Nodal Domain Theorems à la Courant

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


Consider a Schrödinger operator
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 1,2 0 (Ω 0 ) and form core C ∞ 0 (Ω 0 ) and A. Ancona, B. Helffer, T. Hoffmann-Ostenhof 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 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 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.

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 where λ j (Ω 0 ) is the j-th eigenvalue of H(Ω 0 ).We also define and So we always have : with equality when λ is not an eigenvalue.Note that n(λ, Ω 0 ) − 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 ℓ j ) j∈N\{0} (with λ ℓ j ≤ λ ℓ j+1 ).When counting the eigenvalues less than some given λ , we shall for simplicity write and analogously for the quantities with over-, respectively, underbars.
and is proved essentially in the same way.
Finally we show that Courant's nodal theorem is an easy corollary of Theorem 2.1.

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

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 Then, for any subset L ⊂ {1, 2, . . ., k} such that 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 : 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 : ..,k be the family of the nodal domains associated to u, let L be a subset of {1, . . ., k} with #L = ℓ and let where λ j (Ω * L ) are the eigenvalues of Let us first recall some basic tools (see e.g.[17]) which were already vital for the proof of Courant's classical result.

Variational characterization
Let us first recall the variational characterization of eigenvalues.
and (u j ) j≥1 is as before an orthonormal basis of eigenfunctions of H(Ω) associated to (λ j ) j≥1 .Then and 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.

Unique continuation
Next we restate a weak form of the unique continuation property: There are stronger results of this type under weaker assumptions on the potential, see [11].

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 : 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
(5.2) Let ϕ ℓ0 i , i = 1, . . ., n(λ, Ω ℓ0 ), denote the first n ℓ0 eigenfunctions of H(Ω ℓ0 ).The corresponding eigenvalues are strictly smaller than λ.The functions ϕ ℓ0 i and the remaining ℓ =ℓ0 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 n 0 eigenfunctions of H(Ω 0 ).By assumption Φ, 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 ℓ0 i , i < n ℓ0 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 ℓ0 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 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 ℓ j , j ≤ n ℓ , corresponding to the different H(Ω ℓ ) such that Φ⊥U + , Φ ≡ 0, but Φ satisfies and this contradicts (4.3).This proves (2.6).

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 ℓ j , j = 1, . . .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. 2

Proof of Theorem 3.4
For the case that (3.We assume for contradiction that where n * is defined as above.The addition of (5.10) and (5.13) leads to a contradiction. 2 6 Illustrative examples

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 −∆ with Dirichlet boundary condition are given by while those for Ω 1 , and hence for Ω 2 which can be obtained by a translation of Ω 1 , are given by 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 has the following asymptotics as λ tends to infinity: We have not to consider A(λ) but rather Hence we get 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 Note that n i (λ) = A + i (λ), i = 0, 1 .In order to control n i (λ), we observe that, for any ǫ > 0 : ) .( The asymptotic formula (6.4) implies and this shows that (2.6) is never sharp for large λ.
Documenta Mathematica 9 (2004) 283-299 6.2About 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 (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 with corresponding eigenfunctions (x, y) → sin(πm x a )(sin πny).If , and the zeroset of u 4 is given by {(x, y) 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.

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.
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 .

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 : 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 : 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 ℓ0 j of ϕ ℓ0 .Combining the two parts of the proof gives that any nodal set of u is a nodal set of ϕ ℓ and vice-versa.
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 ∞ (U )), we may restrict to functions f ∈ C ∞ (U ) vanishing in B(0, ρ).Then, since we have 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).
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 Therefore, for any δ > 0, we have : Without loss of generality, we can assume that v(x 0 ) > 0. Choose r 1 > 0 so small that v ≥ c 0 := 1 2 v(x 0 ) in B(x 0 , r 1 ).Since u ∈ W 1,2 0 (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 −1 0 u |B(x0,r1) , f n = c −1 0 u n|B(x 0,r1) , we see that B(x 0 , r 1 ) \ D is polar.

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 (−∆ + V )u = λu which vanishes on an open subset ω of Ω is identically zero in the connected component of Ω containing ω.