The nodal count mystery

Transcription

1 The nodal count mystery Peter Kuchment, Texas A&M University Partially supported by NSF 21st Midwest Geometry Conference Wichita State University, KS March

2 Table of contents 1 Nodal patterns of eigenfunctions Minimal partitions and Courant sharp eigenfunctions 5

3 Nodal patterns Nodal patterns (Chladni figures) observed and studied by Leonardo da Vinci ( ), Galileo Galilei ( ), Robert Hooke ( ), Ernst Chladni ( ),...

4 More nodal patterns

5 Even more nodal patterns

6 Even more nodal patterns

7 Mysteries still abound Studied in: math. physics, number theory, medical imaging (sic!). Local structure of nodal sets. Smoothness, intersection angles - easy. Curvature of joint nodal sets - HARD (Bourgain, Rudnick, Agranovsky & Quinto, Ambartsoumian & P.K.,..., Finch et al) How large is the nodal set? Yau s conjecture: H n 1 (N λ ) λ 1/2. Proven in analytic case by Fefferman and Donnely. In smooth case - HARD, work is very active (Colding, Hezari, Mangoubi, Minicozzi, I. Polterovich, Rudnick, Sarnak, Sodin, Zelditch). How many nodal domains does the nth eigenfunction have? (Courant, Helffer & T. Hoffman-Ostenhof, Berkolaiko& P.K. & Smilansky, Bourgain, Bogomolny & Schmit, Nazarov & Sodin) Inverse problems. Band, Hald, Klawonn, J. McLaughlin, Smilansky. Nadirashvili, K. Uhlenbeck,...

8 Nodal count in 1D Sturm s theorem H = d 2 dx 2 + q(x) on [a, b] with Dirichlet boundary conditions u(a) = u(b) = 0. Sturm s Theorem: Let λ 1 λ 2... be the spectrum and ψ n (x) - eigenfunctions of H. Then ψ n changes sign exactly n 1 times inside [a, b]. Thus, there are exactly n nodal domains, where ψ n has constant sign.

9 Nodal count in higher dimensions H = + q(x) in a bounded domain Ω R d with Dirichlet boundary conditions u Ω = 0. (λ 1 (Ω), ψ 1 (x)), (λ 2 (Ω), ψ 2 (x)),... Nodal set of eigenfunction ψ n : Z n := {x Ω ψ n (x) = 0}. Nodal domain of ψ n : a connected component of Ω \ Z n. Nodal count ν n := number of nodal domains. Courant s Theorem: ν n n

10 Nodal count in higher dimensions - continued Courant sharp eigenfunctions are such that ν n = n. E.g., in 1D, all eigenfunctions are Courant sharp (Sturm). λ 1 and λ 2 are always Courant sharp Theorem (Pleijel) ( ) 2 2 For d > 1 and large n, ν n < n 0.691n. (improved by j Bourgain 13, also I. Polterovich) Corollary For d > 1 there are only finitely many Courant sharp eigenfunctions. Definition µ n := n ν n nodal deficiency.

11 Two domains for large n Large n and ν n = 2 (A. Stern, 1925):

12 Nodal count, continued Some basic questions are not answered, e.g. lim sup ν n = n??????????????

13 Nodal Partitions Helffer & Hoffman-Ostenhof : Look at possible partitions P of domain Ω: Q: Which partitions P = {Ω j } can be nodal partitions of eigenfunctions?

14 Nodal Partitions - continued A: Necessarily, Local structure (smooth, correct intersection angles) Bipartite : Equipartition : λ 1 (Ω j ) = λ n (Ω).

15 Nodal Partitions - continued Helffer Hoffman-Ostenhof functional: Λ(P) := max λ 1 (Ω j ) j Theorem (H. H.-O. Terracini, 2009) Minimal bipartite partitions are exactly nodal partitions of Courant sharp eigenfunctions. Why? What about the non-courant-sharp ones?

16 Critical Partitions The answer given for the quantum graph case by R. Band, G. Berkolaiko, H. Raz, U. Smilansky, Comm. Math. Physics In the billiard case, Theorem (G.B. P.K. U.S, GAFA 2012) Among all generic equipartitions, the bipartite critical points of Λ are exactly nodal partitions of eigenfunctions. The Morse index is equal to the nodal deficiency µ n. In particular, at minimal points, Morse index is zero and thus the eigenfunction is Courant sharp. Berkolaiko (2011) (a modification by Colin de Verdiere), in graph case - Morse indices w.r.t. perturbations by magnetic potentials.

17 Genericity

18 Some references Berkolaiko, Gregory; Kuchment, Peter; Smilansky, Uzy Critical partitions and nodal deficiency of billiard eigenfunctions. Geom. Funct. Anal. 22 (2012), no. 6, Qing Han, Fang-Hua Lin Nodal Sets of Solutions of Elliptic Differential Equations, qhan/nodal.pdf Helffer, B.; Hoffmann-Ostenhof, T.; Terracini, S. Nodal domains and spectral minimal partitions. The state of the art in arxiv: Zelditch, Steve Eigenfunctions and nodal sets. Surveys in differential geometry. Geometry and topology, , Surv. Differ. Geom., 18, Int. Press, Somerville, MA, 2013.

19 Thanks Thank you very much for the invitation and patience