On Courant’s nodal domain property for linear combinations of eigenfunctions (after P. B´ erard, P. Charron and B. Helffer). October 2019 Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ On Courant’s nodal domain property for linear combinations of eigenfunctions e de Nantes.
Abstract We revisit Courant’s nodal domain property for linear combinations of eigenfunctions. This property was proven by Sturm (1836) in the case of dimension 1. Although stated as true for the Dirichlet Laplacian in dimension > 1 in a footnote of the celebrated book of Courant-Hilbert (and wrongly attributed to H. Herrmann, a PHD student of R. Courant), it appears to be wrong. This was first observed by V. Arnold in the seventies. This talk which has three parts, Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ On Courant’s nodal domain property for linear combinations of eigenfunctions e de Nantes.
First Part We present simple and explicit counterexamples to this so-called ”Herrmann’s statement” for domains in R d , S 2 or T 2 . We also discuss the existence of a counterexample in a C ∞ , convex domain Ω in R 2 in relation with the analysis of the number of domains delimited by the level sets of a second eigenfunction for the Neumann problem. This work has been done in collaboration with P. B´ erard. Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ On Courant’s nodal domain property for linear combinations of eigenfunctions e de Nantes.
Second Part We then prove that the Extended Courant property is false for the subequilateral triangle and for regular N -gons ( N large), with the Neumann boundary condition. More precisely, we prove that there exists a Neumann eigenfunction u k of the N -gon, with index 4 ≤ k ≤ 6, such that the set { u k � = 1 } has ( N + 1) connected components. Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ On Courant’s nodal domain property for linear combinations of eigenfunctions e de Nantes.
Third part Finally we prove that there exists metrics g on T 2 (resp. on S 2 ) which are arbitrarily close to the flat metric (resp. round metric), and an eigenfunction f of the associated Laplace-Beltrami operator such that the set { f � = 1 } has infinitely many connected components. In particular the Extended Courant property is false for these closed surfaces. These results are strongly motivated by a recent paper by Buhovsky, Logunov and Sodin (2019). As for the positive direction, we prove that the Extended Courant property is true for the isotropic quantum harmonic oscillator in R 2 . The results of Parts II and III have been proven in collaboration with P. B´ erard and P. Charron. Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ On Courant’s nodal domain property for linear combinations of eigenfunctions e de Nantes.
Introduction Let Ω ⊂ R d be a bounded open domain or, more generally, a compact Riemannian manifold with boundary. Consider the eigenvalue problem � − ∆ u = λ u in Ω , (1) B ( u ) = 0 on ∂ Ω , where B ( u ) is some boundary condition on ∂ Ω, so that we have a self-adjoint boundary value problem (including the empty condition if Ω is a closed manifold). � For example, D ( u ) = u ∂ Ω for the Dirichlet boundary condition, or � N ( u ) = ∂ u ∂ν | ∂ Ω for the Neumann boundary condition. Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ On Courant’s nodal domain property for linear combinations of eigenfunctions e de Nantes.
Call H (Ω , B ) the associated self-adjoint extension of − ∆, and list its eigenvalues in nondecreasing order, counting multiplicities, 0 ≤ λ 1 (Ω , B ) < λ 2 (Ω , B ) ≤ λ 3 (Ω , B ) ≤ · · · (2) For any integer n ≥ 1, define the index τ (Ω , B , λ n ) = min { k | λ k (Ω , B ) = λ n (Ω , B ) } . (3) E ( λ n ) will denote the eigenspace associated with λ n . Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ On Courant’s nodal domain property for linear combinations of eigenfunctions e de Nantes.
The Courant nodal theorem For a real continuous function v on Ω, we define its nodal set Z ( v ) = { x ∈ Ω | v ( x ) = 0 } , (4) and call β 0 ( v ) the number of connected components of Ω \ Z ( v ) i.e., the number of nodal domains of v . Courant’s nodal Theorem (1923) For any nonzero u ∈ E ( λ n (Ω , B )) , β 0 ( u ) ≤ τ (Ω , B , λ n ) ≤ n . (5) Courant’s nodal domain theorem can be found in Courant-Hilbert [15]. Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ On Courant’s nodal domain property for linear combinations of eigenfunctions e de Nantes.
The extended Courant nodal property Given r > 0, denote by L (Ω , B , r ) the space � L (Ω , B , r ) = c j u j | c j ∈ R , u j ∈ E λ j (Ω , B ) . (6) λ j (Ω , B ) ≤ r Extended Courant Property:= (ECP) We say that v ∈ L (Ω , B , λ n (Ω , B )) satisfies (ECP) if β 0 ( v ) ≤ τ (Ω , B , λ n ) . (7) A footnote in Courant-Hilbert [15] indicates that this property also holds for any linear combination of the n first eigenfunctions, and refers to the PhD thesis of Horst Herrmann (G¨ ottingen, 1932) [21]. Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ On Courant’s nodal domain property for linear combinations of eigenfunctions e de Nantes.
Historical remarks : Sturm (1836), Pleijel (1956). 1. (ECP) is true for Sturm-Liouville equations. This was first announced by C. Sturm in 1833, [40] and proved in [41]. Other proofs were later on given by J. Liouville and Lord Rayleigh who both cite Sturm explicitly. 2. ˚ A. Pleijel (1956) mentions (ECP) in his well-known paper [36] on the asymptotic behaviour of the number of nodal domains of a Dirichlet eigenfunction associated with the n -th eigenvalue in a plane domain but says that no proof of this property is available. Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ On Courant’s nodal domain property for linear combinations of eigenfunctions e de Nantes.
Historical remarks: V. Arnold (1973-1979) As pointed out by V. Arnold [1], when Ω = S d , (ECP) is 3. related to Hilbert’s 16 − th problem. Arnold [2] mentions that he actually discussed the footnote with R. Courant, that (ECP) cannot be true, and that O. Viro produced in 1979 counter-examples for the 3-sphere S 3 , and any degree larger than or equal to 6, [42]. Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ On Courant’s nodal domain property for linear combinations of eigenfunctions e de Nantes.
Historical remarks: Gladwell-Zhu (2003) 4. In [GZ2003], Gladwell and Zhu refer to (ECP) as the Courant-Herrmann conjecture . They claim that this extension of Courant’s theorem is not stated, let alone proved, in Herrmann’s thesis or subsequent publications. They consider the case in which Ω is a rectangle in R 2 , stating that they were not able to find a counter-example to (ECP) in this case. They also provide numerical evidence that there are counter-examples for more complicated (non convex) domains. They suggest that may be the conjecture could be true in the convex case. Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ On Courant’s nodal domain property for linear combinations of eigenfunctions e de Nantes.
Some geometrical statements–positive results ? There are two statements of V. Arnold for which no proof is available. ◮ ECP is true for the sphere S 2 with its standard metric. ◮ ECP does not hold for other metrics on the sphere. The second claim is very vague ! Shall we add ”for generic metric” ? Outside the (1 D )-case, the only known result is for R P 2 and was obtained by J. Leydold (1996). We will also give at the end of the talk a positive result for the isotropic harmonic oscillator in (2D) (B´ erard-Charron-Helffer (2019)). Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ On Courant’s nodal domain property for linear combinations of eigenfunctions e de Nantes.
Sphere S 2 with cracks There is a similar example in the case of a rectangle with cracks (B´ erard-Helffer (2017)) but let us describe the case of a sphere with cracks. On the round sphere S 2 , we consider the geodesic lines √ √ 1 − z 2 cos θ i , 1 − z 2 sin θ i , z ) through the north ( x , y , z ) �→ ( pole (0 , 0 , 1), with distinct θ i ∈ [0 , π [. Removing the geodesic segments θ 0 = 0 and θ 2 = π 2 with 1 − z ≤ a ≤ 1, we obtain a sphere S 2 a with a crack in the form of a cross. We consider the Neumann condition on the crack. We then easily produce a function in the space generated by the two first eigenspaces of the sphere with a crack having five nodal domains. Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ On Courant’s nodal domain property for linear combinations of eigenfunctions e de Nantes.
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