References
- Sogge, C. (2011). Problems related to the concentration of eigenfunctions. arXiv:1510.077223. J. 63(4):519–538.
- Zelditch, S. (2008). Local and global analysis of eigenfunctions on Riemannian manifolds. In: Handbook of Geometric Analysis. No. 1, Advanced Lectures in Mathematics, 7. Somerville, MA: International Press, pp. 545–658.
- Zelditch, S. (2017). Eigenfunctions of the Laplacian on a Riemannian manifold. CBMS Regional Conference Series in Mathematics, 125. AMS, Providence, RI, p. xiv + 394pp.
- Donnelly, H., Fefferman, C. (1988). Nodal sets of eigenfunctions on Riemannian manifolds. Invent. Math. 93(1):161–183. DOI: https://doi.org/10.1007/BF01393691.
- Logunov, A. (2018). Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture. Ann. Math. 187(1):241–262. DOI: https://doi.org/10.4007/annals.2018.187.1.5.
- Logunov, A. (2018). Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure. Ann. Math. 187(1):221–239. DOI: https://doi.org/10.4007/annals.2018.187.1.4.
- Nazarov, F., Polterovich, L., Sodin, M. (2005). Sign and area in nodal geometry of Laplace eigenfunctions. Am. J. Math. 127(4):879–910. DOI: https://doi.org/10.1353/ajm.2005.0030.
- Roy-Fortin, G. (2015). Nodal sets and growth exponents of Laplace eigenfunctions on surfaces. Anal. PDE. 8(1):223–255. DOI: https://doi.org/10.2140/apde.2015.8.223.
- Sogge, C., Zelditch, S. (2011). Lower bounds on the Hausdorff measure of nodal sets. Math. Res. Lett. 18(1):25–37. DOI: https://doi.org/10.4310/MRL.2011.v18.n1.a3.
- Mukherjee, M. Mass non-concentration at the nodal set and a sharp Wasserstein uncertainty principle. arXiv:2103.11633.
- Steinerberger, S. (2021). Wasserstein distance, Fourier series and applications. Monatsh. Math. 194(2):305–338. DOI: https://doi.org/10.1007/s00605-020-01497-2.
- Sogge, C. (1988). Concerning the Lp norm of spectral clusters for second-order elliptic operators on compact manifolds. J. Funct. Anal. 77(1):123–138. DOI: https://doi.org/10.1016/0022-1236(88)90081-X.
- Monastra, A., Smilansky, U., Gnutzmann, S. (2003). Avoided intersections of nodal lines. J. Phys. A: Math. Gen. 36(7):1845–1853. DOI: https://doi.org/10.1088/0305-4470/36/7/304.
- Steinerberger, S. (2014). Lower bounds on nodal sets of eigenfunctions via the heat flow. Commun. PDE. 39(12):2240–2261. DOI: https://doi.org/10.1080/03605302.2014.942739.
- Mangoubi, D. (2010). The volume of a local nodal domain. J. Topol. Anal. 02(02):259–275. DOI: https://doi.org/10.1142/S1793525310000306.
- Chanillo, S., Logunov, A., Malinnikova, E., Mangoubi, D. Local version of Courant’s nodal domain theorem. arXiv:2008.00677.
- Grebenkov, D., Nguyen, B.-T. (2013). Geometrical structure of Laplace eigenfunctions. SIAM Rev. 55(4):601–667. DOI: https://doi.org/10.1137/120880173.
- Delitsyn, A., Nguyen, B.-T., Grebenkov, D. (2012). Exponential decay of Laplacian eigenfunctions in domains with branches of variable cross-sectional profiles. Eur. Phys. J. B. 85(11):371. DOI: https://doi.org/10.1140/epjb/e2012-30286-8.
- Nguyen, B.-T., Grebenkov, D., Delytsin, A. (2014). On the exponential decay of Laplacian eigenfunctions in planar domains with branches. Contemp. Math., 337–348. DOI: https://doi.org/10.1090/conm/630/12674.
- van den Berg, M., Bolthausen, E. (1999). Estimates for Dirichlet eigenfunctions. J. London Math. Soc. 59(2):607–619. DOI: https://doi.org/10.1112/S0024610799007267.
- Banuelos, R., Davis, B. (1992). Sharp estimates for Dirichlet eigenfunctions on horn-shaped regions. Commun. Math. Phys. 150(1):209–215. DOI: https://doi.org/10.1007/BF02096574.
- Hayman, W. (1978). Some bounds for principal frequency. Applicable Anal. 7(3):247–254. DOI: https://doi.org/10.1080/00036817808839195.
- Georgiev, B., Mukherjee, M. (2018). On maximizing the fundamental frequency of the complement of an obstacle. C. R. Math. Acad. Sci. Paris. 356(4):406–411. DOI: https://doi.org/10.1016/j.crma.2018.01.018.
- Mangoubi, D. (2008). Local asymmetry and the inner radius of nodal domains. Commun. PDE. 33(9):1611–1621. DOI: https://doi.org/10.1080/03605300802038577.
- Maz’ya, V., Shubin, M. (2005). Can one see the fundamental frequency of a drum? Lett. Math. Phys. 74(2):135–151. DOI: https://doi.org/10.1007/s11005-005-0010-1.
- Davies, B., Simon, B. (1984). Ultracontractivity and heat kernels for Schrödinger operators and Dirichlet Laplacians. J. Func. Anal. 59(2):335–395. DOI: https://doi.org/10.1016/0022-1236(84)90076-4.
- Kent, J. (1980). Eigenvalue expansions for diffusion hitting times. Z Wahrscheinlichkeitstheorie Verw. Gebiete. 52(3):309–319. DOI: https://doi.org/10.1007/BF00538895.
- Hardt, R., Simon, L. (1989). Nodal sets for solutions of elliptic equations. J. Diff. Geom. 30(2):505–522.
- Georgiev, B., Mukherjee, M. (2018). Nodal geometry, heat diffusion and Brownian motion. Anal. PDE. 11(1):133–148. DOI: https://doi.org/10.2140/apde.2018.11.133.
- Varadhan, S. (1967). Diffusion processes in a small time interval. Commun. Pure Appl. Math. 20(4):659–685. DOI: https://doi.org/10.1002/cpa.3160200404.
- Cheeger, J., Gromov, M., Taylor, M. (1982). Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differential Geom. 17(1):15–53. DOI: https://doi.org/10.4310/jdg/1214436699.
- Li, P., Yau, S.-T. (1986). On the parabolic kernel of the Schrödinger operator. Acta Math. 156:153–201. DOI: https://doi.org/10.1007/BF02399203.
- Grigor’yan, A. (2009). Heat kernel and analysis on manifolds. In: AMS/IP Studies in Advanced Mathematics, 47. American Mathematical Society. Providence, RI; Boston, MA: International Press.
- Ludewig, M. (2019). Strong short time asymptotics and convolution approximation of the heat kernel. Ann. Glob. Anal. Geom. 55(2):371–394. DOI: https://doi.org/10.1007/s10455-018-9630-4.
- Grigor'yan, A., Saloff-Coste, L. (2002). Hitting probabilities for Brownian motion on Riemannian manifolds. J. Math. Pures Appl. 81(2):115–142. DOI: https://doi.org/10.1016/S0021-7824(01)01244-2.
- Stroock, D. (2000). An introduction to the analysis of paths on a Riemannian manifold. In: Mathematical Surveys and Monographs, Vol. 74. Providence, RI: AMS.
- Rosenberg, S. (2009). The Laplacian on a Riemannian Manifold. An introduction to analysis on manifolds. London Mathematical Society Student Texts, 31. Cambridge: Cambridge University Press, 1997.
- Baudoin, F. Online notes. Available at: https://fabricebaudoin.wordpress.com/2013/09/18/lecture-17-the-parabolic-harnack-inequality/.
- Schoen, R., Yau, S.-T. (1994). Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA.
- Lieb, E. (1983). On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math. 74(3):441–448. DOI: https://doi.org/10.1007/BF01394245.
- Georgiev, B. (2019). On the lower bound of the inner radius of nodal domains. J. Geom. Anal. 29(2):1546–1554. DOI: https://doi.org/10.1007/s12220-018-0050-2.
- Jakobson, D., Mangoubi, D. (2009). Tubular neighborhoods of nodal sets and Diophantine approximation. Amer. J. Math. 131(4):1109–1135.
- Mangoubi, D. (2008). On the inner radius of nodal domains. Can. Math. Bull. 51(2):249–260. DOI: https://doi.org/10.4153/CMB-2008-026-2.
- Nadirashvili, N. (1991). Metric properties of the eigenfunctions of the Laplace operator on manifolds. Ann. Inst. Fourier (Grenoble). 41(1):259–265. DOI: https://doi.org/10.5802/aif.1256.
- Ahlfors, L. (1973). Conformal invariants: topics in geometric function theory. McGraw-Hill Series in Higher Mathematics. New York: McGraw-Hill Book Co.
- Agmon, S. (1982). Lectures on exponential decay of solutions of second order elliptic equation: bounds on eigenfunctions of N-body Schrödinger operators. Math. Notes, Princeton. 29. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 118 pp.
- Hislop, P., Sigal, I. (1996). Introduction to spectral theory. With applications to Schrödinger operators. In: Applied Mathematical Sciences, Vol. 113. New York: Springer-Verlag.
- Funano, K., Sakurai, Y. (2019). Concentration of eigenfunctions of the Laplacian on a closed Riemannian manifold. Proc. Amer. Math. Soc. 147(7):3155–3164.
- Colding, T., Minicozzi, W. II. (2003). Volumes for eigensections. Geometriae Dedicata. 102(1):19–24. DOI: https://doi.org/10.1023/B:GEOM.0000006578.85728.cf.
- Georgiev, B., Mukherjee, M. (2019). Some remarks on nodal geometry in the smooth setting. Calc. Var. 58(3):25. DOI: https://doi.org/10.1007/s00526-019-1541-0.