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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´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Abstract

We revisit Courant’s nodal domain property for linear combinations

• f 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

• bserved by V. Arnold in the seventies.

This talk which has three parts,

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

First Part

We present simple and explicit counterexamples to this so-called ”Herrmann’s statement” for domains in Rd, S2 or T2. We also discuss the existence of a counterexample in a C ∞, convex domain Ω in R2 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´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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 uk of the N-gon, with index 4 ≤ k ≤ 6, such that the set {uk = 1} has (N + 1) connected components.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Third part

Finally we prove that there exists metrics g on T2 (resp. on S2) 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 R2. 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´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Introduction

Let Ω ⊂ Rd be a bounded open domain or, more generally, a compact Riemannian manifold with boundary. Consider the eigenvalue problem

• −∆u

= λ u in Ω , B(u) = 0

• n ∂Ω ,

(1) 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´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Call H(Ω, B) the associated self-adjoint extension of −∆, and list its eigenvalues in nondecreasing order, counting multiplicities, 0 ≤ λ1(Ω, B) < λ2(Ω, B) ≤ λ3(Ω, B) ≤ · · · (2) For any integer n ≥ 1, define the index τ(Ω, B, λn) = min{k | λk(Ω, B) = λn(Ω, B)}. (3) E(λn) will denote the eigenspace associated with λn.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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 β0(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(λn(Ω, B)) , β0(u) ≤ τ(Ω, B, λn) ≤ n . (5) Courant’s nodal domain theorem can be found in Courant-Hilbert [15].

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

The extended Courant nodal property

Given r > 0, denote by L(Ω, B, r) the space L(Ω, B, r) =   

• λj(Ω,B)≤r

cj uj | cj ∈ R, uj ∈ Eλj(Ω,B)    . (6)

Extended Courant Property:= (ECP)

We say that v ∈ L (Ω, B, λn(Ω, B)) satisfies (ECP) if β0(v) ≤ τ(Ω, B, λn) . (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¨

• ttingen, 1932) [21].

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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]
• n 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´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Historical remarks: V. Arnold (1973-1979)

3. As pointed out by V. Arnold [1], when Ω = Sd, (ECP) is 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 S3, and any degree larger than

• r equal to 6, [42].

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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 R2, 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´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Some geometrical statements–positive results ?

There are two statements of V. Arnold for which no proof is available.

◮ ECP is true for the sphere S2 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 (1D)-case, the only known result is for RP2 and was

• btained 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´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Sphere S2 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 S2, we consider the geodesic lines (x, y, z) → ( √ 1 − z2 cos θi, √ 1 − z2 sin θi, z) through the north pole (0, 0, 1), with distinct θi ∈ [0, π[. Removing the geodesic segments θ0 = 0 and θ2 = π

2 with

1 − z ≤ a ≤ 1, we obtain a sphere S2

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´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

The function z is also an eigenfunction of S2

a with eigenvalue 2.

For a small enough, λ4(a) = 2, with eigenfunction z. For 0 < b < a, the linear combination z − b has five nodal domains in S2

a, see Figure below in spherical coordinates.

It follows that (ECP) does not hold on the sphere with cracks.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Figure: Sphere with crack, five nodal domains

• Remark. Removing more geodesic segments around the north

pole, we can obtain a linear combination z − b with as many nodal domains as we want.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

The equilateral triangle (Dirichlet or Neumann)

Let Te denote the equilateral triangle with sides 1, see Figure 2. The eigenvalues and eigenfunctions of Te, with either Dirichlet or Neumann condition on the boundary ∂Te, can be completely described. We show that the equilateral triangle provides a counterexample to the Extended Courant Property for both the Dirichlet and the Neumann boundary condition.

Figure: Equilateral triangle Te = [OAB]

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Neumann boundary condition

The sequence of Neumann eigenvalues of the equilateral triangle Te begins as follows, 0 = λ1(Te, N) < 16π2 9 = λ2(Te, N) = λ3(Te, N) < λ4(Te, N) . (8) The second eigenspace has dimension 2, and contains one invariant eigenfunction ϕN

2 under the mirror symmetry w.r.t OM, and

another anti-invariant eigenfunction ϕN

3 .

ϕN

2 is given by

ϕN

2 (x, y) = 2 cos

2πx 3 cos 2πx 3

• + cos

2πy √ 3

• − 1 . (9)

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

The set {ϕN

2 + 1 = 0} consists of the two line segments

{x = 3

4} ∩ Te and {x +

√ 3y = 3

2} ∩ Te, which meet at the point

( 3

4, √ 3 4 ) on ∂Te. The sets {ϕ2 + a = 0}, with

a ∈ {0, 1 − ε, 1, 1 + ε}, and small positive ε, are shown in Figure 3. When a varies from 1 − ε to 1 + ε, the number of nodal domains of ϕ2 + a in Te jumps from 2 to 3, with the jump occurring for a = 1.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Figure: Levels sets {ϕN

2 + a = 0} for a ∈ {0 ; 0.9 ; 1 ; 1.1}

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

It follows that ϕN

2 + a = 0, for 1 ≤ a ≤ 1.2, provides a

counterexample to the Extended Courant Property for the equilateral triangle with Neumann boundary condition. In the case of Dirichlet one can prove that the level sets of ϕN

2 and

ϕD

2 /ϕD 1 are the same. In this way, we get immediately a counter

example for Dirichlet.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

All the previous examples are in domains which have some singularity (cracks or corners). It is then natural to ask for smooth counterexamples.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

A smooth example (B´ erard-Helffer (2018))

Our theoretical result is the following

Theorem

There exists a one parameter family of C ∞ domains {Ωt, 0 < t < t0} in R2, with the symmetry of the equilateral triangle Te, such that:

• 1. The family is strictly increasing, and Ωt tends to Te, in the

sense of the Hausdorff distance, as t tends to 0.

• 2. For any t ∈]0, t0[, ECP(Ωt) is false.

More precisely, for each t, there exists a linear combination of a symmetric 2nd Neumann eigenfunction and a 1st Neumann eigenfunction of Ωt, with exactly three nodal domains.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

As mentioned before, for the equilateral triangle Te, ECP(Te, a) is false for both the Dirichlet, and the Neumann boundary conditions. The idea of the proof of our theorem is to show that one can find a deformation of Te such that the symmetric second Neumann eigenfunction deforms nicely. For this purpose, we revisit a deformation argument given by Jerison and Nadirashsvili (2000) in the framework of the “hot spots” conjecture1. This argument permits to control with respect to t the nodal deformation of a suitable symmetric eigenfunction. Note that the symmetry considered by Jerison and Nadirashvili is different (two perpendicular axes).

1The ”hot spots” conjecture says that the eigenfunction corresponding to

the second eigenvalue of the Laplacian with Neumann boundary conditions attains its maximum and minimum on the boundary of the domain.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Next natural questions. Can we save (ECP) property by proving a weaker statement ?

We denote by LN(Ω) the space generated by the N-first eigenvalues of the Laplacian in Ω and by β0(w) the number of connected components of the complementary of the zero set. Q1: Fix Ω as above, and N ≥ 2. Can one bound β0(w), for w ∈ LN(Ω), in terms of N and geometric invariants of Ω? Q2: Assume that Ω ⊂ R2 is a convex domain. Can one bound β0(w), for w ∈ LN(Ω), in terms of N, independently of Ω? Q3: Assume that Ω is a simply-connected closed surface. Can one bound β0(w), for w ∈ LN(Ω), in terms of N, independently of Ω?

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

A negative answer to Question Q1 for the 2-torus is given in a revised version of [BLS] (Buhovsky-Logunov-Sodin). In this paper, the authors construct a smooth metric g on T2, and a family of eigenfunctions φj with infinitely many isolated critical points. As a by-product of their construction, they prove that there exist a smooth metric g, a family of eigenfunctions ωj, and a family of real numbers cj such that β0(ωj − cj) = +∞. Remark When the metric is real analytic, an eigenfunction can

• nly have finitely many isolated critical points. In the introduction
• f [BLS], the authors ask:

Q4: Does there exist an asymptotic upper bound for the number

• f critical points of an eigenfunction, in terms of the corresponding

eigenvalue.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Subequilateral triangle, Neumann boundary condition

Let T (b) denote the interior of the triangle with vertices A = ( √ 3, 0), B = (0, b), and c = (0, −b). When b = 1, T (1) is an equilateral triangle with sides of length 2. We assume that 0 < b < 1. The angle at the vertex A is less than

π 3 , and we say T (b) is a subequilateral triangle, see Figure 4.

Figure: 4 Subequilateral triangle, BC < AB = AC

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Call νi(T (b)) the Neumann eigenvalues of T (b), and write them in non-decreasing order, with multiplicities, starting from the index 1, 0 = ν1(T (b)) < ν2(T (b)) ≤ ν3(T (b)) ≤ · · · (10) We recall the following theorems.

Theorem of Laugesen-Siudeja (2010)

Every second Neumann eigenfunction of a subequilateral triangle T (b) is even in y, u(x, −y) = u(x, y).

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Theorem of Miyamoto (2013)

Let T (b) be a subequilateral triangle. Then, the eigenvalue ν2(T (b)) is simple, and an associated eigenfunction u satisfies u(O) = 0, where O is the point O = (0, 0). Normalize u by assuming that u(O) = 1. Then, the following properties hold.

• 1. The partial derivative ux is negative in T (b) \
• BC ∪ {A}
• .
• 2. The partial derivative uy is positive in T (b)+ \
• OA ∪ {B}
• ,

and negative in T (b)− \

• OA ∪ {C}
• .
• 3. The function u has exactly four critical points O, A, B and C

in T.

• 4. The points B and C are the global maxima of u, and

u(B) = u(C) > u(O) > 0.

• 5. The point A is the global minimum of u, and u(A) < 0.
• 6. The point O is the saddle point of u.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

As a direct corollary of these theorems, we have

Proposition [BCH]a

Conjecture (ECP) is false for the subequilateral triangle T (b) (0 < b < 1), with Neumann boundary condition.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Figure: Nodal behaviour for the subequilateral triangle

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Regular N-gon, Neumann boundary condition

Proposition [BCH]b

Let PN denote the regular polygon with N sides, inscribed in the unit disk D. Then, for N large enough, Conjecture (ECP) is false for PN, with the Neumann boundary condition. More precisely, there exist m ≤ 6, an eigenfunction um associated with νm(PN), and a value a such that the function um + a has N + 1 nodal domains.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Sketch of the proof

When N tends to infinity, the polygon PN tends to the disk D in the Hausdorff distance. It follows that, for all j ≥ 1, the Neumann eigenvalue νj(PN) tends to the Neumann eigenvalue νj(D) of the unit disk. The Neumann eigenvalues of the unit disk satisfy ν1(D) < ν2(D) = ν3(D) < ν4(D) = ν5(D) < ν6(D) < ν7(D) · · · (11) and are given respectively by the squares of the zeros of the derivatives of Bessel functions. It follows that, for N large enough, the eigenvalue ν6(PN) is simple.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

From now on, we assume that N is sufficiently large to ensure that ν6(PN) is a simple eigenvalue. Let u6 be an associated

• eigenfunction. We can then use the symmetries together with the

result for the subequilateral triangle to get Proposition [BCH]b.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Figure: 6 P9 and P12, Neumann boundary condition

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Remarks

◮ The above proposition also shows that the regular N-gons,

with the Neumann boundary condition, provides a counterexample to Question Q2, when N is large enough. This is illustrated by Figure 6.

◮ As recalled previously, Conjecture (ECP) is false for the

regular hexagon P6 with Neumann boundary condition. In this case, ν6(P6) = ν7(P6), and has multiplicity 2, with two eigenfunctions associated with different irreducible representations of D6.

◮ Our proposition is probably true for all N ≥ 6. The argument

in the proof of the proposition fails in the cases N = 4 and N = 5 which remain open.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

ECP can not be repaired

There was some hope (E.H. Lieb question) that we can save ECP by replacing the labelling by a power of the labelling. This is excluded by the next examples initiated by Buhovsky-Logunov-Sodin and continued in my work with P. B´ erard and P. Charron. These examples are for the Laplacian on T2 and S2.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Counterexamples on T2

The following result has appeared in [BLS] (2019) and was a motivation of my work with Pierre B´ erard and Philippe Charron.

Proposition [BLS]

There exist a smooth metric g on the torus T2, in the form g = Q(x)(dx2 + dy2), an infinite sequence φj of eigenfunctions of the Laplace-Beltrami operator ∆g, and an infinite sequence cj of real numbers, such that the level sets {(x, y) | φj(x, y) = cj} have infinitely many connected components. This contradicts any extension of Conjecture (ECP) and in particular answers negatively to Q1.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Interested only by Q1, we give an easy proof of Proposition [BLS] in the particular case of one eigenfunction only, avoiding the subtleties of [BLS]. We note that this particular case is sufficient to prove that Conjecture (ECP) is false on (T2, g) for some Liouville metrics which can be chosen arbitrarily close to the flat metric.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Metrics on T2 with a prescribed eigenfunction

As in [BLS], we apply an approach due to Jakobson and Nadirashvili [JaNa1999] for a different problem. Consider the torus T2 = (R/2πZ)2 with the flat metric g0 = dx2 + dy2 , and associated Laplace-Beltrami operator ∆0 = ∂2

x + ∂2 y .

Consider Liouville metrics on T2, of the form Q(x)

• dx2 + dy2

, where Q is a positive C ∞ function on T1 = R/2πZ. For a given function Q > 0, and for k ∈ N, consider the family of eigenvalue problems − y′′(x) + k2y(x) = σ Q(x) y(x) on T1 . (12)

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

An associated complete set of spectral pairs is given by {(σk,j, Fk,j) | j ∈ N∗} . Coming back to our Laplacian Q(x)−1∆0, a complete set of spectral pairs (λ, φ) for the eigenvalue problem − ∆0φ(x, y) = λ Q(x) φ(x, y) on T2 , (13) is given by {(σk,j, Fk,j(x) cos(ky)) , (σk,j, Fk,j(x) sin(ky))

• k ∈ N, j ∈ N∗} .

(14)

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Given a positive C ∞ function F on T2, and m ∈ N∗, define the function Φ(x, y) = F(x) cos(my). Φ is an eigenfunction of the eigenvalue problem above, for some positive function Q, if and

• nly if there exists some λ > 0 such that

Q(x) = m2 λ

• 1 − 1

m2 F ′′(x) F(x)

• ,

with m chosen such that F ′′(x) < m2 F(x) , for all x ∈ R , (15) in order to assure the positivity of Q(x).

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Since the flat metric g0 corresponds to Q ≡ 1, we choose λ to be m2, and Q(x) = 1 − 1 m2 F ′′(x) F(x) . (16) so that the associated metric on T2 appears as a perturbation of g0 (for m large).

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

We prove the following result by describing an explicit construction.

Proposition [BCH]A

There exists a metric gQ = Q(x)

• dx2 + dy2
• n the torus T2, and

an eigenfunction Φ of the associated Laplace-Beltrami operator, −∆QΦ := Q−1 ∆0Φ = Φ, such that the super-level set {Φ > 1} has infinitely many connected components. As a consequence, Conjecture (ECP) is false for (T2, gQ).

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Remarks

◮ The metric gQ in the proposition is smooth, but not real

• analytic. We have a different kind of result for real analytic

metrics.

◮ This proposition also implies that Φ has infinitely many

isolated critical points, a particular case of Theorem 1 in [BLS].

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Proof–Step 1. Construction of Φ

Let φ : [−π, π] → R be a function such that      0 ≤ φ(x) ≤ 1 , supp(φ) ⊂ [− π

2 , π 2 ] ,

φ ≡ 1 on [− π

3 , π 3 ] .

(17) Define the function F1 : [−π, π] → R by F1(x) = φ(x) exp

• − 1

x2

• cos

1 x2

• + 1 − φ(x) .

(18)

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

It is clear that F1 satisfies      |F1(x)| ≤ 1 , |x| > π

2 ⇒ F1(x) = 1 ,

|x| > π

3 ⇒ F1(x) > 0 .

(19)

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

It follows that F1 vanishes only in [− π

3 , π 3 ], with zero set

Z(F1) = {x | F1(x) = 0} given by Z(F1) = {0} ∪ {± 1 π

2 + kπ

• k ∈ N} .

(20) The zero set Z(F1) is an infinite sequence of distinct points with 0 as only accumulation point, and the function F1 changes sign at each zero.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Step 2.

Define F0 to be the function F1 extended as a 2π-periodic function

• n R, and F to be F := 1 + 1
• 2F0. Given m ∈ N, define the

function Φm : T2 → R to be Φm(x, y) = F(x) cos(my) . F and Φm satisfy,            F ∈ C ∞(T2) , F ≥ 1

2 ,

{F0 < 0} × T1 ⊂ {Φm < 1} , {Φm ≥ 1} ⊂ {F0 ≥ 0} × T1 , {F0 ≥ 0} × {0} ⊂ {Φm ≥ 1}. (21)

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

It follows from (20) that {F0 ≥ 0} ⊂ T1 is the union of infinitely many pairwise disjoint closed intervals, Iℓ, ℓ ∈ Z. It follows from (21) that there is at least one connected component of the super-level set {Φm > 1} in each Iℓ × T1. We have constructed a family of functions, Φm, m ∈ N, whose super-level set {Φm > 1} has infinitely many connected components in T2. This is illustrated in Figure 7 (in the figure, m = 1, the red curve is the graph of a function which is rapidly

• scillating, like the function F1 defined in (18), and the closed blue

curves are part of the corresponding level set {Φ = 1}.)

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Figure: 7: Level set {Φ = 1}

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Step 3.

Since F ∈ C ∞(T1), and F ≥ 1

2, the function F ′′ F is bounded from

• above. We choose m such that

m2 > sup

x∈T1

F ′′(x) F(x) , (22) and we define the function Qm : T1 → R, Qm(x) = 1 − 1 m2 F ′′(x) F(x) . (23) Under the condition (22), the function Qm defines a Liouville metric gm on T2, gm = Qm(x)

• dx2 + dy2

(24)

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

This metric can be chosen arbitrarily close to the flat metric dx2 + dy2 as m goes to infinity. For the associated Laplace-Beltrami operator ∆gm, we have − ∆gmΦm(x, y) = m2Φm(x, y) , (25) so that the function Φm is an eigenfunction of ∆gm, with eigenvalue m2. The super-level set {Φm > 1} has infinitely many connected components in T2. In particular, the function Φm − 1 has infinitely many nodal domains.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Example 2

The metric constructed in Proposition [BCH]A is smooth, not real

• analytic. We prove the following result in which we have a real

analytic metric.

Proposition [BCH]B

Let n be any given integer. Then there exists a real analytic Liouville metric g = Q(x)

• dx2 + dy2
• n T2, and an

eigenfunction Φ of the associated Laplace-Beltrami operator, −∆gΦ = Φ, with eigenvalue 1, such that the super-level set {Φ > 1} has at least n connected components. One can choose the metric g arbitrarily close to the flat metric g0. Taking n ≥ 4, and g close enough to g0, the eigenvalue 1 is either the second, the third or the fourth eigenvalue of ∆g.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Remarks

◮

It follows from the proposition that the function Φ − 1 provides a counterexample to Conjecture (ECP) for (T2, g).

◮

The proposition also gives a negative answer to the question raised above. Indeed, given any n ≥ 4, the function Φ given by the proposition is associated with the eigenvalue 1, whose labelling is at most 4, and Φ has at least n isolated critical zeros.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Counterexamples on S2

We prove the following results.

Proposition [BCH]C

There exist C ∞ functions Φ and G on S2, with the following properties.

• 1. The super-level set {Φ > 1} has infinitely many connected

components.

• 2. The function G is positive, and defines a conformal metric

gG = G g0 on S2 with associated Laplace-Beltrami operator ∆G = G −1 ∆0.

• 3. −∆GΦ = 2 Φ.
• 4. The eigenvalue 2 of −∆G has labelling at most 9.

We are not able to propose a choice such that G is close to 1.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Proposition [BCH]D

There exists M > 0 such that, for any m ≥ M, there exist C ∞ functions Φm and Gm on S2 with the following properties.

• 1. The super-level set {Φm > 1} has infinitely many connected

components.

• 2. The function Gm is positive, and defines a conformal metric

gm = Gm g0 on S2 with associated Laplace-Beltrami operator ∆gm = G −1

m ∆0.

• 3. For m ≥ M,
• 1 − 2

m

• ≤ Gm ≤
• 1 + 2

m

• , and
• 4. −∆gmΦm = m(m + 1) Φm.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

These propositions provide counterexamples to Conjecture (ECP) and answer negatively to Questions (Q1)-(Q3). Remark The eigenfunctions on S2 constructed in the above propositions have infinitely many isolated critical points.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

About the proof

The approach is inspired by what we have done for T2 with the following steps.

• 1. Start from a special spherical harmonic Y of the standard

sphere (S2, g0), with eigenvalue m(m + 1).

• 2. Modify Y into a smooth function F, whose super-level set

{F > 1} has infinitely many connected components.

• 3. Construct a conformal metric gQ = Q g0 on S2, whose

associated Laplace-Beltrami has F as eigenfunction, with eigenvalue m(m + 1). We actually use, in spherical coordinates, Y (θ, φ) = sinm θ cos mφ and perturbe sinm(θ) in the variable θ.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Isotropic quantum harmonic oscillator in dimension 2

We finally show that Conjecture (ECP) is true for the harmonic

• scillator H : L2(R2) → L2(R2), H = −∆ + x2 + y2.

Proposition [BCH]E

Let fi be the eigenfunctions of H with eigenvalues ordered in increasing order with multiplicities. Then, for any linear combination f =

n

• i=1

aifi, we have β0(f ) ≤ n .

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

A basis fn of eigenfunctions of H is given by Ha,b(x, y) := e− x2+y2

2

Ha(x)Hb(y) , 0 ≤ a, b ∈ N , where Hn refers to the n-th Hermite polynomial. The associated eigenvalue is given by 2(a + b + 1), with multiplicity a + b + 1. Therefore, counting multiplicities, for each n in the interval [ k(k+1)

2

+ 1, (k+1)(k+2)

2

] for some positive integer k, fn is a polynomial of degree k. For a polynomial f of degree k in 2 variables, we have the following upper bound on the number of its nodal domains:

Lemma

For any polynomial f of degree k in R2, β0(f ) ≤ k(k + 1)/2 + 1 .

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

This is a consequence of a classical theorem by Harnack:

Harnack’s curve theorem

Let f be a real irreducible polynomial in two variables, of degree k. Let ai be the singular points of the nodal set, with order si. We have the following inequality for the number U(f ) of connected components of its nodal set: U(f ) ≤ (k − 1)k 2 −

• i

si(si − 1) 2 + 1 .

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

Thank you for your attention.

Bernard Helffer, Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes. On Courant’s nodal domain property for linear combinations of eigenfunctions

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