Nodal Domain Theorems à la Courant

A. Ancona, B. Helffer, and T. Hoffmann-Ostenhof^{∗}

Received: April 23, 2004 Revised: July 4, 2004

Communicated by Heinz Siedentop

Abstract. Let H(Ω_{0}) = −Δ + V be a Schrödinger operator on a bounded domain Ω_{0} ⊂ ℝ^{d} (d ≥ 2) with Dirichlet boundary condition. Suppose that Ω_{ℓ} (ℓ ∈ {1, …, k}) are some pairwise disjoint subsets of Ω_{0} and that H(Ω_{ℓ}) are the corresponding Schrödinger operators again with Dirichlet boundary condition. We investigate the relations between the spectrum of H(Ω_{0}) and the spectra of the H(Ω_{ℓ}). In particular, we derive some inequalities for the associated spectral counting functions which can be interpreted as generalizations of Courant’s nodal theorem. For the case where equality is achieved we prove converse results. In particular, we use potential theoretic methods to relate the Ω_{ℓ} to the nodal domains of some eigenfunction of H(Ω_{0}).

2000 Mathematics Subject Classification: 35B05

1 Introduction

Consider a Schrödinger operator

H = −Δ + V  (1.1)

on a bounded domain Ω_{0} ⊂ ℝ^{d} with Dirichlet boundary condition. Further we assume that V is real valued and satisfies V ∈ L^{∞}(Ω_{0}). (We could relax this condition and extend our results to the case V ∈ L^{β}(Ω_{0}) for some β > d/2 using [11].) The operator H is selfadjoint if viewed as the Friedrichs extension of the quadratic form of H with form domain W_{0}^{1,2}(Ω_{0}) and form core C_{0}^{∞}(Ω_{0}) and we denote it by H(Ω_{0}). Further H(Ω_{0}) has compact resolvent. So the spectrum of H(Ω_{0}), σ(H(Ω_{0})), can be described by an increasing sequence of eigenvalues

∗Supported by Ministerium für Bildung, Wissenschaft und Kunst der Republik Österreich

Documenta Mathematica 9 (2004) 283–299

λ_{1} < λ_{2} ≤ λ_{3} ≤ ⋯ ≤ λ_{j} ≤ λ_{j+1} ≤ …  (1.2)

tending to +∞, such that the associated eigenfunctions u_{j} form an orthonormal basis of L^{2}(Ω_{0}). We can assume that these eigenfunctions u_{j} are real valued and by elliptic regularity, [9] (Corollary 8.36), u_{j} belongs to C^{1,α}(Ω_{0}) for every α < 1. Moreover λ_{1} is simple and the corresponding eigenfunction u_{1} can be chosen to satisfy, see e.g. [17],

u_{1}(x) > 0 , for all x ∈ Ω_{0} .  (1.3)

For a bounded domain D we let H(D) be the corresponding selfadjoint operator, with Dirichlet boundary condition on ∂D. Its lowest eigenvalue will be denoted by λ(D). We denote the zero set of an eigenfunction u by

N(u) = \overline{\{x ∈ Ω_{0} | u(x) = 0\}}.  (1.4)

The nodal domains of u, which are by definition the connected components of Ω_{0} \ N(u), will be denoted by D_{j}, j = 1, …, μ(u), where μ(u) denotes the number of nodal domains of u. Suppose that Ω_{ℓ} (ℓ = 1, 2, …, k) are k open pairwise disjoint subsets of Ω_{0}. In this paper we shall investigate relations between the spectrum of H(Ω_{0}) and the spectra of the H(Ω_{ℓ}). Roughly speaking, we shall derive an inequality between the counting function of H(Ω_{0}) and those of the H(Ω_{ℓ}). This inequality can be interpreted as a generalization of Courant’s classical nodal domain theorem. For the case where equality is achieved this will lead to a partial characterization of the Ω_{ℓ} which will turn out to be related to the nodal domains of one of the eigenfunctions of H(Ω_{0}). These results will be given in sections 2 and 3. From these results some natural questions of potential theoretic nature arise which will be analyzed and answered in section 7. The proofs of the results stated in sections 2 and 3 are given in sections 4 and 5. In section 6 some illustrative explicit examples are given.

2 Main results

We start with a result which will turn out to be a generalization of Courant’s nodal theorem. We consider again (1.1) on a bounded domain Ω_{0} and the corresponding eigenfunctions and eigenvalues. We first introduce

\overline{n}(λ, Ω_{0}) = #\{j | λ_{j}(Ω_{0}) ≤ λ\},  (2.1)

where λ_{j}(Ω_{0}) is the j-th eigenvalue of H(Ω_{0}). We also define

\underline{n}(λ, Ω_{0}) = #\{j | λ_{j}(Ω_{0}) < λ\},  (2.2)

and

n(λ, Ω_{0}) = \begin{cases}\underline{n}(λ, Ω_{0}) & if λ ∉ σ(H(Ω_{0})) \\ \underline{n}(λ, Ω_{0}) + 1 & if λ ∈ σ(H(Ω_{0})). \end{cases}  (2.3)

So we always have :

\underline{n}(λ, Ω_{0}) ≤ n(λ, Ω_{0}) ≤ \overline{n}(λ, Ω_{0}),  (2.4)

with equality when λ is not an eigenvalue. Note that \overline{n}(λ, Ω_{0}) − \underline{n}(λ, Ω_{0}) is the multiplicity of λ when λ is an eigenvalue of H(Ω_{0}), i.e. the dimension of the eigenspace associated to λ. We shall consider a family of k open sets Ω_{ℓ} (ℓ = 1, …, k) contained in Ω_{0} and the corresponding Dirichlet realizations H(Ω_{ℓ}). For each H(Ω_{ℓ}) the corresponding eigenvalues counted with multiplicity are denoted by (λ_{j}^{ℓ})_{j∈ℕ\{0}} (with λ_{j}^{ℓ} ≤ λ_{j+1}^{ℓ}). When counting the eigenvalues less than some given λ , we shall for simplicity write

n_{ℓ} = n_{ℓ}(λ) = n(λ, Ω_{ℓ}),  (2.5)

and analogously for the quantities with over-, respectively, underbars.

Theorem 2.1 Suppose Ω_{0} ⊂ ℝ^{d} is a bounded domain and that λ ∈ σ(H(Ω_{0})). Suppose that the sets Ω_{ℓ} (ℓ = 1, …, k) are pairwise disjoint open subsets of Ω_{0}. Then

∑_{ℓ=1}^{k} \overline{n}_{ℓ} ≤ n_{0} + min_{ℓ≥0} (\overline{n}_{ℓ} − n_{ℓ}) .  (2.6)

A direct weaker consequence of (2.6) is the more standard

Corollary 2.2 Under the assumptions of Theorem 2.1, we have

∑_{ℓ=1}^{k} \overline{n}_{ℓ} ≤ \overline{n}_{0} .  (2.7)

This corollary is actually present in the proofs of the asymptotics of the counting function (see for example the Dirichlet-Neumann bracketing in Lieb-Simon [14]).

Remark 2.3 Inequality (2.6) is also true if λ ∉ σ(H(Ω_{0})). The statement becomes

∑_{ℓ=1}^{k} \overline{n}_{ℓ} ≤ n_{0} ,

and is proved essentially in the same way.

Remark 2.4 The assumption that Ω_{0} is connected is necessary. Indeed, suppose Ω_{1} and Ω_{2} are connected and assume that Ω_{0} = Ω_{1} ∪ Ω_{2} with Ω_{1} ∩ Ω_{2} = ∅ and that λ = λ_{1}(Ω_{1}) = λ_{1}(Ω_{2}). Then λ_{1}(Ω_{0}) = λ_{2}(Ω_{0}) and we deduce n(λ, Ω_{0}) = 1. If we no longer assume the connectedness of Ω_{0}, we in general just have Corollary 2.2.

Finally we show that Courant’s nodal theorem is an easy corollary of Theorem 2.1.

Corollary 2.5 : Courant’s nodal theorem If Ω_{0} is connected and if u is an eigenfunction of H(Ω_{0}) associated to some eigenvalue λ, then

μ(u) ≤ n(λ, Ω_{0}) .

Proof. We now simply apply Theorem 2.1 by taking Ω_{1}, …, Ω_{μ(u)} as the nodal domains associated to u. We just have to use (1.3) for each Ω_{ℓ}, ℓ = 1, …, μ(u), which gives \overline{n}_{ℓ} = n_{ℓ} = 1. □

Remark 2.6 Courant’s nodal theorem is one of the basic results in spectral theory of Schrödinger-type operators. It is the natural generalization of Sturm’s oscillation theorem for second order ODE’s. For recent investigations see for instance [1] and [4].

3 Converse results.

In this section we consider some results that are converse to Theorem 2.1.

Theorem 3.1 Suppose that the Ω_{ℓ}, 1 ≤ ℓ ≤ k, are pairwise disjoint open subsets of Ω_{0}. If λ ∈ σ(H(Ω_{0})) and

∑_{ℓ=1}^{k} \overline{n}_{ℓ} ≥ n_{0} ,  (3.1)

then λ ∈ σ(H(Ω_{ℓ})) for each Ω_{ℓ}. If U_{ℓ}(λ) denotes the eigenspace of H(Ω_{ℓ}) associated to the eigenvalue λ, then there is an eigenfunction u of H(Ω_{0}) with eigenvalue λ such that

u = ∑_{ℓ=1}^{k} φ_{ℓ} in W_{0}^{1,2}(Ω_{0}) ,  (3.2)

where each φ_{ℓ} belongs to U_{ℓ}(λ) \ {0} and is identified with its extension by 0 outside Ω_{ℓ}.

Remark 3.2 One can naturally think that formula (3.2) has immediate consequences on the family Ω_{ℓ}, which should for example have some covering property. The question is a bit more subtle because we do not a priori want to assume strong regularity properties for the boundaries of the Ω_{ℓ}. We shall discuss this point in detail in the last section.

Another consequence of equalities in Theorems 2.1 or 3.1 is given by the following result.

Theorem 3.3 Suppose that, for some bounded domain Ω_{0} in ℝ^{d}, some λ ∈ σ(H(Ω_{0})) and some family of pairwise disjoint open sets Ω_{ℓ} ⊂ Ω_{0}, 0 < ℓ ≤ k, we have

∑_{ℓ=1}^{k} \overline{n}_{ℓ} = n_{0} + min_{ℓ≥0} (\overline{n}_{ℓ} − n_{ℓ}) .  (3.3)

Then, for any subset L ⊂ \{1, 2, …, k\} such that Ω_{L}^{∗} = Int ( ∪_{ℓ∈L} \overline{Ω_{ℓ}}) \ ∂Ω_{0} is connected, we have

∑_{ℓ∈L} \overline{n}_{ℓ} = n(λ, Ω_{L}^{∗}) + min ( min_{ℓ∈L} (\overline{n}_{ℓ} − n_{ℓ}), \overline{n}(λ, Ω_{L}^{∗}) − n(λ, Ω_{L}^{∗})) . (3.4)

A simpler variant is the following :

Theorem 3.4 Suppose (3.1) holds and that Ω_{L}^{∗} is defined as above. Then we have the inequality :

∑_{ℓ∈L} \overline{n}_{ℓ} ≥ n(λ, Ω_{L}^{∗}) .  (3.5)

On the sharpness of Courant’s nodal theorem It is well known that Courant’s nodal theorem is sharp only for finitely many k’s [15]. Let Ω_{0} be connected. We will say that an eigenfunction u associated to an eigenvalue λ of H(Ω_{0}) is Courant-sharp if μ(u) = n(λ, Ω_{0}). Theorem 3.3 now implies :

Corollary 3.5

i) Let u be a Courant-sharp eigenfunction of H(Ω_{0}) with μ(u) = k. Let {D_{i}}_{i=1,...,k} be the family of the nodal domains associated to u, let L be a subset of \{1, …, k\} with #L = ℓ and let Ω_{L}^{∗} = Int (\overline{∪_{i∈L}D_{i}}) \ ∂Ω_{0}. Then

λ_{ℓ}(Ω_{L}^{∗}) = λ_{k} ,  (3.6)

where λ_{j}(Ω_{L}^{∗}) are the eigenvalues of H(Ω_{L}^{∗}). ii) Moreover, if Ω_{L}^{∗} is connected, and if ℓ < k, then u|_{Ω_{L}^{∗}} is Courant-sharp and λ_{ℓ}(Ω_{L}^{∗}) is simple.

4 Basic tools

Let us first recall some basic tools (see e.g. [17]) which were already vital for the proof of Courant’s classical result.

4.1 Variational characterization

Let us first recall the variational characterization of eigenvalues.

Proposition 4.1 Let Ω be a bounded open set in ℝ^{d} and let V ∈ L^{∞}(Ω) be real-valued. Suppose λ ∈ σ(H(Ω)) and let U_{±} = span ⟨u_{1}, …, u_{k_{±}}⟩ where

k_{−} = \underline{n}(λ, Ω) , k_{+} = \overline{n}(λ, Ω) ,  (4.1)

and (u_{j})_{j≥1} is as before an orthonormal basis of eigenfunctions of H(Ω) associated to (λ_{j})_{j≥1}. Then

λ = inf_{φ⊥U_{−}, φ∈W_{0}^{1,2}(Ω)} \frac{⟨φ, H(Ω)φ⟩}{∥φ∥^{2}}  (4.2)

and

λ < λ_{\overline{n}(λ, Ω)+1} = inf_{φ⊥U_{+}, φ ∈W_{0}^{1,2}(Ω)} \frac{⟨φ, H(Ω)φ⟩}{∥φ∥^{2}}.  (4.3)

If equality is achieved in (4.2) for some φ ≢ 0, then φ is an eigenfunction in the eigenspace of λ.

Note that (4.2) and (4.3) are actually the same statement. We just stated them separately for later reference. Note that we have not assumed that Ω is connected.

4.2 Unique continuation

Next we restate a weak form of the unique continuation property:

Theorem 4.2 Let Ω be an open set in ℝ^{d} and let V ∈ L_{loc}^{∞}(Ω) be real-valued. Then any distributional solution in Ω to (−Δ + V)u = λu which vanishes on an open subset ω of Ω is identically zero in the connected component of Ω containing ω.

There are stronger results of this type under weaker assumptions on the potential, see [11].

4.3 A consequence of Harnack’s inequality

The standard Harnack’s inequality (see e.g. Theorem 8.20 in [9]), together with the unique continuation theorem leads to the following theorem :

Theorem 4.3 If Ω is a bounded domain in ℝ^{d} and u is an eigenfunction of H(Ω), then for any x in N(u) ∩ Ω and any ball B(x, r) (r > 0), there exist y_{±} ∈ B(x, r) ∩ Ω such that ±u(y_{±}) > 0.

5 Proof of the main theorems

5.1 Proof of Theorem 2.1

Assume first for contradiction that

∑_{ℓ≥1} \overline{n}_{ℓ} > n_{0} + min_{ℓ≥0} (\overline{n}_{ℓ} − n_{ℓ}) ,  (5.1)

and recall that we assume that λ ∈ σ(H(Ω_{0})). Pick some ℓ_{0} such that

\overline{n}_{ℓ_{0}} − n_{ℓ_{0}} = min_{ℓ≥0} (\overline{n}_{ℓ} − n_{ℓ}) .

Suppose first that ℓ_{0} ≥ 1. We can rewrite (5.1) to obtain

∑_{ℓ≠ℓ_{0}, ℓ≥1} \overline{n}_{ℓ} + n_{ℓ_{0}} > n_{0} .  (5.2)

Let φ_{i}^{ℓ_{0}}, i = 1, …, \underline{n}(λ, Ω_{ℓ_{0}}), denote the first \underline{n}_{ℓ_{0}} eigenfunctions of H(Ω_{ℓ_{0}}). The corresponding eigenvalues are strictly smaller than λ. The functions φ_{i}^{ℓ_{0}} and the remaining ∑_{ℓ≠ℓ_{0}} \overline{n}_{ℓ} eigenfunctions associated to the other H(Ω_{ℓ}) span a space of dimension at least n_{0}. We can pick a linear combination Φ ≢ 0 of these functions which is orthogonal to the \underline{n}_{0} eigenfunctions of H(Ω_{0}). By assumption

\frac{⟨Φ, H(Ω_{0})Φ⟩}{∥Φ∥^{2}} ≤ λ,  (5.3)

hence Φ must by the variational principle be an eigenfunction and there must be equality in (5.3). There are two possibilities: either some φ_{i}^{ℓ_{0}}, i < n_{ℓ_{0}} contributes to the linear combination which makes up Φ or not. In the first case this means that the left hand side of (5.3) is strictly smaller than λ, contradicting the variational characterization of λ. In the other case we obtain a contradiction to unique continuation, since then Φ ≡ 0 in Ω_{ℓ_{0}} and hence Φ vanishes identically in all of Ω_{0}.

Consider now the case when ℓ_{0} = 0. We have to show that the assumption

∑_{ℓ≥1} \overline{n}_{ℓ} > \overline{n}_{0} ,  (5.4)

leads to a contradiction. To this end it suffices to apply (4.3). Indeed, we can find a linear combination Φ of the eigenfunctions φ_{j}^{ℓ}, j ≤ \overline{n}_{ℓ}, corresponding to the different H(Ω_{ℓ}) such that Φ⊥U_{+}, Φ ≢ 0, but Φ satisfies

\frac{⟨Φ, H(Ω_{0}) Φ⟩}{∥Φ∥^{2}} ≤ λ = λ_{\overline{n}_{0}} ,

and this contradicts (4.3). This proves (2.6).

5.2 Proof of Theorem 3.1

The inequality (3.1) implies that we can find a non zero u⊥U_{−} in the span of the eigenfunctions φ_{j}^{ℓ}, j = 1, … \overline{n}_{ℓ}, of the different H(Ω_{ℓ}). Again by the variational characterization, (4.2) and (5.3) hold and hence u must be an eigenfunction. □

5.3 Proof of Theorem 3.3

We assume (3.3). Without loss we might assume that we have labeled the Ω_{ℓ} such that L = {1, …, K}, with K ≤ k. Let n_{∗} = n(λ, Ω_{L}^{∗}). We apply Theorem 2.1 to the family Ω_{ℓ} (ℓ ∈ L) and replace Ω_{0} by Ω_{L}^{∗} and obtain :

∑_{1≤ℓ≤K} \overline{n}_{ℓ} ≤ n_{∗} + min (\overline{n}_{∗} − n_{∗}, min_{1≤ℓ≤K}(\overline{n}_{ℓ} − n_{ℓ})) .  (5.5)

We assume for contradiction that

∑_{1≤ℓ≤K} \overline{n}_{ℓ} < n_{∗} + min (\overline{n}_{∗} − n_{∗}, min_{1≤ℓ≤K}(\overline{n}_{ℓ} − n_{ℓ})) .  (5.6)

This implies

∑_{1≤ℓ≤K} \overline{n}_{ℓ} < \overline{n}_{∗} ,  (5.7)

and

∑_{1≤ℓ≤K} \overline{n}_{ℓ} < n_{∗} + min_{1≤ℓ≤K}(\overline{n}_{ℓ} − n_{ℓ}) .  (5.8)

Theorem 2.1, applied to the family Ω_{L}^{∗}, Ω_{ℓ} (ℓ > K), implies that

\overline{n}_{∗} + ∑_{K<ℓ≤k} \overline{n}_{ℓ} ≤ n_{0} + min (\overline{n}_{0} − n_{0}, min_{K<ℓ≤k}(\overline{n}_{ℓ} − n_{ℓ})) ,  (5.9)

and

n_{∗} + ∑_{K<ℓ≤k} \overline{n}_{ℓ} ≤ n_{0} .  (5.10)

By adding (5.7) and (5.9), we get :

∑_{1≤ℓ≤k} \overline{n}_{ℓ} < n_{0} + min (\overline{n}_{0} − n_{0}, min_{K<ℓ≤k}(\overline{n}_{ℓ} − n_{ℓ})) .  (5.11)

By adding (5.8) and (5.10), we obtain

∑_{1≤ℓ≤k} \overline{n}_{ℓ} < n_{0} + min_{1≤ℓ≤K}(\overline{n}_{ℓ} − n_{ℓ}) .  (5.12)

The combination of (5.11) and (5.12) is in contradiction with (3.3).

5.4 Proof of Theorem 3.4

For the case that (3.1) holds, (3.5) can be shown similarly. (3.1) reads

∑_{1≤ℓ≤k} \overline{n}_{ℓ} ≥ n_{0} .

We assume for contradiction that

∑_{1≤ℓ≤K} \overline{n}_{ℓ} < n_{∗} ,  (5.13)

where n_{∗} is defined as above. The addition of (5.10) and (5.13) leads to a contradiction. □

6 Illustrative examples

6.1 Examples for a rectangle

We illustrate Theorem 2.1 by the analysis of various examples in rectangles. Pick a rectangle Ω_{0} = (0, 2π) × (0, π) and take Ω_{1} = (0, π) × (0, π) and consequently Ω_{2} = (π, 2π) × (0, π). The eigenvalues corresponding to Ω_{0} for −Δ with Dirichlet boundary condition are given by

σ(H(Ω_{0})) = {λ ∈ ℝ | λ = m^{2}/4 + n^{2}, (m, n) ∈ ℤ^{2}, m, n > 0} ,  (6.1)

while those for Ω_{1}, and hence for Ω_{2} which can be obtained by a translation of Ω_{1}, are given by

σ(H(Ω_{1})) = σ(H(Ω_{2})) = {λ ∈ ℝ|λ = m^{2}+n^{2}, (m, n) ∈ ℤ^{2}, m, n > 0}. (6.2)

Denote the eigenvalues associated to Ω_{0} by {λ_{i}} and those to Ω_{1} by {ν_{i}}. We easily check that λ_{5} = λ_{6} = ν_{2} = ν_{3} = 5, λ_{11} = λ_{12} = ν_{5} = ν_{6} = 10 so that Theorem 2.1 is sharp for these cases. One could ask whether there are arbitrarily high eigenvalues cases for which we have equality in (2.6). This is not the case, as can be seen from the following standard number theoretical considerations. We have (see [18] and for more recent contributions [16] and [2]) the following asymptotic estimate for the number of lattice points in an ellipse. Let a, b > 0, then

A(λ) := #{(m, n) ∈ ℤ^{2} | am^{2} + bn^{2} ≤ λ}  (6.3)

has the following asymptotics as λ tends to infinity:

A(λ) = \frac{π}{\sqrt{ab}}λ + O(λ^{1/3}).  (6.4)

We have not to consider A(λ) but rather

A^{+} = #{(m, n) ∈ ℤ^{2}, m, n > 0 | am^{2} + bn^{2} ≤ λ} .  (6.5)

Hence we get

\begin{matrix}A(λ) = 4A^{+}(λ) + 2#\{m ∈ ℕ, m > 0 | m ≤ [(λ/a)^{1/2}]\} \\ +2#\{n ∈ ℕ, n > 0 | n ≤ [(λ/b)^{1/2}]\} + 1 . \end{matrix}  (6.6)

If we apply this to A^{+} with a = 1/4, b = 1 (in this case denoted by A_{0}^{+}) and to A^{+} with a = 1, b = 1 (in this case denoted by A_{1}^{+}), we get asymptotically

A_{0}^{+}(λ) − 2A_{1}^{+}(λ) = \frac{1}{2}\sqrt{λ} + o (\sqrt{λ}) .  (6.7)

Note that

\overline{n}_{i}(λ) = A_{i}^{+}(λ), i = 0, 1 .

In order to control n_{i}(λ), we observe that, for any ϵ > 0 :

\overline{n}_{i}(λ − ϵ) ≤ n_{i}(λ) ≤ \overline{n}_{i}(λ) .

This implies

\overline{n}_{i}(λ) − n_{i}(λ) = O(λ^{\frac{1}{3}} ) .  (6.8)

The asymptotic formula (6.4) implies

\overline{n}_{i}(λ) − n_{i}(λ) = o(\sqrt{λ}) ,  (6.9)

and this shows that (2.6) is never sharp for large λ.

6.2 About Corollary 3.5

One can ask whether there is a converse to Corollary 3.5 in the following sense. Suppose we have an eigenfunction u with k nodal domains and eigenvalue λ. For each pair of neighboring nodal domains of u, say, D_{i} and D_{j}, let Ω_{i,j} = Int (\overline{D_{i} ∪ D_{j}}) and suppose that λ = λ_{2}(Ω_{i,j}). Does this imply that λ = λ_{k}? The answer to the question is negative, as the following easy example shows : Consider the rectangle Q = (0, a) × (0, 1) ⊂ ℝ^{2} and consider H_{0}(Q). We can work out the eigenvalues explicitly as

\{π^{2}(\frac{m^{2}}{a^{2}} + n^{2})\}, for m, n ∈ ℕ \ 0,  (6.10)

with corresponding eigenfunctions (x, y) ↦ sin(πm\frac{x}{a})(sin πny). If

a^{2} ∈ (\frac{9}{4}, \frac{8}{3}) ,  (6.11)

then

λ_{3}(Q) = π^{2}(\frac{1}{a^{2}} + 4) < λ_{4}(Q) = π^{2}(\frac{9}{a^{2}} + 1) ,

and the zeroset of u_{4} is given by {(x, y) ∈ Q | x = a/3, x = 2a/3}. For u_{4} we have Ω_{1,2} = Q ∩ {0 < x < 2a/3}. If 2a/3 > 1 (which is the case under assumption (6.11)), then λ_{2}(Ω_{1,2}) = λ_{4}(Q). We have consequently an example with k = 3.

7 Converse theorems in the case of regular open sets

7.1 Preliminaries

As a consequence of Theorem 3.1 and using (1.3), we get that each nodal domain D_{kℓ} of φ_{ℓ} is included in a nodal domain D_{j0} of u. Using a result of Gesztesy and Zhao ([8], Theorem 1), this implies also that the capacity (see next subsection) of D_{j0} \ D_{kℓ} (hence the Lebesgue-measure) is 0. We now would like to show that under some extra condition the nodal domains of u are those of the φ_{ℓ}. This is easy when it is assumed that the boundaries of the Ω_{ℓ} are C^{1,α}. However, this regularity assumption is rather strong. A natural weaker regularity condition involving the notion of capacity will be given in this section.

7.2 Capacity

There are various equivalent definitions of polar sets and capacity (see e.g. [5], [7], [10], [13]). If U is a bounded open subset of ℝ^{d}, we denote by ∥.∥_{W_{0}^{1,2}(U)}

the Hilbert norm on W_{0}^{1,2}(U) :

∥u∥_{W_{0}^{1,2}(U)} := (∫_{U} |∇u|^{2} dx)^{\frac{1}{2}} .

The capacity in U of A ⊂ U is defined^{†} as

Cap_{U}(A) := inf{∥s∥_{W_{0}^{1,2}(U)}^{2} ; s ∈ W_{0}^{1,2}(U)

and s ≥ 1 a.e. in some neighborhood of A } .

It is easily checked that if K is compact and K ⊂ U ∩ V, where V is also open and bounded in ℝ^{d}, then there is a c = c(K, U, V) such that Cap_{U}(A) ≤ c Cap_{V} (A) for A ⊂ K. So Cap_{U}(A) = 0 for some bounded open U ⊃ A iff for each a ∈ A there exists an r > 0 and a bounded domain V such that V ⊃ B(a, r) and Cap_{V} (B(a, r) ∩ A) = 0. In this case we may simply write Cap(A) = 0 without referring to U.

7.3 Converse theorem

We are now able to formulate our definition of a regular point.

Definition 7.1 Let D be an open set in ℝ^{d}. We shall say that a point x ∈ ∂D is (capacity)- regular (for D) if, for any r > 0, the capacity of B(x, r)∩∁D is strictly positive.

Theorem 7.2 Under the assumptions of Theorem 3.1, any point x ∈ ∂Ω_{ℓ} ∩ Ω_{0} which is (capacity)-regular with respect to Ω_{ℓ} (for some ℓ) is in the nodal set of u.

This theorem admits the following corollary :

Corollary 7.3 Under the assumptions of Theorem 3.1 and if, for all ℓ, every point in (∂Ω_{ℓ})∩Ω_{0} is (capacity)-regular for Ω_{ℓ}, then the family of the nodal domains of u coincides with the union over ℓ of the family of the nodal domains of the φ_{ℓ}, where u and φ_{ℓ} are introduced in (3.2).

Proof of corollary It is clear that any nodal domain of φ_{ℓ} is contained in a unique nodal domain of u. Conversely, let D be a nodal domain of u and let ℓ ∈ {1, …, k}. Then, by combining the assumption on ∂Ω_{ℓ}, Proposition 7.4 and (3.2), we obtain the property :

∂Ω_{ℓ} ∩ D = ∅ .

Now, D being connected, either Ω_{ℓ} ∩ D = ∅ or D ⊂ Ω_{ℓ}. Moreover the second case should occur for at least one ℓ, say ℓ = ℓ_{0}. Coming back to the definition of a nodal set and (3.2), we observe that D is necessarily contained in a nodal domain D_{j}^{ℓ_{0}} of φ_{ℓ_{0}}. Combining the two parts of the proof gives that any nodal set of u is a nodal set of φ_{ℓ} and vice-versa.

†For d ≥ 3 the restriction that U is bounded can be removed and one may take U = ℝ^{d}.

Documenta Mathematica 9 (2004) 283–299

7.4 Proof of Theorem 7.2

The proof is a consequence of (3.2) and of the following proposition :

Proposition 7.4 Let D, Ω ⊂ ℝ^{d} be open sets such that D ⊂ Ω, and let x_{0} ∈ ∂D ∩ Ω. Assume that, for some given r_{0} > 0 such that B(x_{0}, r_{0}) ⊂ Ω, there exists u ∈ W_{0}^{1,2}(D) and v ∈ C^{0}(B(x_{0}, r_{0})) such that :

u_{|D∩B(x_{0},r_{0})} = v_{|D∩B(x_{0},r_{0})} a.e. in D ∩ B(x_{0}, r_{0}) .

Then if v(x_{0}) ≠ 0, there exists a ball B(x_{0}, r_{1}) (r_{1} > 0), such that B(x_{0}, r_{1})\D is polar, that is, of capacity 0.

Remark 7.5 Using some standard potential theoretic arguments, Proposition 7.4 can be deduced from Théorème 5.1 in [6] which characterizes, in the case where d ≥ 3, those u ∈ W^{1,2}(Ω) that belong to W_{0}^{1,2}(Ω). The proof below should be more elementary in character.

Remark 7.6 Given an open subset D ⊂ ℝ^{d} and a ball B = B(x, r), x ∈ ∂D, the difference set B \ D is polar if and only if B ∩ ∂D is polar. This follows from the fact that a polar subset of B does not disconnect B [3].

Remark 7.7 If D is a nodal domain of an eigenfunction u of H(Ω), then any point of ∂D∩Ω is capacity-regular for D. This is an immediate consequence of Theorem 4.3 (it also follows from the preceding remark). Indeed, if x is in ∂D ∩ Ω, then for any r > 0, one can find a ball B(y, r′) in ∁D ∩ B(x, r).

To prove Proposition 7.4 we require some well-known facts stated in the next three lemmas.

Lemma 7.8 Let U be a bounded convex domain in ℝ^{d} and let B(a, ρ), ρ > 0 be a ball such that \overline{B}(a, ρ) ⊂ U. There exists a positive constant c = c(a, ρ, U) such that, for every f ∈ W^{1,2}(U) vanishing a.e. in B(a, ρ),

∥f∥_{W^{1,2}(U)} ≤ c ∥∇f∥_{L^{2}(U)} .

Proof of Lemma 7.8 We can assume without loss of generality that a = 0 and let U′ = U \ B(0, ρ). Fix R so large that U ⊂ B(0, R). By approximating f by smooth functions (e.g. regularize the function x ↦ f((1−δ)x) for δ > 0 and small to get f_{1} ∈ C^{∞}(\overline{U})), we may restrict to functions f ∈ C^{∞}(U) vanishing in B(0, ρ). Then, since

|f(x)|^{2} = | ∫_{0}^{1} x · ∇f(sx) ds|^{2} ≤ R^{2} ∫ 1_{\frac{ρ}{|x|}} |∇f(sx)|^{2} ds for x ∈ U′ ,

we have

\begin{aligned}∫_{U′} |f(x)|^{2} dx & ≤ R^{2} ∫∫_{x∈U′, \frac{ρ}{|x|} ≤s≤1} |∇f(sx)|^{2} dx ds \\ & ≤ R^{2} ∫∫_{z∈sU′, ρ≤|z|, s≤1} |∇f(z)|^{2} dz \frac{ds}{s} \\ & ≤ \frac{R^{3}}{ρ} ∫_{U′} |∇f(x)|^{2} dx, \end{aligned}  (7.1)

and the lemma follows.

Lemma 7.9 Let U be a domain in ℝ^{d}. For every real-valued f ∈ W^{1,2}(U) the function g = f_{+} is also in W^{1,2}(U), with ∥g∥_{W^{1,2}(U)} ≤ ∥f∥_{W^{1,2}(U)}. Moreover the map f ↦ g from W^{1,2}(U) into itself is continuous (in the norm topology).

Remark 7.10 Since inf{f_{n}, 1} = 1 − (1 − f_{n})_{+}, it follows from the lemma that inf{f_{n}, 1} → inf{f, 1} in W^{1,2}(U) whenever f_{n} → f in W^{1,2}(U).

Proof of Lemma 7.9 For the first two facts we refer to [12] or [13], where it is moreover shown that the weak partial derivatives ∂_{j}f_{+} and ∂_{j}f satisfy

∂_{j}f_{+} = 1_{{f>0}} ∂_{j}f = 1_{{f≥0}} ∂_{j}f a.e. in U.

Therefore, for any δ > 0, we have :

\begin{aligned}∥∇[f_{n}]_{+} & − ∇f_{+}∥_{L^{2}} \\ & = ∥1_{\{f_{n}>0\}}∇f_{n} − 1_{\{f>0\}}∇f∥_{L^{2}} \\ & ≤ ∥1_{\{f_{n}>0\}}(∇f_{n} − ∇f)∥_{L^{2}} + ∥(1_{\{f>0\}} − 1_{\{f_{n}>0\}})∇f∥_{L^{2}} \\ & ≤ ∥∇f_{n} − ∇f∥_{L^{2}} + ∥(1_{\{f>0;f_{n}≤0\}} + 1_{\{f≤0;f_{n}>0\}})∇f∥_{L^{2}} \\ & ≤ ∥∇f_{n} − ∇f∥_{L^{2}} + ∥1_{\{0≤|f|≤δ\}}∇f∥_{L^{2}} + 2∥1_{\{|f_{n}−f|≥δ\}}∇f∥_{L^{2}}. \end{aligned} (7.2)

Given ε > 0, fix δ > 0 so that ∥1_{{0≤|f|≤δ}}∇f∥_{L^{2}} ≤ ε (recall that ∇f = 0 a.e.

in \{f = 0\}). Since ∇f ∈ L^{2}(U) and ∥1_{\{|f−f_{n}|≥δ\}}∥_{L^{1}} ≤ \frac{∥f_{n}−f∥_{L^{2}}^{2}}{δ^{2}} , it follows

that lim_{n→∞} ∥(1_{{|f−f_{n}|≥δ}})∇f∥_{L^{2}} = 0. Therefore lim sup_{n→∞} ∥∇[f_{n}]_{+} − ∇f_{+}∥_{L^{2}} ≤ ε,

which proves that [f_{n}]_{+} → f_{+} in W^{1,2}(U), if f_{n} → f in W^{1,2}(U).

Lemma 7.11 Let ω be open in ℝ^{d} and let {f_{n}} be a sequence of functions continuous in ω such that f_{n} ∈ W^{1,2}(ω) for each n ≥ 1 and lim_{n→∞} ∥f_{n}∥_{W^{1,2}(ω)} = 0. Then the set F = {x ∈ ω; lim inf_{n→∞} |f_{n}(x)| > 0 } is polar.

Proof of Lemma 7.11 It suffices to show that cap_{ω}(F ∩ K) = 0 for any compact subset K of ω. Let φ ∈ C_{0}^{∞}(ℝ^{d}) be such that 0 ≤ φ ≤ 1 in ℝ^{d}, φ = 1 in K and supp(φ) ⊂ ω. Then g_{n} = f_{n}φ → 0 in W_{0}^{1,2}(ω) and g_{n} = f_{n} in K. Set F_{ν} = {x ∈ ω; |g_{n}(x)| ≥ 2^{−ν} for all n ≥ ν}. By the definition of the capacity, we have Cap_{ω}(F_{ν}) ≤ 2^{2ν}∥∇g_{n}∥_{L^{2}}^{2} for all n ≥ ν and cap(F_{ν}) = 0. Therefore cap_{ω}(⋃_{ν≥1} F_{ν}) = 0 and cap_{ω}(F ⋂ K) = 0, since F ⋂ K ⊂ ⋃_{ν≥1} F_{ν}.

Proposition 7.12 Let U be a non-empty open subset of the ball B = B(a, r) in ℝ^{d}. Suppose there exist a function f continuous in U and a sequence {f_{n}} of functions continuous in B such that

(i) f ≥ 1 in U and f ∈ W^{1,2}(U), (ii) f_{n} = 0 in a neighborhood of B \ U and f_{n} ∈ W^{1,2}(U) for each n ≥ 1,

(iii) lim_{n→∞} ∥f − f_{n}∥_{W^{1,2}(U)} = 0.

Then the set F := B \ U is polar.

Proof of Proposition 7.12 Replacing f by inf{f, 1} and f_{n} by inf{f_{n}, 1}, we see^{‡} from Lemma 7.9 that we may assume that f = 1 in U. So

lim_{n→∞} ∥∇f_{n}∥_{L^{2}(U)} = 0 and lim_{n→∞} ∥1 − f_{n}∥_{L^{2}(U)} = 0 .

Fix a ball \overline{B}(z_{0}, 2ρ) ⊂ U, ρ > 0, and a cut-off function α ∈ C^{∞}(ℝ^{d}) such that α = 1 in B(z_{0}, ρ), α = 0 in ℝ^{d} \ B(z_{0}, 2ρ). Set g = 1 − α, g_{n} = (1 − α)f_{n}. Then g, g_{n} belong to W^{1,2}(B), ∇g = ∇g_{n} = 0 a.e. in F and

lim_{n→∞} ∥∇(g − g_{n})∥_{L^{2}(B)} = lim_{n→∞} ∥∇(g − g_{n})∥_{L^{2}(U)} = 0.

So, by Lemma 7.8, lim_{n→∞} ∥g−g_{n}∥_{W^{1,2}(B)} = 0. But g−g_{n} ≥ 1 in F and it follows from Lemma 7.11 that F is polar.

‡The weak convergence inf{f_{n}, 1} →^{w} inf{f, 1} suffices here. It allows the approximation of 1 = inf{f, 1} in the norm topology in W^{1,2}(U) by finite convex combination of the inf{f_{n}, 1}.

So we are again left with the case when f = 1 in U.

Documenta Mathematica 9 (2004) 283–299

Proof of Proposition 7.4 Without loss of generality, we can assume that v(x_{0}) > 0. Choose r_{1} > 0 so small that v ≥ c_{0} := \frac{1}{2}v(x_{0}) in B(x_{0}, r_{1}). Since u ∈ W_{0}^{1,2}(D), there is a sequence {u_{n}} in C_{0}^{∞}(ℝ^{d}) such that supp(u_{n}) ⊂ D and u_{n} → u in W^{1,2}(ℝ^{d}). Applying Proposition 7.12 to the ball B(x_{0}, r_{1}) and the functions f = c_{0}^{−1}u_{|B(x_{0},r_{1})}, f_{n} = c_{0}^{−1}u_{n}_{|B(x_{0},r_{1})}, we see that B(x_{0}, r_{1}) \ D is polar.

Acknowledgements. It is a pleasure to thank Fritz Gesztesy and Vladimir Maz’ya for useful discussions. We thank also the referee for his careful reading. The two last authors were partly supported by the European Science Foundation Programme Spectral Theory and Partial Differential Equations (SPECT) and the EU IHP network Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems HPRN-CT-2002-00277.

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A. Ancona Département de Mathématiques Bat. 425 Université Paris-Sud 91 405 Orsay Cedex, France Alano.Ancona@math.u-psud.fr B. Helffer Département de Mathématiques Bat. 425 Université Paris-Sud 91 405 Orsay Cedex, France Bernard.Helffer@math.u-psud.fr

T. Hoffmann-Ostenhof Institut für Theoretische Chemie Universität Wien Währinger Strasse 17 A-1090 Wien, Austria and International Erwin Schrödinger Institute for Mathematical Physics Boltzmanngasse 9, A-1090 Wien, Austria thoffman@esi.ac.at 300

Documenta Mathematica 9 (2004)