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CRITICAL PARTITIONS AND NODAL DEFICIENCY

OF BILLIARD EIGENFUNCTIONS

GREGORY BERKOLAIKO1, PETER KUCHMENT1, AND UZYSMILANSKY2,3

Abstract. The paper addresses the nodal count (i.e., the num-ber of nodal domains) for eigenfunctions of Schrodinger operatorswith Dirichlet boundary conditions in bounded domains. The clas-sical Sturm theorem claims that in dimension one, the nodal andeigenfunction counts coincide: the nth eigenfunction has n nodaldomains. The Courant Nodal Theorem claims that in any dimen-sion, the number of nodal domains of the nth eigenfunction cannotexceed n. However, it follows from a stronger upper bound by Plei-jel that in dimensions higher than 1 the equality can hold for onlyfinitely many eigenfunctions. Thus, in most cases a “nodal defi-ciency” arises. Moreover, examples are known of eigenfunctionswith arbitrarily large index n that have just two nodal domains.One can say that the nature of the nodal deficiency has not beenunderstood.

It was suggested in the recent years to look at the partitionsof the domain, rather than eigenfunctions, to try to figure outwhich of them can correspond to eigenfunctions, and what wouldbe the corresponding deficiency. One notices that if a partitiondoes correspond to an eigenfunction, then the ground state ener-gies of all the nodal domains are the same, i.e., it is an equiparti-tion. Additionally, the graph of any nodal partition must be bipar-tite. It was shown in a recent paper by Helffer, Hoffmann-Ostenhofand Terracini that (under some natural conditions) bipartite par-titions minimizing the maximum of the ground-state energies insub-domains of the partition, correspond to the “Courant sharp”eigenfunctions, i.e. to those with zero nodal deficiency.

In this paper, the authors show that, under some genericityconditions, among the bipartite equipartitions, the nodal ones cor-respond exactly to the critical points of an analogous functional,with the nodal deficiency being equal to the Morse index at thispoint. This explains, in particular, why all the minimal partitionsmust be Courant sharp.

1

2 GREGORY BERKOLAIKO1, PETER KUCHMENT1, AND UZY SMILANSKY2,3

1. Introduction

We consider a Schrodinger operator

(1) H = −∆+ V (x)

with Dirichlet conditions in a bounded domain Ω ⊂ Rd. We will assume

that the domain has a smooth boundary and that the real potentialV is also smooth. This is a severe overkill, but we do not want toaggravate the considerations with less significant details. See the finalsection for additional remarks.The operator H can be defined via its quadratic form

h[u, u] =

∫

Ω

|∇u(x)|2dx+∫

Ω

V (x)|u(x)|2dx

with the domain H10(Ω). Thus defined, it is self-adjoint in L2(Ω) and

has real discrete spectrum of finite multiplicity

λ1 < λ2 ≤ λ3 ≤ . . . ,

where limn→∞

λn = ∞. It has an orthonormal basis of real-valued eigen-

functions ψn such that ψ1(x) ≥ 0. We will sometimes use the notationsH(Ω) and λj(Ω), when we need to emphasize the dependence of theoperator and its spectrum on the domain.

Definition 1. For a function f(x) we will be interested in its nodal(zero) set

Z(f) := f−1(0) = x ∈ Ω | f(x) = 0.The complement Ω \N(f) is the union of connected open sub-domainsD1, . . . , Dν of Ω, which we will call nodal domains. The nodal do-mains form the nodal partition P (f) = Dj corresponding to thefunction.

In most cases, we will be interested in the case when f(x) is aneigenfunction ψn, and thus in its nodal set

Z(ψn) := ψ−1n (0) = x ∈ Ω | ψn(x) = 0,

its nodal domains D1, . . . , Dν of Ω, and its nodal partition P (ψn) =Dj.A lot of attention has been paid to the nodal structure of eigenfunc-

tions (e.g., [10–14, 24, 34, 37] and references therein), and in particularto the number ν (or νψn

, if one wants to emphasize dependence on theeigenfunction) of nodal domains of the n-th eigenfunction ψn of H .

NODAL DEFICIENCY OF BILLIARD EIGENFUNCTIONS 3

In spite of more than 300 years history of this topic1, open questionsstill abound. We will discuss here one of them, the issue of the socalled nodal deficiency. The classical Sturm theorem claims that indimension one, the nodal and eigenfunction counts coincide: νψn

= n.The Courant Nodal Theorem [9, Vol. I, Sec. V.5, VI.6] claims that inany dimension, the upper bound on the number of nodal domains stillholds:

νψn≤ n.

While the equality νψ1= 1 does hold due to positivity of ψ1, it follows

from a stronger upper bound for νψnby Pleijel [40] that in higher

dimensions the equality νψn= n can hold only for a finitely many

values of the index n. Moreover, examples are known of eigenfunctionswith arbitrarily large index n that have just two nodal domains (i.e.,νψn

= 2 is possible for an arbitrarily large n).The eigenfunctions ψn for which νψn

= n, are sometimes calledCourant sharp. The non-negative difference

µn := n− νψn

is said to be the nodal deficiency of an eigenfunction. As far as it isknown to the authors, it has not been understood what (if anything)the nodal deficiency signifies. It is the goal of this text to introducesome clarity into this issue.An important new approach has been developed in the last several

years in the series of papers [8,25,26]. Namely, instead of concentratingon an eigenfunction, one can look at a partition by connected opendomains Dj and try to determine whether a given partition can bethe nodal partition of an eigenfunction, and if yes, what could be thecorresponding nodal deficiency.One necessary condition on the partition is not hard to find. Indeed,

one can introduce the graph of the partition, with each partition domainDj serving as a vertex and two nodal domains that have a (d − 1)-dimensional part of the common boundary being connected by an edge.The graph of the nodal partition of an eigenfunction will be calledits nodal graph. Since, due to standard uniqueness theorems for theCauchy problem, the eigenfunction must change its sign when crossingany (d − 1)-dimensional piece of the boundary of two adjacent nodaldomains, we conclude that the following well known property holds:

1Robert Hooke observed on 8 July 1680 the nodal patterns on vibrating glassplates, running a bow along the edge of a glass plate covered with flour [38]. Ahundred years later, the same effect was systematically studied by E. Chladni [7].In fact, such patterns were known to Galileo Galilei [18]. See also some historicaldiscussion in [34].

4 GREGORY BERKOLAIKO1, PETER KUCHMENT1, AND UZY SMILANSKY2,3

Proposition 2. The nodal graph corresponding to an eigenfunction isbipartite.

Another important simple observation is:

Proposition 3. If Dj is the nodal partition corresponding to theeigenfunction ψn with the eigenvalue λn, then for each nodal domainDj, one has

λ1(Dj) = λn(Ω).

(Here, as before, λk(D) denotes the kth Dirichlet eigenvalue in thedomain D.)

Indeed, by definition, ψn does not change sign in Dj and thus isproportional to the groundstate for H(Dj).This observation leads to the following notion that plays a crucial

role in what follows:

Definition 4. A partition P = Pj is said to be an equipartition,if the lowest eigenvalues of all operators H(Pj) are the same, i.e.

(2) λ1(P1) = λ1(P2) = · · · = λ1(Pν).

As we have already mentioned, every nodal partition (i.e., the par-tition corresponding to an eigenfunction) is an equipartition.This observation has lead to the following construction: given a nat-

ural number ν, consider the “space” of “arbitrary” (with some naturalrestrictions) ν-partitions Dj, i.e. partitions with ν sub-domains. Letus also introduce the functional

Λ(Dj) := maxj=1,...,ν

λ1(Dj)

on this space.One can look now at the minimal partitions, i.e. such that realize

the minimum

minν−partitionsDjνj=1

Λ(Dj).

Such minimal partitions are known to exist [8] and the following im-portant result holds:

Theorem 5. [25]Minimal bipartite partitions are exactly the nodal partitions of Courantsharp eigenfunctions.

Our aim is to understand whether there is something that distin-guishes the nodal partitions of the eigenfunctions that are not Courantsharp (which is the overwhelming majority) and what determines their

NODAL DEFICIENCY OF BILLIARD EIGENFUNCTIONS 5

nodal deficiencies. It was shown in a recent work [5] that in the quan-tum graph situation this question can be answered. Inspired by thisdevelopment, we address here the eigenfunctions of the operator H de-fined above. As we will see, one has to look not only at the minima,but at all critical points of an appropriate functional on the “manifold”of equipartitions, and the Morse index determines (in fact, coincideswith) the nodal deficiency.Let us introduce some notation first. For a bounded open domain D

we denote by ψD the normalized positive ground state of the operatorH(D) in L2(D). We also denote by Z(f) the nodal (or zero) set of afunction f and by

P (f) = Pjνj=1

the nodal partition corresponding to a real valued function f .Let us return to the domain Ω ⊂ R

d and a real-valued eigenfunctionψn. It is known (see [2, 3, 30, 35–37, 42] and references therein) thatgenerically with respect to perturbations of the potential V (x) and/orthe domain, the following genericity conditions are satisfied:

(1) The eigenvalue λn is simple(2) Zero is a regular value of the eigenfunction ψn inside Ω (i.e.,

∇ψn(x) 6= 0 whenever ψn(x) = 0). The normal derivative∂ψn/∂n of the eigenfunction ψn on the boundary of Ω has zeroas its regular value (i.e., the tangential to ∂Ω gradient of ∂ψn/∂ndoes not vanish whenever ∂ψn/∂n(x) = 0).

(3) The nodal set N(ψn) is the finite union of non-intersectingsmooth hyper-surfaces

N(ψn) =

(

⋃

k

Ck

)

⋃

(

⋃

l

Bl

)

(see Fig.. 1), where(4) Each Ck is a closed smooth surface in Ω.(5) Each Bl is an arc (in 2D) or a smooth hyper-surface, intersect-

ing transversally the boundary ∂Ω.

Remark 6. In fact, the intersections of Bl with ∂Ω are orthogonal,but we will not need to use this information.

We will be dealing with the generic situation only, in the sensethat all conditions (1)-(5) above are satisfied. In particular,

Definition 7. A partition P = Dj of Ω will be called generic, if itsatisfies the following analogs of the conditions (3)-(5):

6 GREGORY BERKOLAIKO1, PETER KUCHMENT1, AND UZY SMILANSKY2,3

Figure 1. A generic partition

3: The set N = Ω− (⋃

Dj) is the finite union of non-intersectingsmooth hyper-surfaces

N =

(

⋃

k

Ck

)

⋃

(

⋃

l

Bl

)

,

4: Each Ck is a closed smooth surface in Ω.5: Each Bl is an arc (in 2D) or a smooth hyper-surface, inter-secting transversally the boundary ∂Ω.

We now briefly describe the constructions and results of the paper.Given a generic ν-partition P and a sufficiently small positive number

ρ > 0, we will introduce the set Pρ of ν-partitions that are “close” to Pin an appropriate sense. Here ρ will introduce a measure of closeness.The set Pρ will be equipped with the structure of a real Hilbert manifoldby identifying it with a ball in an appropriate Hilbert function space.We will denote by Eρ the subset of Pρ that consists of equipartitions.On the set Eρ of equipartitions one can consider the functional

Λ : E 7→ R

that maps a partition P into the (common) lowest energy λ1(Pj) ofany sub-domain Pj. The notation Λ does not contradict to the oneused previously for the maximum of groundstate energies over all sub-domains, since on equipartitions the two functionals obviously coincide.We will also need some other extensions of the functional Λ from theset Eρ of equipartitions to the whole Pρ. Let c = (c1, . . . , cν) ∈ R

ν be aunit simplex vector, i.e. such that cj ≥ 0 and

∑

cj = 1. We define thefunctional Λc on Pρ as follows:

Λc(P ) =∑

cjλ1(Pj),

NODAL DEFICIENCY OF BILLIARD EIGENFUNCTIONS 7

where P ∈ Pρ and Pj are the sub-domains of this partition.It is obvious that for any unit simplex vector c the restriction of Λc

to Eρ coincides with Λ.We will need the following auxiliary result:

Proposition 8.

(1) For any c, the functional Λc on Pρ is C∞-smooth.(2) For a sufficiently small ρ, Eρ is a smooth sub-manifold of Pρ of

co-dimension ν − 1.(3) The functional Λ on Eρ is C∞ smooth.

This allows us now to formulate the main results of the paper:

Theorem 9. Let P be a generic bipartite equipartition of a smoothdomain Ω. Then, the following statements are equivalent:

(1) P is nodal (i.e., P is the nodal partition of an eigenfunction ψof H).

(2) There exists a vector c = (c1, . . . , cν) ∈ Rν, cj ≥ 0,

∑

cj = 1,such that P is a critical point of the functional Λc on Pρ. Inthis case,

cj = ‖ψ‖2L2(Pj)=

∫

Pj

|ψ(x)|2dx.

(3) P is a critical point of the functional Λ on Eρ.Remark 10. If the domain Ω is simply-connected, the partition graphof a generic partition is a tree and thus bipartite automatically. More-over, in this case one can add another equivalent statement to thestatements (1), (2), and (3) of the Theorem 9:

(4) At any of the boundary surfaces Cj and Bl, the normal derivativesof the groundstates for the two adjacent sub-domains are proportional.

Finally, we offer the following interpretation of the nodal deficiency:

Theorem 11. Let ψn be a generic (i.e., satisfying conditions (1) -(5)) eigenfunction of H and P be its nodal partition. Then P is a non-degenerate critical point of Λ and the nodal deficiency µn = n− νψn

isequal to the Morse index of Λ at the point P .Here the Morse index is the number of negative eigenvalues of the

Hessian of the functional at the point P .

Remark 12. In other words, instead of looking at the minimal points,one has to look at the critical min-max points, where the maximum

8 GREGORY BERKOLAIKO1, PETER KUCHMENT1, AND UZY SMILANSKY2,3

is taken over a subspace of dimension equal to the nodal deficiency µ.This explains, in particular, why the minimal partitions correspond theCourant sharp eigenfunctions only.

The structure of this article is as follows. Section 2 contains a briefexposition of the well known Rayleigh-Hadamard formula for the de-rivative of an eigenvalue with respect to the domain variation. Themanifolds Pρ and Eρ are introduced in Section 3, where also the Propo-sition 8 is proven. Theorems 9 and 11 are proven in sections 4 and5. Section 6 contains final remarks and conclusions. In particular, itoffers various possible generalizations of the results of this paper.

2. Domain variation formulas

In this section we provide the formulas for eigenvalue perturbationdue to domain variation, which will be important for our considera-tions. Such formulas have a long history, going back to J. Rayleigh [41]and J. Hadamard [23] and are still being developed (see [15–17, 19, 20,22, 27, 30, 31, 39] for further results and references).Let D be a proper sub-domain of Ω (i.e., the closure of D belongs

to Ω) with a smooth boundary C = ∂D. Later on in this text D willbe one of the sub-domains of a partition. Let also ψ1(x) and λ1(D) be,as before, the positive groundstate and the corresponding eigenvalue ofH(D).We are interested in the variation of λ1(D) with respect to infinites-

imal smooth deformations of the boundary C. To make it precise, letus consider the unit external normal vector field on C and extend itinto a neighborhood U of C to a smooth unit length vector field N(x)whose trajectories are the normals to C. Let us now also have a suf-ficiently smooth real valued function f(x) in U . Consider the normalto C vector-field f(x)N(x) and the corresponding evolution operatorsGt of the “time” t shift along the trajectories of this field. They aredefined for sufficiently small values of t and produce deformed sur-faces Ct = GtC and the variable domains Dt bound by these surfaces.Correspondingly, one has a variation λ1(Dt) of the ground state eigen-value. The following result (Rayleigh-Hadamard formula) is well known(e.g. [15–17, 19, 20, 22, 23, 30, 31, 39, 41]) for the case when V (x) = 0.However, its proof (e.g. the one in [22]) is valid for non-zero potentialsas well.

NODAL DEFICIENCY OF BILLIARD EIGENFUNCTIONS 9

Theorem 13. The t-derivative at t = 0 of the eigenvalue λ1(Dt) isgiven by the formula

(3) λ′1 = −∫

C

(

∂ψ1(x)

∂n

)2

f(x)dS,

where ∂/∂n denotes the external normal derivative on C and ψ1(x) is,as before, the normalized Dirichlet ground-state.

One can use instead of the unit normal vector-field N(x) an arbitrarynon-tangential vector field M(x) and describe domain perturbationsalong this field rather than N . Then the result modifies as follows:

Remark 14. If one uses a smooth non-tangential vector field M(x)instead of the unit normal vector-field N(x), an analog of the formula(3) for the t-derivative at t = 0 of the eigenvalue λ1(Dt) is given by

(4) λ′1 = −∫

C

(

∂ψ1(x)

∂n

)2

f(x)M(x) ·N(x)dS,

where ∂/∂n denotes the external normal derivative on C and M ·N isthe inner product of vectors M and N .

Some of the nodal sub-domains will reach the boundary. We thusalso need to consider the case of D ⊂ Ω that is cut out from Ω by asmooth surface (curve when d = 2) B transversal to the boundary ∂Ω(Fig. 2). Consider again the external unit normal vector field N(x)

Figure 2. Domain D cut out from Ω by B.

to B. We can assume that this field is smoothly modified near theboundary ∂Ω to a smooth vector field M(x) non-tangential to B thatis tangential to ∂Ω on B ∩ ∂Ω. It can be extended to a smooth vector

10GREGORY BERKOLAIKO1, PETER KUCHMENT1, AND UZY SMILANSKY2,3

field M near B such that the trajectories that start at the boundarypoints (i.e., points of B ∩ ∂Ω) stay on the boundary ∂Ω and such thatthe level sets GtB for small values of t are transversal to ∂Ω. Letf ∈ Hs(B) (for a sufficiently large s) be a real valued function on B.Then, similarly to the case of an internal part of Ω considered above,one can deform the surface B by a vector-field fM , which will definesub-domains Dt of Ω with their boundaries Bt transversal to ∂Ω. Thefollowing formula, analogous to (4) holds:

Theorem 15. The t-derivative at t = 0 of the eigenvalue λ1(Dt) isgiven by the formula

(5) λ′1 = −∫

B

(

∂ψ1(x)

∂n

)2

f(x)M(x) ·N(x)dS,

where ∂/∂n denotes the external normal derivative on B.

Although in this case the variation formula may or may not havebeen written before, the proof of Theorem 13 given, for instance, in [22]carries through without any change.

Remark 16.

(1) If the boundary of the domain D is not connected, then thesame variation formula holds, involving the sum of integralsover each connected component Cj and Bl of the boundary.

(2) Sometimes we will be facing the situation when the field M isdirected towards the interior rather than exterior of D. In thiscase, clearly, one needs to change the sign in the correspondingvariation formulas (4) and (5).

3. Manifolds of partitions. Proof of Proposition 8

Let us now have a generic ν-partition P = Pjνj=1 of Ω. We need tointroduce a manifold structure into the space of “nearby” ν-partitions.The previous section suggests a simple way of doing so. Namely, letCk, Bl be the smooth connected surfaces constituting the bound-aries between the sub-domains Pj inside Ω (∂Ω also contributes to theboundaries of some of the sub-domains, but is not taken into account,since it is not going to be changed). Let us fix a smooth non-tangentialto Ck, Bl vector field M in a neighborhood U of (∪kCk)

⋃

(∪lBl),such that it satisfies the conditions imposed in the previous section(e.g., one can assume that outside of a neighborhood of ∂Ω, this isjust the unit normal field N(x), which is smoothly modified near ∂Ωto be tangential to ∂Ω). Such a field always exists under the genericitycondition imposed on the partition P .

NODAL DEFICIENCY OF BILLIARD EIGENFUNCTIONS 11

Let us pick a sufficiently large positive number s (e.g., s > (d+4)/2will suffice) and consider the space

(6) F :=

(

⊕kHs(Ck)

)

⊕

(

⊕lHs(Bl)

)

,

where Hs is the standard Sobolev space of order s. We also introducea continuous linear extension operator

(7) E : F → Hs+1/2(U).

In other words, the restriction of F(f) to Ck coincides with f . Such anextension operator is well known to exist (see, e.g. [33]). We will alsoassume that all the extended functions F(f) vanish outside of a smallneighborhood of the nodal set, which can be achieved by multiplicationby an appropriate smooth cut-off function.Consider the ball Bρ of radius ρ > 0 around the origin in the space

F . Let f ∈ Bρ and Gf be the shift by time t = 1 along the trajectoriesof the vector field E(f)(x)M(x). For a sufficiently small ρ > 0, Gf

is a diffeomorphism, which preserves the boundary ∂Ω. Its action onthe surfaces Ck and Bl leads to another ν-partition GfP of Ω, which isclose to the original partition P . We will denote this set of partitionsby Pρ and identify it with the ball Bρ in the Hilbert space F . This,in particular, introduces the structure of a Hilbert manifold on Pρ.Notice also that we will use consistent numbering of the sub-domainsof partitions in Pρ. Namely,

(GfP )j = Gf(Pj),

where, as before, for a partition V we denote by Vj its jth sub-domain,and Pj are the sub-domains of the original (unperturbed) partition P .We introduce now a mapping Ξ from Pρ into R

ν as follows: for apartition V we define

(8) Ξ(V ) := (λ1(V1), λ1(V2), . . . , λ1(Vν)).

Let

∆ := (λ, λ, . . . , λ)

be the diagonal in Rν .

Definition 17. The set Eρ consists of all equipartitions in Pρ.In other words, V ∈ Pρ is in Eρ when

λ1(V1) = λ1(V2) = · · · = λ1(Vν).

Or, to put it differently,

(9) Eρ = Ξ−1 (∆) .

12GREGORY BERKOLAIKO1, PETER KUCHMENT1, AND UZY SMILANSKY2,3

One notices that the restriction of the mapping Ξ to Eρ is essentiallythe functional Λ of the Introduction. More precisely,

Ξ(V ) = (Λ(V ), . . . ,Λ(V )).

We are ready now to prove the Proposition 8.

3.1. Proof of the Proposition 8. We prove first the following auxil-iary result, which immediately implies the first statement of the Propo-sition 8.

Lemma 18. The mapping Ξ : Pρ → Rν is C∞.

Proof. It is sufficient to prove that for any j, the mapping

f → λ1(Gf (Pj))

is smooth as a mapping from the ball Bρ to R, if ρ is sufficiently small.Thus, one can restrict attention to a single sub-domain D. The oftenemployed in such circumstances idea is to replace domain dependencewith varying the coefficients of the differential operator in a fixed do-main. Then the smooth dependence of λ1 becomes a standard pertur-bation theory result (e.g., [20, 28]).Let us consider first the case of a sub-domain D that does not touch

the boundary. Consider the mapping

Φf : x→ y := x+ F(f)(x)M(x)

of the domain D into Ω. For sufficiently small ρ > 0, it is a C2-diffeomorphism of D onto a “nearby” sub-domain D∗ of Ω. The qua-dratic form of the operator H(D∗) is given as

∫

D∗

(

∣

∣

∣

∣

∂u

∂y

∣

∣

∣

∣

2

+ V (y)|u(y)|2)

dy.

Changing the variables back from y to x, we arrive to an operatorHf(D) in the fixed spatial domain D, while with variable coefficientsnow:

(Hfu)(x) = ∇ · Af(x)∇u(x) + V (x)u(x),

where the matrix valued function A(x) is of the class C1 and f → Afis a C∞-mapping from Hs to the space of C1-matrix functions.We thus have replaced the domain dependence with the smooth co-

efficients dependence on f . The operator Hf acts continuously fromH2(Ω) to L2(Ω) and for f = 0 coincides with H(D). Moreover, themapping Af → Hf is a continuous linear mapping from the space ofC1-matrix functions to the space of bounded operators from H2(Ω)to L2(Ω). In fact, one can see that so far all dependencies on f are

NODAL DEFICIENCY OF BILLIARD EIGENFUNCTIONS 13

analytic. Thus, for a sufficiently small ρ we get an analytic familyof Fredholm operators between the aforementioned spaces. Due tothe simplicity of λ1(D), the standard perturbation theory (or analyticFredholm theory, e.g. [43, Theorem 4.11]) shows that λ1(Hf ) dependsanalytically on f , for a sufficiently small radius ρ.A similar consideration works when D reaches the boundary of Ω,

i.e. at least one of the boundaries Bl is involved. Without loss of gener-ality, we can assume that only one such Bl is involved. Introducing anappropriate smooth coordinate change, one can reduce considerationto the cylinder B × (−ε, ε) for small ε > 0, with ∂B × (−ε, ε) as thecorresponding part of ∂Ω. Then the same reduction to a fixed domainbut varying operator as before is possible, which again implies smoothdependence of λ1 on f .

Let us now address the second statement of the Proposition, thatEρ is a smooth sub-manifold of Pρ of co-dimension ν − 1. We willemploy for this purpose the formula (9) in conjunction with the do-main variation formulas (Theorems 13 and 15 and Remark 14) and atransversality theorem.According to Lemma 18, the mapping Ξ : Pρ → R

ν is a smoothmapping of Banach manifolds. The pre-image of the diagonal ∆ ⊂ R

ν

coincides, according to (9), with Eρ. We would like to know whether thispre-image is a smooth sub-manifold, and of what co-dimension. This isexactly the question tackled by the transversality theorems. Namely,if we can show that the mapping Ξ is transversal to the diagonal one-dimensional sub-manifold ∆ of Rν , this will prove that the pre-imageof ∆ is a smooth sub-manifold of co-dimension ν − 1.

Definition 19. (e.g., [1,6,32]) The mapping Ξ : Pρ → Rν is transver-

sal to ∆, if at any point ζ ∈ ∆ that also belongs to the image of Ξ,i.e. ζ = Ξ(v) for some v ∈ Pρ, the vector sum of the tangent spaceTζ∆ to ∆ at ζ and of the range DΞ(TvPρ) of the differential DΞ onthe tangent space TvPρ is the whole space R

ν.

Theorem 20. (e.g., [1, Sect. 3, Theorem 2] or [6, Sect. 5.11.7])If Ξ is transversal to ∆, then Eρ = Ξ−1(∆) is a smooth sub-manifold

of Pρ of co-dimension ν − 1.

Thus, to finish the proof of the second statement of the Proposition8, it only remains to prove the transversality of the mapping Ξ to thediagonal ∆. The Rayleigh-Hadamard domain variation formulas arehelpful here.Consider the partition graph Γ that corresponds to an (automatically

generic) ν-partition P ∈ Pρ. The vertices of the graph correspond to

14GREGORY BERKOLAIKO1, PETER KUCHMENT1, AND UZY SMILANSKY2,3

the sub-domains Pj and the edges to the interfaces Cj, Bl. We willidentify the target space R

ν of the mapping Ξ with the space of realvalued functions on the set V of vertices of the graph Γ. Consider a pairof adjacent sub-domains Pi, Pj with the common part of their boundaryS (one of Cj, Bl). We consider the corresponding vertices vi, vj and theedge s of Γ. Let us pick a function f ∈ F that is non-zero on S onlyand find the corresponding directional (Gateaux) derivative of Ξ at Pin the direction of this f . The formulas (3) and (5) show that the onlynon-zero components of this derivative correspond to vertices vi andvj. These components are

±∫

S

(

∂ψ(Pi)

∂n

)2

f(x)M(x) ·N(x)dx

and

∓∫

S

(

∂ψ(Pj)

∂n

)2

f(x)M(x) ·N(x)dx.

Consider two possibilities:1: The normal derivatives ∂ψ/∂n on S from both sides are linearly

dependent (i.e., proportional. In this case, if

∂ψ(Pj)

∂n|S = αij

∂ψ(Pi)

∂n|S,

then the vector in Rν that this Gateaux derivative produces is propor-

tional to the vector having the ith component equal to 1, the jth oneequal to −α2

ij , and all others are equal to zero. Moreover, using thearbitrariness of the choice of a smooth function f on S, one can achieveexactly this vector. Notice that αij 6= 0, due to the uniqueness theoremfor the Cauchy problem.2: The derivatives are linearly independent as functions on S. Then,

one can find a function f on S such that it is orthogonal to∂ψ(Pj )

∂n|S, but

not to ∂ψ(Pi)∂n

|S. This will lead to a vector that has the ith componentequal to 1 and all others equal to zero. One can similarly achieve thevector with the jth component equal to one and all others to zero.We see that the range of the Frechet differential of Ξ at P contains

a vector with the ith component equal to 1 and the jth one equal to−α2

ij for each case 1 boundary and two vectors for each case 2 boundary,namely, a vector with its ith or jth (but not both) component equal to 1and all others equal to zero. Thus, the range of the differential containsthe range of the “incidence” matrix J which has all these vectors ascolumns. The range of this matrix is the orthogonal complement of thekernel of the transposed matrix J t. It is easy to see what is this kernel.It consists of all vectors (a1, . . . , aν) such that aj = α2

ijai for all edges

NODAL DEFICIENCY OF BILLIARD EIGENFUNCTIONS 15

(vi, vj) of the first type and aj = bj = 0 for each edge of the secondtype. If there is at least one 2nd type edge, due to connectedness ofthe graph Γ, the zero value at the corresponding two vertices vi, vjpropagates and forces the whole vector to vanish. Thus, in this casethe kernel of the dual matrix is zero and thus the range of the Frechetdifferential is the whole space. Suppose that there is no 2nd typeedge. Then, again due to connectedness, a value ai at a single vertexdetermines all other values (with the signs of all aj being the same). Itis still conceivable that the only such vector is zero. Correspondingly,the range of J is either ν − 1-dimensional, or ν-dimensional. In thelatter case, the transversality to ∆ is guaranteed. In the first one,it remains to show that the vector (1, 1, . . . , 1) is not orthogonal tothe one-dimensional kernel just described. Since the kernel vector, asit was mentioned before, has all components aj of the same sign, itcannot be orthogonal to the vector (1, 1, . . . , 1). This finishes the proofof transversality and thus of the second statement of the Proposition.Since the first two claims of the proposition imply the third one, theproof is completed.

4. Proof of Theorem 9

4.1. Proof of the equivalence (1) ⇔ (2). If P is the nodal partitionof a real valued eigenfunction ψ, then, as we have already mentionedbefore (Proposition 3), the restrictions of ψ to the nodal domains areproportional to the groundstates in these domains. Denote these pro-portionality constants by aj . Since the eigenfunction ψ is continuouslydifferentiable, the groundstates ψ1(Pj) scaled with the correspondingfactors aj have matching normal derivatives at the common boundaries:

(10) ai∂ψ1(Pi)

∂nS = aj

∂ψ1(Pj)

∂nS,

where sub-domains Pi and Pj have the common boundary S, and n isa normal vector to S.Now let ck = a2k and consider the Gateaux derivative of the functional

Λc in the direction f that is non-zero only on a single boundary betweenthe sub-domains Pi and Pj . Since the only affected terms in Λc are λ(Pi)and λ(Pj), the derivative is

(11)

∫

S

(

a2j

(

∂ψ1(Pj)

∂n

)2

− a2i

(

∂ψ1(Pi)

∂n

)2)

f(x)M(x) ·N(x)dS,

where we applied the Rayleigh-Hadamard formula (Theorems 13 and15 and Remark 14). The difference in signs is explained in the Remark

16GREGORY BERKOLAIKO1, PETER KUCHMENT1, AND UZY SMILANSKY2,3

16. Now we observe that, due to (10), the integrand is identically equalto zero and thus the Gateaux derivative in the direction of f is equal tozero. The same is obviously true for arbitrary variations f , involvingany number of boundaries. Thus, the nodal partition is a critical pointof the functional Λc.Conversely, if a partition Pρ is a critical point of Λc, we get that the

Gateaux derivative of Λc is zero in any direction f(x). This implies theequality

ci

(

∂ψ1(Pi)

∂n

)2

= cj

(

∂ψ1(Pj)

∂n

)2

on the common boundary S of any two neighboring domains Pi and Pj.Setting αk = ±√

ck and choosing the signs so that any two neighboringdomains have different signs (possible due to bipartiteness) ensures that(10) is satisfied. Then the function ψ defined by

ψ |Pk= akψ1(Pk)

is an eigenstate of H .

4.2. Proof of the equivalence (2) ⇔ (3). If P is a critical point onPρ of the functional Λc, then the restriction of Λc to Eρ is a criticalpoint on Eρ. But on Eρ any functional Λc coincides with Λ.Conversely, assume that P is a critical point of Λ on Eρ. We can

extend the functional Λ to the whole Pρ as λ1(P1). Since Eρ can begiven by the smooth relations λ1(P1) − λ1(Pj) = 0 for j = 2, 3, . . . , ν,the Lagrange multiplier method implies that P must be a critical pointof a non-trivial linear combination Λb :=

∑

bjλ1(Pj). All bj are of thesame sign: otherwise there are two neighboring domains with bj ofdifferent signs and the variation of Λb with respect to the boundarybetween the two domains cannot be zero for a sign-definite f(x), seeequation (11). Thus the vector of coefficients bj can be normalized tobe a unit simplex vector. This finishes the proof of Theorem 9.

5. Proof of Theorem 11

Let us present first the strategy of the proof. As we have alreadyseen, it is sometimes useful to play with different extensions of thefunctional Λ from the (local patch of the) space Eρ of equipartitionsto a larger manifold. While previously it was the (local patch of the)space of all partitions, now we need some further extension. Indeed, wewould like to compare somehow the nodal count ν (which is fixed) withthe consecutive number n of an eigenfunction ψn. It is hard to observewhere the information about n is hidden in the spaces of partitionsthemselves. On the other hand, the quadratic form (or Rayleigh ratio)

NODAL DEFICIENCY OF BILLIARD EIGENFUNCTIONS 17

contains this information. We thus will extend the functional Λ toa larger space that is a functional space, not just a set of domainpartitions. Then we will have to restrict back in order to comparethe Morse indices of Λ and of its extension. We also discover that inorder to prove the required equality through the tight upper and lowerbounds, one benefits from using various extensions of Λ.Before implementing this program, in the following sub-sections we

start proving some auxiliary statements that will come handy later on.

5.1. Critical points on direct products.

Theorem 21. Let X = Y ⊕ Y ⊥ be an orthogonal decomposition of aHilbert space, and dim Y ⊥ = n (in the 1st statement below, n = ∞is allowed). Let also f : X → R be a smooth functional such that(0, 0) ∈ X is its non-degenerate critical point of Morse index m.

(1) Let Y be the locus of minima of f over the affine subspacesy0 × Y ⊥, i.e.

(12) (y0, 0) = arg minx∈y0×Y ⊥

f(x),

for any y0 in a neighborhood of zero in Y . Then the Morseindex of 0 as a critical point of the restriction f |Y is equal tom (i.e., the same as the Morse index of this point on the wholeneighborhood of zero in X)

(2) Let Y be the locus of maxima of f over the affine subspacesy0 × Y ⊥, i.e.

(13) (y0, 0) = arg maxx∈y0×Y ⊥

f(x),

for any y0 in a neighborhood of zero in Y . Then the Morseindex of 0 as a critical point of the restriction f |Y is equal tom− n.

Proof. We start with an auxiliary statement:

Lemma 22. (1) Let X = Y ⊕ Y ⊥ be an orthogonal decompositionof a Hilbert space and f : X → R be a smooth functional suchthat (0, 0) ∈ X is its critical point. If for any y0 in a neigh-borhood of zero in Y , the point (y0, 0) is a critical point of fover the affine subspace y0×Y ⊥, then the Hessian F2 of f atthe origin, as an operator in X, is reduced by the decompositionX = Y ⊕ Y ⊥.

(2) Under the same conditions, if the origin is a non-degeneratecritical point of f on X, it is also non-degenerate for the re-striction f |Y .

18GREGORY BERKOLAIKO1, PETER KUCHMENT1, AND UZY SMILANSKY2,3

Proof of the Lemma. The second statement of the Lemma imme-diately follows from the first one. Indeed, due to the reducibility ofthe Hessian, if the restriction to Y had zero in the spectrum, the samewould be true for the functional on the whole space X . So, we concen-trate on proving the first statement.We can assume, without loss of generality, that f(0, 0) = 0. Using

this and the condition of the criticality of the origin, the Taylor formulaof the second order for f on X near the origin looks as follows:

f(y, y′) = (Ay, y) + (By, y′) + (Cy′, y′) + higher order terms.

Here A is a bounded symmetric linear operator in Y , B is a boundedlinear operator from Y to Y ⊥, C is a bounded symmetric linear operatorin Y ⊥, and the parentheses (·, ·) denote the inner product in X . Thenthe gradient F of f at the origin has components in the directions of Yand Y ⊥ that are equal to 2Ay +B∗y′ and By + 2Cy′ correspondingly.The condition that y′ = 0 is a critical point for any fixed y easily impliesthat B = 0. This immediately leads to the following structure of theHessian:

(

2A 00 2C

)

,

which proves the statement of the lemma.

We can now prove the statements of Theorem 21. We notice firstthat in both cases of the Theorem, the Lemma implies that zero is anon-degenerate critical point of the restriction f |Y and that the HessianF2 of f at this point is reduced by the direct decomposition at hand.This means that the spectrum of the Hessian is the union of the spectraof its restrictions to Y and to Y ⊥. Thus, in the 1st case of the Theorem,the spectrum of F2|Y ⊥ is strictly positive, and thus the whole negativepart must reside in Y . This proves the 1st statement. Analogously, inthe second case, the whole spectrum of F2|Y ⊥ must be negative, andthus the dimension of the negative spectral subspace in Y is m−n.

Before moving on to the proof of the theorem, we still need to intro-duce some auxiliary bundles and manifolds.

5.2. Some objects needed for the proof. All the considerationsbelow are needed only locally, near a generic eigenfunction ψn indi-cated in the statement of the theorem, and correspondingly near itsnodal partition. So, all objects below are only of interest in a smallneighborhood of ψn, which we will accept without much of a furthermentioning.The basic notions assumed below concerning finite- or infinite- di-

mensional vector-bundles and their sphere bundles can be found in

NODAL DEFICIENCY OF BILLIARD EIGENFUNCTIONS 19

many standard sources on topology (e.g., [4, 29]) or in the survey [43],where such bundles are studied in relation to the operator theory.We will be considering again the (local) manifolds Pρ of partitions

“close” to the nodal partition P (ψn) and its sub-manifold Eρ of codi-mension ν−1 that consists of equipartitions only. In the trivial bundlePρ×H1

0 (Ω) over Pρ, we consider a fibered sub-space B that has the fiberover a partition P consisting of only such functions, which vanish on thepartition’s interfaces Z. In other words, this fiber is H1

P :=⊕

j H10 (Pj).

Lemma 23. B is a smooth sub-bundle of the trivial bundle

Pρ ×H10 (Ω) 7→ Pρ.

Proof. The proof follows the same line as the one of Lemma 18. Namely,the dependence on the partition P is replaced, using a smooth familyof diffeomorphisms, with a fixed partition, but the operator now hascoefficients varying smoothly with P . Then B becomes just the trivial

sub-bundle Pρ ×(

⊕

j H10 (Pj)

)

in Pρ × H10 (Ω) 7→ Pρ. Inverting the

diffeomorphisms provides a smooth trivialization of B, which provesthe lemma.

Definition 24. We denote by SB the locally-trivial bundle of the unit(in L2-norm) spheres of the fibers of B.The restrictions of B and SB to Eρ (clearly locally-trivial bundles)

will be denoted by BE and SBE correspondingly.

We will now restrict the bundle B further.

Definition 25. We denote by C the fibered space whose fiber over apartition P consists of functions of the form

∑

j cjψ1(Pj), where cj are

real constants and ψ1(Pj) is the normalized positive groundstate on thesub-domain Pj.Correspondingly, SC is the fiber-bundle of the unit (in L2-norm)

spheres of C and CE and SCE are restrictions of the correspondingfiber-bundles to Eρ.Lemma 26. C is a smooth ν-dimensional sub-bundle of B, and thusof the trivial bundle

Pρ ×H10 (Ω) 7→ Pρ.

Proof. The proof follows the same line as in Lemmas 18 and 23. Namely,after applying a smooth family of diffeomorphisms, one deals with afixed partition Π, but instead with the operator whose coefficients de-pend smoothly on P . The perturbation theory shows that the corre-sponding ground-state fj in each sub-domain Πj depends smoothly onP , as a vector in H1

0 (Πj). We extend it, without changing the notation,

20GREGORY BERKOLAIKO1, PETER KUCHMENT1, AND UZY SMILANSKY2,3

by zero to the whole domain Ω. Then fj is a smoothly dependent onP frame of ν linearly independent vectors in H1

0 (Ω). Thus, this framespans a smooth finite-dimensional vector-bundle. After applying theinverses of the diffeomorphisms, we get the claim of the lemma.

5.3. Some Morse indices estimates. Consider the quadratic formon H1

0 (Ω)

Q[f ] :=

∫

Ω

|∇f(x)|2dx.

It can, by restricting to each fiber, be defined as a smooth functionalon the vector bundle B and its sub-bundles that we considered above.

Lemma 27. The point (P (ψn), ψn) in SB is a critical point of Q ofMorse index µ = n− 1. The same holds true on SC.

Proof. It is clear that Q on H10 (Ω) has an n− 1-dimensional subspace

on which its Hessian at ψn is negative. Namely, this is the subspacegenerated by the eigenfunctions ψ1, ..., ψn−1. If we show that thesedirections are among the tangential ones to SB at (P (ψn), ψn), thiswill prove that µ ≥ n− 1.Due to the locally-trivial structure of SB, there are two main ways to

get tangential vectors to SB: one is to vary the partition P (which willgive “horizontal” tangent vectors to the bundle SB), and the second- to keep the partition fixed, while varying the function ψn, keepingthe nodal set fixed (“vertical” tangent vectors). The vertical tangentvectors are just arbitrary functions in H1

0 (Ω) that vanish on the nodalset Z of the partition P (ψn) (we have previously denoted this spaceH1P ). The horizontal tangential vectors look as follows:

E(f)Mψn = E(f)Nψn(x)(M(x) ·N(x)),

where f ∈ Hs on the nodal set of ψn is the function defining this set’sinfinitesimal variation. Also, the notation Xg for a vector field X anda function g (e.g., Mψn and Nψn) mean the derivative of the functiong along the field X . Notice that, since M(x) is non-tangential to Z,M(x) ·N(x) is a smooth separated from zero function on the nodal set.We now show that any ψj with j < n can be represented as a sum of

vertical and horizontal vectors. We notice that the genericity conditionrequires in particular that zero is a regular value of the normal deriv-ative of ψn on ∂Ω. This implies that the derivative Nψn is a smoothfunction that has a non-degenerate zero at ∂Ω on the nodal set Z.Since ψj is a smooth function vanishing on ∂Ω, the function

f(x) :=ψj(x)

(Nψn)M(x) ·N(x)

NODAL DEFICIENCY OF BILLIARD EIGENFUNCTIONS 21

belongs to Hs on the nodal set. Hence, the horizontal tangent vectorin the direction of f

h := (Nψn)E(f)M(x) ·N(x)

coincides with ψj on the nodal set. This means that the differenceg := ψj − h belongs to H1

P , and thus is a vertical tangent vector. Thisshows that each eigenfunction ψ1, ..., ψn−1 can be represented as thesum of a vertical and horizontal vectors and thus is tangent to SB.This proves the estimate µ ≥ n− 1 for SB.We will now prove that µ cannot exceed n−1. Suppose that there is

an n-dimensional subspace L in the tangent space to SB at (P (ψn), ψn),where the Hessian of Q is non-positive. Since each fiber of SB consistsof functions from the space H1

0 (Ω), there is a tautological mapping(P, f) 7→ f from SB intoH1

0 (Ω) (in fact, into the set of functions of unitL2-norm). If the Frechet derivative of the tautological mapping has zerokernel, then the subspace L will produce an n-dimensional subspacetransversal to ψn, where the Hessian of Q at ψn is non-positive, which isa contradiction. So, let us show that the kernel of the Frechet derivativeis zero. Due to the local trivial structure of SB, one sees that the imageof any tangent vector under the Frechet derivative has the form

g = E(f)(x) (Mψn) + h(x),

where the function f ∈ F is responsible for the infinitesimal variationof the nodal set of ψn, E is the previously introduced extension oper-ator from the nodal set to Ω, and a function h ∈ H1

P (Ω) correspondsto the infinitesimal variation in the fiber direction (i.e., the pair (f, h)describes a tangent vector to SB). Suppose now that g = 0. In partic-ular, g|Z = 0. Taking into the account that h|Z = 0 (which is true forany function from the space H1

0 ), we conclude that

E(f)(x) (Mψn) |Z = f(x)M(x) · ∇ψn(x)|Z = 0.

Since ∇ψn(x)|Z 6= 0 and M is transversal to Z, we see that

M(x) · ∇ψn(x)|Z 6= 0.

This implies that f(x) (defined on Z only) vanishes identically. Dueto the linearity of the extension operator, the extension E(f) vanisheseverywhere in Ω. Hence, g = h. Since g = 0, we conclude that h = 0and thus the Frechet derivative of the tautological mapping is injective.This finishes the proof of the lemma for the case of SB.

22GREGORY BERKOLAIKO1, PETER KUCHMENT1, AND UZY SMILANSKY2,3

Let us move now to the case of SC. We know that the eigenfunctionψn(x) has the form

ψn(x) =ν∑

j=1

cjψ1(Dj)(x),

where Dj is the nodal partition of ψn and each ground state ψ1(Dj)is extended as zero to the whole domain Ω,

∑

c2j = 1, and none of cjare equal to zero. A similar presentation (with different real coefficientscj) holds in a neighborhood U of ψn in SC.As we have already seen, C is a smooth ν-dimensional sub-bundle

of B and thus SC is a smooth sub-manifold of SB. We will introducenow near the point (P (ψn), ψn) a smooth foliation of SB by manifoldstransversal to SC, such that SC will be the locus of minima of Qover the leaves of this foliation. Then, after a smooth local changeof coordinates, we will be in the situation of Theorem 21. Thus, theMorse index of Q at (P (ψn), ψn) on SC will be equal to the one onSB, which will prove the lemma.So, let us finish the proof by constructing such a foliation. Consider

the following mapping Υ from a neighborhood U of (P (ψn), ψn) ∈ SBto Pρ × Sν−1, where Sν−1 is the unit sphere in R

ν :

Υ(P, f) = (P, ‖f‖L2(P1), . . . , ‖f‖L2(Pν)).

Notice that none of the components ‖f‖L2(Pj) of the vector Υ(P, f)vanishes (since this is the case for Υ(P (ψn), ψn)).By the arguments provided before, this is a smooth mapping. It is

also clear that it is a submersion.Let w = (P,

∑

cjψ1(Pj)) be a point in U ∩ SC near (P (ψn), ψn).Consider the leaf Lw = Υ−1(P, c1, c2, . . . , cν). Due to the submersionproperty of Υ, the leaves Lw form near (P (ψn), ψn) a smooth fibration.Since the differential of Υ on SC is surjective, this foliation is transver-sal to SC. Moreover, the representation of the groundstate ψ1(Pj) bymeans or the minimum of the Rayleigh ratio implies that the minimalvalue of Q on the leaf Lw is attained exactly when fj = cjψ1(Pj) forall j, i.e. on SC.

Lemma 28. The point (P (ψn), ψn) in SBE is a critical point of Q ofMorse index not less than n−ν. The same holds true for the restrictionof Q to SCE.

Proof. Indeed, according to Proposition 8, SBE (correspondingly, SCE)is a smooth sub-manifold of SB (correspondingly, SC) of co-dimensionν−1. Hence, the Morse index cannot drop during the restriction morethan ν − 1, which, together with Lemma 27, proves the claim.

NODAL DEFICIENCY OF BILLIARD EIGENFUNCTIONS 23

Lemma 29. The restriction of the quadratic form Q to the fibers ofSCE is the pull-back of the functional Λ from the base Eρ.In other words, if π : SCE 7→ Eρ is the bundle projection, then for

any x ∈ SCE,Q(x) = Λ(π(x)).

Proof. Let f =∑

j cjψ1(Pj) be an element of the fiber of SCE over a

partition P = Pj. ThenQ(f) =

∑

|cj|2λ1(Pj).Since P is an equipartition, all the values λ1(Pj) are equal to the samevalue Λ(P ). Taking into account that

∑

|cj|2 = ‖f‖2L2(Ω) = 1, we get

Q(f) =∑

|cj|2λ1(Pj) = Λ(P )(

∑

|cj|2)

= Λ(P ),

which proves the statement of the lemma.

After this preparation, we can start proving Theorem 11. The strat-egy is to use extensions of Λ and the results just proven to obtaintwo-sided estimates for the Morse index: from below, µ ≥ n−νψn

, andfrom above, µ ≤ n− νψn

.

5.4. Proof of the estimate from below: µ ≥ n−νψn. The required

estimate from belowµ ≥ n− ν

now follows. Indeed, since the pull-back of Λ to SCE has Morse indexat least n − ν at the point (P (ψn), ψn), the same holds true for Λ atψn. Since the quadratic functional Q is constant along the fibers ofπ : SCE 7→ Eρ, the subspace of dimension at least (n − ν), wherethe Hessian of Q is strictly negatively defined, must come from such asubspace for the Hessian of Λ.

5.5. Proof of the estimate from above: µ ≤ n − νψn. We start

with the following auxiliary statement:

Lemma 30. The point (P (ψn), ψn) in SCE is a critical point of Q of“index” at most n− 1, where the “index” is understood as the numberof non-positive eigenvalues of the Hessian2.

Proof. The statement of the lemma follows from Lemma 27 and themore general property: the index in this extended sense cannot increaseduring restriction of the Hessian to a sub-space. Indeed, let one have aq-dimensional spectral subspace for the Hessian of the restriction, thenthe quadratic form of the Hessian is non-positive there. Now, using

2The Morse index involves the negative eigenvalues only.

24GREGORY BERKOLAIKO1, PETER KUCHMENT1, AND UZY SMILANSKY2,3

this subspace in conjunction with the min-max property of eigenvaluesshows that the full Hessian must have a spectral subspace of at leastthe same dimension, where it is non-positive.

Now we use Lemma 29 again. Since the functional Q is constanton each of the (ν − 1)-dimensional fibers of SCE , the statement ofLemma 30 shows that the Morse index of Λ at P (ψn) cannot exceed(n− 1)− (ν − 1) = n− ν. This finishes the proof of the estimate fromabove and thus of Theorem 11.

6. Final remarks and conclusions

(1) The results of this paper (Theorems 9 and 11) transfer withoutany changes in their proofs to the case when Ω is a compactsmooth Riemannian manifold with or without boundary.

(2) Smoothness conditions imposed on the domain, potential, andpartition interfaces, can certainly be weakened. In this text, wehave decided not to do so, in order not to complicate consider-ations unnecessarily.

(3) The manifolds P and E of partitions that we considered involvedonly generic partitions, which allows introduction on each theirconnected component of a structure of an infinite dimensionalsmooth manifold and the consequent considerations of the text.It is clear, however, that these smooth pieces are joined intosingle singular “varieties” P and E , where the junctions occurwhen partition interfaces start meeting each other. It would beinteresting to see whether one could prove an analog of The-orems 9 and 11 for such non-generic partitions. The authorsbelieve that something of this nature can be done.

(4) The results of this work were presented at the internationalSpectral Geometry Conference in Dartmouth College, at theAnalysis on Graphs and Applications conference at the IsaacNewton Institute (Cambridge), both in July 2010, at the Follow-up workshop to the 2009 ESI programme on ”Selected topics inspectral theory” at Erwin Schrodinger International Institutefor Mathematical Physics in January 2011, and at the AppliedInverse Problems conference at Texas A&M University in May2011.

Acknowledgments

The authors thank R. Band, L. Friedlander, B. Helffer, H. Hezari,T. Hoffmann-Ostenhof, H. Raz, P. Sarnak, and S. Zelditch for usefuldiscussions and references. The work of G. B. was partially supported

NODAL DEFICIENCY OF BILLIARD EIGENFUNCTIONS 25

by the NSF grant DMS-0907968. The work of P. K. was partially sup-ported by MSRI and IAMCS. The work of U. S. was partially supportedby The Wales Institute of Mathematical and Computational Sciences(WIMCS), EPSRC (grant EP/G021287), and ISF (grant 166/09). Theauthors express their gratitude for this support.

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1 Department of Mathematics, Texas A&M University, College Sta-

tion, TX 77843-3368 USA.

2 Department of Physics of Complex Systems, Weizmann Institute

of Science, Rehovot 76100, Israel.

3 School of Mathematics, Cardiff University, Cardiff, Wales, UK