Mat. Iberoam., Volume 32 (2016) no. 3, pp. 995-999 | DOI | MR | Zbl

[25] Wyman, Emmett L. Explicit bounds on integrals of eigenfunctions over curves in surfaces of nonpositive curvature (2017) (https://arxiv.org/abs/1705.01688)

[26] Wyman, Emmett L. Looping directions and integrals of eigenfunctions over submanifolds (2017) (https://arxiv.org/abs/1706.06717)

[27] Wyman, Emmett L. Period integrals in non-positively curved manifolds (2018) (https://arxiv.org/abs/1806.01424)

[28] Zelditch, Steven Kuznecov sum formulae and Szegő limit formulae on manifolds, Commun. Partial Differ. Equations, Volume 17 (1992) no. 1-2, pp. 221-260 | DOI | MR | Zbl

[29] Zworski, Maciej Semiclassical analysis, Graduate Studies in Mathematics, 138, American Mathematical Society, 2012, xii+431 pages | DOI | MR