Sturm's theorem on the zeros of sums of eigenfunctions: Gelfand's strategy implemented
SSturm’s theorem on the zeros of sums ofeigenfunctions: Gelfand’s strategy implemented
Pierre B´erard and Bernard HelfferNovember 15, 2018
Keywords: Zeros of eigenfunction, Nodal domain, Courant nodal domain theorem,Sturm theorem.MSC 2010: 35P99, 35Q99, 58J50.
Version: berard-helffer-ecp-gelfand-181112.tex.
Abstract
In the second section “Courant-Gelfand theorem” of his last published paper(Topological properties of eigenoscillations in mathematical physics, Proc. SteklovInstitute Math. 273 (2011) 25–34), Arnold recounts Gelfand’s strategy to provethat the zeros of any linear combination of the n first eigenfunctions of the Sturm-Liouville problem − y (cid:48)(cid:48) ( s ) + q ( x ) y ( x ) = λ y ( x ) in ]0 , , with y (0) = y (1) = 0 , divide the interval into at most n connected components, and concludes that “thelack of a published formal text with a rigorous proof . . . is still distressing.”Inspired by Quantum mechanics, Gelfand’s strategy consists in replacing the ana-lysis of linear combinations of the n first eigenfunctions by that of their Slaterdeterminant which is the first eigenfunction of the associated n -particle operatoracting on Fermions.In the present paper, we implement Gelfand’s strategy, and give a complete proofof the above assertion. As a matter of fact, refining Gelfand’s strategy, we provea stronger property taking the multiplicity of zeros into account, a result whichactually goes back to Sturm (1836). On September 30, 1833, C. Sturm presented a memoir on second order linear differentialequations to the Paris Academy of Sciences. The main results are summarized in [23, 24],and were later published in the first volume of Liouville’s journal, [25, 26]. We refer to[5] for more details. In this paper, we shall consider the following particular case. Jacques Charles Fran¸cois
Sturm (1803–1855) a r X i v : . [ m a t h . A P ] N ov heorem 1.1 (Sturm, 1836) . Let q be a smooth real valued function defined is a neigh-borhood of the interval [0 , . The Dirichlet eigenvalue problem (1) (cid:40) − y (cid:48)(cid:48) ( x ) + q ( x ) y ( x ) = λ y ( x ) in ]0 , ,y (0) = y (1) = 0 , has the following properties.1. There exists an infinite sequence of (simple) eigenvalues λ < λ < · · · (cid:37) ∞ , with an associated orthonormal family of eigenfunctions { h j , j ≥ } .2. For any j ≥ , the eigenfunction h j has exactly ( j − zeros in the interval ]0 , .3. For any ≤ m ≤ n , let U = (cid:80) nk = m a k h k be any nontrivial real linear combinationof eigenfunctions. Then,(a) U has at most ( n − zeros in ]0 , , counted with multiplicities,(b) U changes sign at least ( m − times in ]0 , . Sturm’s motivations came from mathematical physics. He took a novel point of view,looking for qualitative behavior of solutions rather than for explicit solutions. To proveAssertions 1 and 2, he introduced the comparison and oscillation theorems which todaybear his name. Assertion 3 first appeared as a corollary of Sturm’s investigation ofthe evolution of zeros of a solution u ( t, x ) of the associated heat equation, with initialcondition U , as times goes to infinity (in direct line with his motivations). Assertions 1and 2 can be found in most textbooks on Sturm-Liouville theory. This is not the casefor Assertion 3. In Section 2, we provide Liouville’s proof, which is based on the soleordinary differential equation. We refer to [5] for historical details. Remarks 1.2. (i) In the framework of Fourier series, Assertion 3b is often referred toas the Sturm-Hurwitz theorem. See [21] for a quite recent qualitative version of thisassertion.(ii) Sturm’s theorem applies to more general operators, with more general boundaryconditions; we refer to [5] for more details.R. Courant partly generalized Assertion 2, in Sturm’s theorem, to higher dimensions. Theorem 1.3.
Let < λ < λ ≤ λ · · · (cid:37) ∞ be the Dirichlet eigenvalues of − ∆ in a bounded domain of R d , listed in nondecreasing order, with multiplicities. Let u beany nontrivial eigenfunction associated with the eigenvalue λ n , and let β ( u ) denote thenumber of connected components of Ω \ u − (0) ( nodal domains ). Then, β ( u ) ≤ n . In a footnote of [10, p. 454], Courant and Hilbert make the following statement. Richard
Courant (1888–1972). tatement 1.4. Any linear combination of the first n eigenfunctions divides the domain,by means of its nodes, into no more than n subdomains. See the G¨ottingen dissertationof H. Herrmann, Beitr¨age zur Theorie der Eigenwerten und Eigenfunktionen, 1932. In the literature, Statement 1.4 is referred to as the “Courant-Herrmann theorem”,“Courant-Herrmann conjecture”, “Herrmann’s theorem”, or “Courant generalized the-orem”. In [6, 7], we call it the
Extended Courant property.
Remarks 1.5.
Some remarks are in order.1. It is easy to see that Courant’s upper bound is not sharp. This is indeed the casewhenever the eigenvalue λ n is not simple. More generally, it can be shown that thenumber β ( u ) is asymptotically smaller than γ ( n ) n when n tends to infinity, where γ ( n ) < n . It is interestingto investigate the eigenvalues for which Courant’s upper bound is sharp, see thereview article [9]. For this research topic, we also refer to the surprising results inthe recent paper [13].2. In dimension greater than or equal to 2, there is no general lower bound for β ( u ),except the trivial ones (1 for λ , and 2 for λ k , k ≥ noticed that Statement 1.4, would provide a partial answerto one of the problems formulated by D. Hilbert .Citation from Arnold [3, p. 27].I immediately deduced from the generalized Courant theorem [Statement 1.4]new results in Hilbert’s famous (16th) problem. . . . And then it turnedout that the results of the topology of algebraic curves that I had derivedfrom the generalized Courant theorem contradict the results of quantum fieldtheory. . . . Hence, the statement of the generalized Courant theorem is nottrue (explicit counterexamples were soon produced by Viro). Courant died in1972 and could not have known about this counterexample .Arnold was very much intrigued by Statement 1.4, as is illustrated by [3], his last pub-lished paper, where he in particular relates a discussion with I. Gelfand , which we tran-scribe below, using Arnold’s words, in the form of an imaginary dialog.(Gelfand) I thought that, except for me, nobody paid attention to Courant’s remarkableassertion. But I was so surprised that I delved into it and found a proof. (Arnold is quite surprised, but does not have time to mention the counterexamples beforeGelfand continues.)
However, I could prove this theorem of Courant only for oscillations of one-dimensionalmedia, where m = 1 . (Arnold) Where could I read it? Vladimir Igorevich
Arnold (1937-2010). David
Hilbert (1862–1943). As far as we know, the first paper of Arnold on this subject is [1], published in 1973. Israel Moiseevich
Gelfand (1913-2009).
I never write proofs. I just discover new interesting things. Finding proofs(and writing articles) is up to my students.
Arnold then recounts Gelfand’s strategy to prove Statement 1.4 in the one-dimensionalcase . Quotations from [3, Abstract and Section 2].Nevertheless, the one-dimensional version of Courant’s theorem is apparentlyvalid. . . . Gelfand’s idea was to replace the analysis of the system of n eigen-functions of the one-particle quantum-mechanical problem by the analysisof the first eigenfunction of the n -particle problem (considering as particles,fermions rather than bosons). . . .Unfortunately, [Gelfand’s hints] do not yet provide a proof for this generalizedtheorem: many facts are still to be proved. . . .Gelfand did not publish anything concerning this: he only told me that hehoped his students would correct this drawback of his theory. . . .Viktor Borisovich Lidskii told me that “he knows how to prove all this”. . . .Although [Lidskii’s] arguments look convincing, the lack of a published formaltext with a proof of the Courant-Gelfand theorem is still distressing.In [14], Kuznetsov refers to Statement 1.4 as Herrmann’s theorem , and relates thatGelfand’s approach so attracted Arnold that he included Herrmann’s theorem for eigen-functions of problem [ (1) ] together with Gelfand’s hint into the 3rd Russian edition of hisOrdinary Differential Equations , see Problem 9 in the “Supplementary problems” at theend of [2].More precisely, Arnold’s Problem 9 proposes to prove the following statement, which isthe one-dimensional analogue of Statement 1.4.
Statement 1.6.
The zeros of any linear combination of the n first eigenfunctions of theSturm-Liouville problem (1) divide the interval into at most n connected components. This statement is equivalent to saying that any linear combination of the n first eigen-functions of (1) has at most ( n −
1) zeros in the open interval. This is a weak form ofSturm’s upper bound, Assertion 3a in Theorem 1.1.In the present paper, we implement Gelfand’s strategy to prove Statement 1.6, and weextend this strategy to take the multiplicities of zeros into account, and to prove Asser-tion 3a in Theorem 1.1. Inspired by Quantum mechanics, Gelfand’s strategy consists inreplacing the analysis of linear combinations of the n first eigenfunctions by that of theirSlater determinant which is the first eigenfunction of the associated n -particle operatoracting on Fermions. We give more details in Section 5. Note that Assertion 3b canactually be deduced from Assertion 3a, see Section 2.The paper is organized as follows. In Section 2, we give J. Liouville’s Joseph
Liouville (1809–1882).
Acknowledgements.
The authors would like to thank E. Lieb and N. Kuznetsov foruseful comments on a first version of this paper.
Assertions 1 and 2 in Theorem 1.1 are well-known, and can be found in many textbooks.This is not the case for Assertion 3. In this section, we give a short proof, based on thearguments of Liouville [18], and Rayleigh [22, § Proof of Assertion 3a.
Write equation (1) for h and for h k , multiply the first one by h k ,the second by ( − h ) and add to obtain the relation (cid:0) h h (cid:48) k − h (cid:48) h k (cid:1) (cid:48) = ( λ − λ k ) h h k . Multiply by a k , and sum from k = m to k = n to obtain(2) (cid:0) h U (cid:48) − h (cid:48) U (cid:1) (cid:48) = h U , where U = (cid:80) nk = m ( λ − λ k ) a k h k .Integrating this relation from 0 to x , and using the Dirichlet boundary condition, gives h ( x ) U (cid:48) ( x ) − h (cid:48) ( x ) U ( x ) = (cid:90) x h ( t ) U ( t ) dt . Note that the left hand side can be rewritten as h ( x ) ddx Uh ( x ) in ]0 , U has N zeros in ]0 , Uh , so that, byRolle’s theorem, ddx Uh has a least ( N −
1) zeros in ]0 , x (cid:55)→ (cid:82) x h ( t ) U ( t ) dt has at least ( N −
1) zeros in ]0 , h j form an orthonormal family. By Rolle’s theorem again,we conclude that its derivative, h U , has at least N zeros in ]0 , U and U have the same form, we can repeat the argument, and conclude that, for any (cid:96) ≥ U (cid:96) = (cid:80) nk = m ( λ − λ k ) (cid:96) a k h k has at least N zeros in ]0 , (cid:96) tend toinfinity, using the fact that the eigenvalues λ k are simple, and the fact that h n has ( n − , N ≤ ( n − Proof of Assertion 3b.
Assume that U changes sign exactly M times at the points z < · · · < z M in the interval ]0 , M < ( m − M ≤ ( m − V ( x ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ( z ) . . . h ( z M ) h ( x )... ... ... h n ( z ) . . . h n ( z M ) h n ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) John William
Strutt , Lord
Rayleigh (1842–1919).
5t is easy to prove that the function V is not identically zero (see Lemma 6.1). It clearlyvanishes at the points z j , ≤ j ≤ M , and it is a linear combination of the eigenfunctions h , . . . , h M (develop the determinant with respect to the last column). According toAssertion 3a in Theorem 1.1, V does not have any other zero, and each z j has order 1, sothat V changes sign exactly at the points z j . Since M ≤ ( m − U and V are orthogonal, and their product U V does not change sign in ]0 , U V vanishes identically, a contradiction.
Remark 2.1.
With the above notation, we can rewrite (2) as(3) h U = h U (cid:48)(cid:48) + ( λ − q ) h U .
A similar relation holds between U (cid:96) +1 and U (cid:96) . Using these relations, and letting (cid:96) tendto infinity as in the preceding proof, we obtain the following lemma which is interestingin itself. Lemma 2.2.
The nonzero linear combination U cannot vanish at infinite order at anypoint in [0 , . In particular, its zeros are isolated. Let n be an integer, n ≥
1, and J ⊂ R an interval. Given n points x , . . . , x n in J, wedenote the corresponding vector by (cid:126)x = ( x , . . . , x n ) ∈ J n . Generally speaking, we denoteby (cid:126)k = ( k , · · · , k n ) a vector with positive integer entries.We use the notation (cid:42) c = ( c , . . . , c n − ) for an ( n − n real continuous functions f , . . . , f n defined on J, we denote by (cid:126)f the vector-valued function (cid:0) f , · · · , f n (cid:1) , and we introduce the determinant(4) (cid:12)(cid:12)(cid:12) (cid:126)f ( x ) . . . (cid:126)f ( x n ) (cid:12)(cid:12)(cid:12) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) f ( x ) . . . f ( x n ) f ( x ) f ( x ) . . . f ( x n )... ... . . . ... f n ( x ) f n ( x ) . . . f n ( x n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Given a vector (cid:126)b = ( b , . . . , b n ) ∈ R n , we denote by(5) S (cid:126)b ( x ) = n (cid:88) j =1 b j f j ( x ) , the linear combination of f , . . . , f n , with coefficients b j ’s.Let (cid:126)c ∈ J n be a vector of the form(6) (cid:126)c = (¯ c , . . . , ¯ c , ¯ c , . . . , ¯ c , . . . , ¯ c p , . . . , ¯ c p ) , with ¯ c repeated k times, . . . , ¯ c p repeated k p times, 1 ≤ p ≤ n , k + · · · + k p = n , andwith ¯ c < ¯ c < · · · < ¯ c p . 6t will be convenient to relabel the variables (cid:126)x = ( x , . . . , x n ) according to the structureof (cid:126)c , as follows,(7) (cid:126)x = ( x , , . . . , x ,k , x , , . . . , x ,k , . . . , x p, , . . . , x p,k p ) , so that,(8) (cid:40) x , = x , . . . , x ,k = x k and, for 2 ≤ i ≤ p ,x i, = x k + ··· + k i − +1 , . . . , x i,k i = x k + ··· + k i − + k i . In this case, we will also write the vector (cid:126)x as(9) (cid:126)x = (cid:0) x (1) , . . . , x ( p ) (cid:1) , with x ( i ) = ( x i, , . . . , x i,k i ), for 1 ≤ i ≤ p .We shall usually use both ways of labeling inside a formula, there should not be anyconfusion.We introduce the real polynomials(10) (cid:40) Q ( x ) = 1 , and, for n ≥ ,Q n ( x , . . . , x n ) = (cid:81) nj =2 ( x − x j ) , and(11) (cid:40) P ( x ) = 1 , and, for n ≥ ,P n ( x , . . . , x n ) = (cid:81) ≤ i The polynomial P n , defined in (11) , is up to sign a Vandermonde deter-minant (12) P n ( x , . . . , x n ) = ( − n ( n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . . . x . . . x n ... ... x n − . . . x n − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Furthermore,1. P n is anti-symmetric under the action of the group of permutations s n , and homoge-nous of degree n ( n − .2. As a function of x , . . . , x n , P n is harmonic, ∆ P n = 0 , and satisfies (13) ∂ n − x n ∂ n − x n − · · · ∂ x ∂ x P n = ( n − n − . . . . Alexandre Th´eophile Vandermonde (1735–1796). roof. The identity (12) is well-known, and readily implies Assertion 1. The polynomial P n being anti-symmetric, its Laplacian is also anti-symmetric, and hence, must be divi-sible by P n . Being of degree less than P n , ∆ P n must be zero. The identity (13) followsimmediately from the multi-linearity of the determinant, or by induction on n . Notation 4.2. When (cid:126)x = ( x , . . . , x n ), we will also write P n ( (cid:126)x ) for P n ( x , . . . , x n ) . Wewill denote by D n ( ∂ (cid:126)x ) the differential operator which appears in (13)(14) D n ( ∂ (cid:126)x ) := ∂ n − x n ∂ n − x n − · · · ∂ x ∂ x , so that(15) D n ( ∂ (cid:126)x ) P n ( (cid:126)x ) = ( n − n − . . . . Notation 4.3. In the sequel, we use ω ( (cid:126)c, (cid:126)ξ ) as a generic notation for a function whichdepends on (cid:126)c, (cid:126)ξ , and tends to zero as (cid:126)ξ tends to zero. Lemma 4.4. Given (cid:126)x = ( (cid:126)y, (cid:126)z ) ∈ R p × R q , the function (cid:126)x (cid:55)→ P p ( (cid:126)y ) P q ( (cid:126)z ) is harmonic as a function on R p + q . We shall now describe the local behaviour of the harmonic polynomial P n near a point (cid:126)c ∈ R n at which it vanishes. We first treat two simple examples. Example 4.5. Let n = 5, and (cid:126)c = (¯ c , ¯ c , ¯ c , ¯ c , c ), with ¯ c < ¯ c < c . Then, P ( (cid:126)c ) = 0.Write (cid:126)x = (cid:126)c + (cid:126)ξ . An easy computation gives,(16) P ( (cid:126)c + (cid:126)ξ ) = P ( ξ , ξ ) P ( ξ , ξ ) (cid:110) ρ ( (cid:126)c ) + ω ( (cid:126)c, (cid:126)ξ ) (cid:111) , where ρ ( (cid:126)c ) = (¯ c − ¯ c ) (¯ c − c ) (¯ c − c ) is a nonzero constant. Example 4.6. Let n = 5. Let (cid:126)c = (¯ c , ¯ c , ¯ c , c , c ), with ¯ c < c < c . Then, P ( (cid:126)c ) = 0.Write (cid:126)x = (cid:126)c + (cid:126)ξ . An easy computation gives,(17) P ( (cid:126)c + (cid:126)ξ ) = P ( ξ , ξ , ξ ) (cid:110) ρ ( (cid:126)c ) + ω ( (cid:126)c, (cid:126)ξ ) (cid:111) , where ρ ( (cid:126)c ) = (¯ c − c ) (¯ c − c ) ( c − c ) is a nonzero constant. Remark 4.7. In both examples, the leading term on the right hand side of P n ( (cid:126)c + (cid:126)ξ ) isa homogeneous harmonic polynomial is some of the variables ξ j ’s, as we can expect fromBers’s theorem, [8]. Furthermore, ω ( (cid:126)c, (cid:126)ξ ) is actually a polynomial in the ( ξ i − ξ j )’s, withcoefficients depending on (cid:126)c , and without constant term.In the following lemma, we use both the standard coordinates names and their relabeling(7)–(9), for both variables (cid:126)x and (cid:126)ξ . 8 emma 4.8. Let p be an integer, ≤ p ≤ n , and ( k , . . . , k p ) be a p -tuple of positiveintegers, such that k + · · · + k p = n . Let (¯ c , . . . , ¯ c p ) be a p -tuple, such that ¯ c < · · · < ¯ c p .Let (cid:126)c be the n -vector (18) (cid:126)c = (¯ c , . . . , ¯ c , . . . , ¯ c p , . . . , ¯ c p ) , where each ¯ c j is repeated k j times, ≤ j ≤ p . Writing (cid:126)x = (cid:126)c + (cid:126)ξ , and relabeling thecoordinates of the vectors (cid:126)x and (cid:126)ξ as in (7) – (9) , we have the following relation, (19) P n ( (cid:126)c + (cid:126)ξ ) = ρ ( (cid:126)c ) P k ( ξ , , . . . , ξ ,k ) . . . P k p ( ξ p, , . . . , ξ p,k p ) (cid:16) ω ( (cid:126)c, (cid:126)ξ ) (cid:17) , where ρ ( (cid:126)c ) is a nonzero constant depending only on (cid:126)c , and where ω ( (cid:126)c, (cid:126)ξ ) is actually apolynomial in the variables ( ξ i − ξ j ) ’s, with coefficients depending on the c j ’s, withoutconstant term.Proof. From the definition of P n , and using the relabeling of the variables (cid:126)x and (cid:126)ξ , asindicated in (7)–(9), we obtain the following relations.(20) P n ( (cid:126)c + (cid:126)ξ ) = (cid:32) k (cid:89) i =1 Q n +1 − i ( c i + ξ i , . . . , c n + ξ n ) (cid:33) n (cid:89) i = k +1 Q n +1 − i ( c i + ξ i , . . . , c n + ξ n ) , (21) P n ( (cid:126)c + (cid:126)ξ ) = (cid:32) k (cid:89) i =1 Q n +1 − i ( c i + ξ i , . . . , c n + ξ n ) (cid:33) P n − k ( c , + ξ , , . . . , c p,k p + ξ p,k p ) , Developing the factors Q n +1 − i for i ≤ k , we obtain,(22) k (cid:89) i =1 Q n +1 − i ( c i + ξ i , . . . , c n + ξ n ) = ρ ( (cid:126)c ) P k ( ξ , , . . . , ξ ,k ) (cid:16) ω ( (cid:126)c, (cid:126)ξ ) (cid:17) , where(23) ρ ( (cid:126)c ) = (cid:104) (¯ c − ¯ c ) k . . . (¯ c − ¯ c p ) k p (cid:105) k (cid:54) = 0 , and ω as in Notation 4.3. Finally, we have P n ( (cid:126)c + (cid:126)ξ ) = ρ ( (cid:126)c ) P k ( ξ , , . . . , ξ ,k ) P n − k ( c , + ξ , , . . . , c p,k p + ξ p,k p ) (cid:16) ω ( (cid:126)c, (cid:126)ξ ) (cid:17) , or, more concisely,(24) P n ( (cid:126)c + (cid:126)ξ ) = ρ ( (cid:126)c ) P k (cid:0) ξ (1) (cid:1) P n − k (cid:0) c (2) + ξ (2) , . . . , c ( p ) + ξ ( p ) (cid:1) (cid:16) ω ( (cid:126)c, (cid:126)ξ ) (cid:17) . We can then apply the same kind of computation to the factor P n − k , and repeat theoperation until we finally obtain the desired formula, with(25) ρ ( (cid:126)c ) = ρ ( (cid:126)c ) · · · ρ p ( (cid:126)c ) (cid:54) = 0 . We conclude this section with a technical lemma, which will play a key role later on.9 emma 4.9 (Division lemma) . Let P, Q be polynomials in R [ X , . . . , X n ] . Assume that Q is harmonic and homogenous. If the set of real zeros of Q is contained in the set ofreal zeros of P , { x ∈ R n | Q ( x ) = 0 } ⊂ { x ∈ R n | P ( x ) = 0 } , then Q divides P , i.e. there exists R in R [ X , . . . , X n ] such that P = QR . This lemma follows from Theorem 2 and Lemma 4 in [20]. It is stated as Lemma 2.1 in[19], with a proof given in [19, Section 5.3]. In this section, we explain Gelfand’s strategy to prove Statement 1.6, in the particularcase of the harmonic oscillator. We also show how one can extend it to obtain a proof ofAssertion 3a in Theorem 1.1.Let H (1) denote the 1-particle harmonic oscillator (26) H (1) := − d dx + x on the line. The eigenvalues are given by { λ n = 2 n − , n ≥ } , they are simple, withassociated orthonormal basis of eigenfunctions { h n , n ≥ } ,(27) h n ( x ) = γ n − H n − ( x ) exp( − x / , where H m is the m -th Hermite polynomial, and γ m a normalizing constant [15, Chap. 3].The polynomial H m ( x ) has degree m , with leading coefficient 2 m , and satisfies the differ-ential equation,(28) y (cid:48)(cid:48) ( x ) − x y (cid:48) ( x ) + 2 m y ( x ) = 0on the line R .We consider the n -particle Hamiltonian in R n ,(29) H ( n ) := n (cid:88) j =1 (cid:18) − ∂ ∂x j + x j (cid:19) = − ∆ + | (cid:126)x | . Gelfand’s strategy is to look at H ( n ) F , the operator H ( n ) restricted to Fermions , i.e., tofunctions which are anti-invariant under the action of the permutation group s n on R n ,(30) L F ( R n ) = (cid:8) f ∈ L ( R n ) | f (cid:0) x σ (1) , . . . , x σ ( n ) (cid:1) = ε ( σ ) f ( x , . . . , x n ) , ∀ σ ∈ s n (cid:9) . Equivalently, we consider the Dirichlet realization H ( n ) F of H ( n ) in(31) Ω n = { ( x , . . . , x n ) ∈ R n | x < x · · · < x n } . Slater determinant (32) S n ( (cid:126)x ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ( x ) . . . h ( x n )... ... h n ( x ) . . . h n ( x n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = A n exp( −| (cid:126)x | / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H ( x ) . . . H ( x n )... ... H n − ( x ) . . . H n − ( x n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Using the properties of Hermite polynomials, we find that(33) S n ( (cid:126)x ) = B n exp( −| (cid:126)x | / P n ( (cid:126)x ) . In the preceding equalities, A n and B n are nonzero constants depending only on n .According to Arnold [3, Section 2], Gelfand noticed the following two facts. A . The (antisymmetric) eigenfunction [ S n ] of the operator [ h ( n ) ] is the firsteigenfunction for this operator (on functions satisfying the Dirichlet conditionin the fundamental domain [Ω n ]). B . Choosing the locations [( c , . . . , c n )] of the other electrons (except for thefirst one), one can obtain any linear combination of the first n eigenfunctionsof the one-electron problem as a linear combination [ S n ( x, c , . . . , c n )] (up tomultiplication by a nonzero constant).Observe however that B is true only for linear combinations of the n first eigenfunctionswhich have ( n − 1) distinct zeros.In the case of the harmonic oscillator, the proof of facts A and B is easy. More precisely,we have the following proposition which implies Statement 1.6 in this particular case. Proposition 5.1. Recall the notation (cid:126)h ( c ) = (cid:0) h ( c ) , . . . , h n ( c ) (cid:1) .1. The function S n ( (cid:126)x ) is the first Dirichlet eigenfunction of − ∆ + | (cid:126)x | in Ω n .2. For any (cid:42) c = ( c , . . . , c n − ) ∈ Ω n − , the vectors (cid:126)h ( c ) , . . . , (cid:126)h ( c n − ) , are linearlyindependent.3. Given (cid:126)b ∈ R n \{ } , the linear combination S (cid:126)b ( x ) = n (cid:88) j =1 b j h j ( x ) has at most ( n − distinct zeros. Furthermore, if the function S (cid:126)b has exactly ( n − distinct zeros c < c < · · · < c n − , then there exists a nonzero constant C suchthat S (cid:126)b ( x ) = C S n ( c , . . . , c n − , x ) for all x ∈ R . 4. The function S n ( c , . . . , c n − , x ) vanishes at order at each c j , ≤ j ≤ ( n − ,and does not have any other zero. John Clark Slater (1900–1976). roof. Assertion 1 . It is clear that S n is an eigenfunction of − ∆ + | (cid:126)x | , and that itvanishes on ∂ Ω n . From (12) and (33), we see that it does not vanish in Ω n , so that S n must be the first Dirichlet eigenfunction for − ∆ + | (cid:126)x | in Ω n . Assertion 2 . If the vectors (cid:126)h ( c ) , . . . , (cid:126)h ( c n − ), were dependent, S n ( c , . . . , c n − , x ) wouldbe identically zero. Developing this determinant with respect to the last column, wewould have S n − ( c , . . . , c n − ) h n ( x ) + · · · ≡ . This is impossible because the h j ’s are linearly independent and S n − ( c , . . . , c n − ) (cid:54) = 0 . Assertion 3 . Assume that S (cid:126)b has at least n distinct zeros c < · · · < c n . The n components b j , ≤ j ≤ n would satisfy a system of n equations, whose determinant S n ( c , . . . , c n )is nonzero. This would imply that (cid:126)b = (cid:126) 0. Assume that S (cid:126)b has exactly ( n − 1) ze-ros, c < · · · < c n − . The function x (cid:55)→ S n ( c , . . . , c n − , x ) can be written as a linearcombination S (cid:126)s ( (cid:42) c ) ( x ), with coefficients s j ( (cid:42) c ) , ≤ j ≤ n given by Slater like determi-nants. Both vectors (cid:126)b and (cid:126)s ( (cid:42) c ) would then be orthogonal to the ( n − 1) independentvectors (cid:126)h ( c ) , . . . , (cid:126)h ( c n − ). This implies that there exists a nonzero constant C such that (cid:126)b = C (cid:126)s ( (cid:42) c ). Assertion 4 . It suffices to consider the case of c . Up to sign, we look at the local behaviorof the function x (cid:55)→ S ( x, c , . . . , c n ) near c . Consider (cid:126)c = ( c , c , c , . . . , c n − ), and write S n ( (cid:126)c + (cid:126)ξ ) = B n exp( −| (cid:126)c + (cid:126)ξ | / P n ( (cid:126)c + (cid:126)ξ ) . Using Notation 4.3 and Lemma 4.8, we conclude that S n ( (cid:126)c + (cid:126)ξ ) = α ( (cid:126)c ) ( ξ − ξ ) (cid:16) ω ( (cid:126)c, (cid:126)ξ ) (cid:17) , for some nonzero constant α ( (cid:126)c ) depending on (cid:126)c .It follows that S n ( c + ξ, c , . . . , c n − ) = α ( (cid:126)c ) ξ (1 + ω ( (cid:126)c, ξ )) , so that this function vanishes precisely at order 1 at c . Remark 5.2. It is standard in Quantum mechanics (except that the usual context forthe one-particle Hamiltonian is a 3D-space) that the ground state energy of the n -particleHamiltonian is the sum of the n first eigenvalues of the one-particle Hamiltonian, a conse-quence of the Pauli exclusion principle. In a context closer to our paper (see Section 6),but with a different motivation, this sum associated with a one-particle Hamiltonian inan interval, and the properties of the corresponding ground state, are considered in [16] atthe beginning of the sixties. Later on, this sum appears in the celebrated Lieb-Thirring’sinequality in connection with the analysis of the stability of matter (see for example [17])and references therein.The following lemma allows us to extend Gelfand’s strategy in order to take care of themultiplicity of zeros, and to achieve a proof of Sturm’s upper bound. Wolfgang Ernst Pauli (1900–1958). emma 5.3. Let (cid:42) c = (¯ c , . . . , ¯ c , . . . , ¯ c p , . . . , ¯ c p ) , where ¯ c j is repeated k j times, with ¯ c < · · · < ¯ c p , and k + · · · + k p = n − . Let (cid:126)k = ( k , . . . , k p ) . Define the function (34) S (cid:126)k ( x ) = (cid:12)(cid:12)(cid:12) (cid:126)h ( c ) . . . (cid:126)h ( k − ( c ) . . . (cid:126)h ( c p ) . . . (cid:126)h ( k p − ( c p ) (cid:126)h ( x ) (cid:12)(cid:12)(cid:12) , where (cid:126)h ( m ) ( x ) is the vector (cid:0) h ( m )1 ( x ) , . . . , h ( m ) n ( x ) (cid:1) , and where the superscript ( m ) denotesthe m -th derivative.The function S (cid:126)k is not identically zero, and vanishes at exactly order k j at ¯ c j . Fur-thermore, the vectors (cid:126)h ( c ) , . . . , (cid:126)h ( k − ( c ) , . . . , (cid:126)h ( c p ) , . . . , (cid:126)h ( k p − ( c p ) , are linearly inde-pendent.Proof. It suffices to consider the case of ¯ c . Clearly, S (cid:126)k vanishes at least at order k at¯ c . It is sufficient to prove that the k -th derivative of this function does not vanish at c . We have S ( k ) (cid:126)k ( x ) = ± (cid:12)(cid:12)(cid:12) (cid:126)h ( k ) ( x ) (cid:126)h ( c ) . . . (cid:126)h ( k − ( c ) (cid:126)h ( c ) . . . (cid:126)h ( k − ( c ) . . . (cid:126)h ( c p ) . . . (cid:126)h ( k p − ( c p ) (cid:12)(cid:12)(cid:12) . Claim: The value of this determinant at x = ¯ c is different from zero . Indeed, considerthe vector (cid:126)c = (¯ c , . . . , ¯ c , . . . , ¯ c p , . . . , ¯ c p ), where ¯ c is repeated k + 1 times, and for2 ≤ j ≤ p , ¯ c j is repeated k j times. Then S ( k ) (cid:126)k (¯ c ) is a higher order derivative of S n at (cid:126)c . More precisely, using the relabeling of variables associated with (cid:126)c , as given in (7)–(9), S ( k ) (cid:126)k (¯ c ) is, up to sign, the derivative (cid:16) ∂ k ξ ,k . . . ∂ ξ , (cid:17) (cid:16) ∂ k − ξ ,k . . . ∂ ξ , (cid:17) . . . (cid:16) ∂ k p − ξ p,kp . . . ∂ ξ p, (cid:17) S n ( (cid:126)c + (cid:126)ξ ) (cid:12)(cid:12)(cid:12) (cid:126)ξ =0 , or, using the notation (14), D k ( ∂ ξ (1) ) . . . D k p ( ∂ ξ ( p ) ) S n ( (cid:126)c + (cid:126)ξ ) (cid:12)(cid:12)(cid:12) (cid:126)ξ =0 . The claim then follows from Lemma 4.1, Equation (13) and Lemma 4.8, Equation (19).The second assertion follows immediately.As a by product of the preceding proof, we have, Corollary 5.4. Given, p , ≤ p ≤ n , let k , . . . , k p be p positive integers such that k + · · · + k p = n . Let ¯ c < · · · < ¯ c p be real numbers. Then, the determinant (35) (cid:12)(cid:12)(cid:12) (cid:126)h ( c ) . . . (cid:126)h ( k − ( c ) (cid:126)h ( c ) . . . (cid:126)h ( k − ( c ) . . . (cid:126)h ( c p ) . . . (cid:126)h ( k p − ( c p ) (cid:12)(cid:12)(cid:12) is nonzero, so that the corresponding vectors are linearly independent. Proposition 5.5. For any n ≥ , a nontrivial linear combination S (cid:126)b of the eigenfunctions h , . . . , h n of the harmonic operator H (1) has at most ( n − zeros on the real line, countedwith multiplicities. Assume that S (cid:126)b has p zeros, c < · · · < c p on the real line, withmultiplicities k j ’s, such that k + · · · + k p = n − . Then, there exists a nonzero constant C such that S (cid:126)b ( x ) = C (cid:12)(cid:12)(cid:12) (cid:126)h ( c ) . . . (cid:126)h ( k − ( c ) . . . (cid:126)h ( c p ) . . . (cid:126)h ( k p − ( c p ) (cid:126)h ( x ) (cid:12)(cid:12)(cid:12) . roof. The first assertion is Sturm’s upper bound, Theorem 1.1, in the particular case ofthe harmonic oscillator on the line. The function S (cid:126)b is a linear combination of the Hermitepolynomials H , . . . , H n − , times the positive function exp( −| (cid:126)x | / S (cid:126)b on the real line, counted with multiplicities, is atmost ( n − S (cid:126)b has a least n zeros on the real line, counted withmultiplicities. From these zeros, one can determine some positive integer p , and sequences¯ c < · · · < ¯ c p , k , . . . , k p satisfying the assumptions of Corollary 5.4, and such that S (cid:126)b vanishes at order (at least) k j at ¯ c j , 1 ≤ j ≤ p . This last condition implies that the n entries of the vector (cid:126)b satisfy a system of n equations, whose determinant is precisely | (cid:126)h ( c ) . . . (cid:126)h ( k − ( c ) (cid:126)h ( c ) . . . (cid:126)h ( k − ( c ) . . . (cid:126)h ( c p ) . . . (cid:126)h ( k p − ( c p ) | . Corollary 5.4 then implies that (cid:126)b = 0, so that a nontrivial linear combination S (cid:126)b can haveat most ( n − 1) zeros on the real line, counted with multiplicities.The second assertion is a consequence of (the proof of) Lemma 5.3. In this section, we show how Gelfand’s strategy, see Section 5, can be applied to thegeneral Dirichlet Sturm-Liouville problem (1). Let q be a C ∞ real function defined in a neighborhood of the interval I :=]0 , h (1) := − d dx + q ( x ) , and, more precisely, its Dirichlet realization in I, i.e. the Dirichlet boundary value problem(37) (cid:40) − d ydx + q y = λ y ,y (0) = y (1) = 0 . Let { ( λ j , h j ) , j ≥ } be the eigenpairs of h (1) , with(38) λ < λ < λ < · · · , and { h j , j ≥ } an associated orthonormal basis of eigenfunctions.We also consider the Dirichlet realization h ( n ) of the n -particle operator in I n ,(39) h ( n ) := − n (cid:88) j =1 (cid:0) ∂ ∂x j + q ( x j ) (cid:1) = − ∆ + Q , Q ( x , . . . , x n ) = q ( x ) + · · · + q ( x n ).Denote by (cid:126)k = ( k , · · · , k n ) a vector with positive integer entries, and by (cid:126)x = ( x , · · · , x n )a vector in I n . The eigenpairs of h ( n ) are the (Λ (cid:126)k , H (cid:126)k ), with(40) (cid:40) Λ (cid:126)k = λ k + · · · + λ k n , and H (cid:126)k ( (cid:126)x ) = h k ( x ) · · · h k n ( x n ) , where H (cid:126)k is seen as a function in L (I n , dx ) identified with (cid:99)(cid:78) L (I , dx j ).The symmetric group s n acts on I n by σ ( (cid:126)x ) = ( x σ (1) , · · · , x σ ( n ) ), if (cid:126)x = ( x , · · · , x n ). Itconsequently acts on L (I n ), and on the functions H (cid:126)k as well. A fundamental domain ofthe action of s n on I n is the n -simplex(41) Ω I n := { < x < x < · · · < x n < } . In analogy with (32), we introduce the Slater determinant S n defined by,(42) S n ( x , . . . , x n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ( x ) h ( x ) . . . h ( x n ) h ( x ) h ( x ) . . . h ( x n )... ... ... h n ( x ) h n ( x ) . . . h n ( x n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Let (cid:42) c = ( c , . . . , c n − ) ∈ I n − . We consider the function x (cid:55)→ S n ( c , . . . , c n − , x ). De-veloping the determinant with respect to the last column, we see that this function is alinear combination of the functions h , . . . , h n , which we write as(43) S s ( (cid:42) c ) ( x ) = n (cid:88) j =1 s j ( (cid:42) c ) h j ( x )where s ( (cid:42) c ) = (cid:16) s ( (cid:42) c ) , . . . , s n ( (cid:42) c ) (cid:17) , and(44) s j ( (cid:42) c ) = s j ( c , . . . , c n − ) = ( − n + j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ( c ) . . . h ( c n − )... ... h j − ( c ) . . . h j − ( c n − ) h j +1 ( c ) . . . h j +1 ( c n − )... ... h n ( c ) . . . h n ( c n − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) so that s ( (cid:42) c ) is computed in terms of Slater determinants of size ( n − × ( n − We now prove Statement 1.6 using Gelfand’s strategy, as explained in Section 5. Lemma 6.1. The function S n is not identically zero. roof. The proof relies on the fact that the functions h j , 1 ≤ j ≤ n are linearly in-dependent. Clearly, S ( x ) = h ( x ) (cid:54)≡ 0. We now use induction on n . Assume that S n − ( x , . . . , x n − ) (cid:54)≡ 0. Develop the determinant S n ( x , . . . , x n ) with respect to the lastcolumn, S n ( x , . . . , x n ) = S n − ( x , . . . , x n − ) h n ( x ) + · · · . By the induction hypothesis, there exists ( x , . . . , x n − ) ∈ I n − , such that S n − ( x , . . . , x n − ) (cid:54) = 0. Then, S n ( x , . . . , x n − , x n ) (cid:54)≡ h j ’s are linearlyindependent, and the lemma follows. Lemma 6.2. The function S n is the first Dirichlet eigenfunction of h ( n ) in Ω I n , withcorresponding eigenvalue Λ ( n ) := λ + · · · + λ n . In particular, the function S n does notvanish in Ω I n . More precisely, one can choose the signs of the functions h j , ≤ j ≤ n ,such that S k is positive in Ω I k for ≤ k ≤ n . As a consequence, for any c < · · · < c n in I , the vectors (cid:126)h ( c ) , . . . , (cid:126)h ( c n ) , are linearly independent. Proof . An eigenfunction Ψ of h ( n ) F is given by a (finite) linear combination Ψ = (cid:80) α (cid:126)k H (cid:126)k of eigenfunctions of h ( n ) , such that the corresponding Λ (cid:126)k are equal, and such that Ψ isantisymmetric. If (cid:126)k = ( k , · · · , k n ) is such that k i = k j for some pair i (cid:54) = j , using thepermutation which exchanges i and j , we see that the corresponding α (cid:126)k vanishes. Itfollows that the eigenvalues of h ( n ) F are the Λ (cid:126)k such that the entries of (cid:126)k are all different.It then follows that the ground state energy of h ( n ) F is Λ ( n ) .It is clear that S n vanishes on ∂ Ω I n . Its restriction S Ω I n to Ω I n satisfies the Dirichletcondition on ∂ Ω I n , and is an eigenfunction of h ( n ) F corresponding to Λ ( n ) . Suppose that S Ω I n is not the ground state. Then, it has a nodal domain ω strictly included in Ω n .Define the function U which is equal to S Ω I n in ω , and to 0 elsewhere in I n . It is clearlyin H (Ω I n ). Using s n , extend the function U to a Fermi state U F on I n . Its energy is Λ ( n ) which is the bottom of the spectrum of h ( n ) F . It follows that U F is an eigenfunction of h ( n ) F , and a fortiori of h ( n ) . This would imply that S n is identically zero, a contradictionwith Lemma 6.1.The fact that one can choose the S n to be positive in Ω I n follows immediately.If the vectors (cid:126)h ( c ) , . . . , (cid:126)h ( c n ) were linearly dependent, the function S n would vanish at( c , . . . , c n ) ∈ I , a contradiction.The following proposition provides a weak form of Sturm’s upper bound on the numberof zeros of a linear combination of eigenfunctions of (37) (“weak” in the sense that themultiplicities of zeros are not accounted for). Proposition 6.3. Let (cid:126)b ∈ R n , with (cid:126)b (cid:54) = (cid:126) . Then, the linear combination S (cid:126)b has a most ( n − distinct zeros in I =]0 , . If S (cid:126)b has exactly ( n − zeros in I , c < · · · < c n − ,then there exists a nonzero constant C such that S (cid:126)b ( x ) = C S n ( c , . . . , c n − , x ) . Furthermore, each zero c j has order . roof. Given (cid:126)b , assume that S (cid:126)b has at least n distinct zeros c < · · · < c n in I. Thismeans that the n components b j , ≤ j ≤ n , satisfy the system of n equations, b h ( c ) + · · · + b n h n ( c ) = 0 , · · · b h ( c n ) + · · · + b n h n ( c n ) = 0 . By Lemma 6.2, the determinant of this system is positive, and hence the unique possiblesolution is (cid:126) 0. This proves the first assertion.Assume that S (cid:126)b has precisely ( n − 1) distinct zeros, c < · · · < c n − , in I. By Lemma 6.2,the vectors (cid:126)h ( c ) , . . . , (cid:126)h ( c n − ), are linearly independent. Then, x (cid:55)→ S n ( c , . . . , c n − , x )can be written as the linear combination S (cid:126)s ( (cid:42) c ) , where the vector (cid:126)s ( (cid:42) c ) is given by (44). Itfollows that the vectors (cid:126)b and (cid:126)s ( (cid:42) c ) are both orthogonal to the family (cid:126)h ( c ) , . . . , (cid:126)h ( c n − ),and must therefore be proportional. This proves the second assertion.Assume that x (cid:55)→ S n ( c , . . . , c n − , x ) vanishes at order at least 2 at c . Then ddx (cid:12)(cid:12)(cid:12) x = c S n ( x, c , c , . . . , c n − ) = 0 . This implies that ∂ S n ∂x ( c , c , c , . . . , c n − ) = 0, and hence that ∂ S n ∂ν ( c , c , c , . . . , c n − ),where ν is the unit normal to the boundary ∂ Ω I n , which contradicts Hopf’s lemma. Thisproves the last assertion, as well as the corollary.For completeness, we state the following immediate corollaries. Corollary 6.4. Given c < · · · < c n − in I , the function x (cid:55)→ S n ( c , . . . , c n − , x ) , vanishes exactly at order , changes sign at each c j , and does not vanish elsewhere in I . Corollary 6.5. Let (cid:126)b ∈ R n \{ } . If the linear combination S (cid:126)b has k distinct zeros, andif one of the zeros has order at least , then k ≤ n − . Remark 6.6. Note that for x ∈ ] c j , c j +1 [, 1 ≤ j ≤ n − S n ( c , . . . , c n − , x ) = ( − n − − j S n ( c , . . . , c j , x, c j +1 , . . . , c n − ) , so that, according to Lemma 6.2, it has the sign of ( − n − − j . This also shows that thisfunction of x changes sign when x passes one of the c j ’s. S n near a zero We begin by treating two particular examples which are similar to Examples 4.5 and 4.6.We then deal with the general case.Consider S . Let (cid:126)c ∈ ∂ Ω I5 be a boundary point. Write (cid:126)x = (cid:126)c + (cid:126)ξ , with (cid:126)ξ close to 0. Thefunction S is an eigenfunction of the operator − ∆ + Q , and vanishes at the point (cid:126)c ∈ I n .17y Bers’ theorem [8], there exists a harmonic homogeneous polynomial (cid:98) P k , of degree k ,such that(45) S ( (cid:126)c + (cid:126)ξ ) = (cid:98) P k ( (cid:126)ξ ) + ω k +1 ( (cid:126)c, (cid:126)ξ ) , where ω k +1 ( (cid:126)c, (cid:126)ξ ) is a function of (cid:126)ξ , depending on (cid:126)c , such that ω k +1 ( (cid:126)c, t(cid:126)ξ ) = O ( t k +1 ). Notethat, for the time being, we have no a priori information on the degree k . In this example, we take (cid:126)c = (¯ c , ¯ c , ¯ c , ¯ c , c ), with ¯ c < ¯ c < c . Call (cid:98) P k the polynomialgiven by (45) for this particular case. Lemma 6.7. The polynomial (cid:98) P k is given by (46) (cid:98) P k ( (cid:126)ξ ) = ρ ( ξ − ξ )( ξ − ξ ) , where ρ is a nonzero constant, and (47) S ( (cid:126)c + (cid:126)ξ ) = ρ P ( ξ , ξ ) P ( ξ , ξ ) (cid:0) ω ( (cid:126)c, (cid:126)ξ ) (cid:1) , where ω tends to zero when (cid:126)ξ tends to zero, see Notation 4.3.Proof. According to (45), we have S (¯ c + ξ , ¯ c + ξ , ¯ c + ξ , ¯ c + ξ , c + ξ ) = (cid:98) P k ( ξ , ξ , ξ , ξ , ξ ) + ω k +1 ( (cid:126)c, (cid:126)ξ ) . Using the anti-symmetry of S , taking (cid:126)ξ = t (cid:126)η , using the fact that ω k +1 ( (cid:126)c, t (cid:126)η ) is of order k + 1, and letting t tend to zero, we see that (cid:98) P k is anti-symmetric with respect to thepair ( ξ , ξ ). A similar argument applies to the pair ( ξ , ξ ). This proves that(48) (cid:98) P k ( ξ , ξ , ξ , ξ , ξ ) = − (cid:98) P k ( ξ , ξ , ξ , ξ , ξ ) = − (cid:98) P k ( ξ , ξ , ξ , ξ , ξ ) , and hence, that (cid:98) P k ( ξ , ξ , ξ , ξ , ξ ) = 0 when ( ξ − ξ )( ξ − ξ ) = 0.We claim that the converse statement is true in a neighborhood of 0. Indeed, assumethat (cid:98) P k ( (cid:126)η ) = 0, where η (cid:54) = η and η (cid:54) = η . Using (48), we can assume that η < η and η < η . Because (cid:98) P k is a nonzero harmonic polynomial which vanishes at (cid:126)η , in anyneighborhood of (cid:126)η , there exist points (cid:126)η ± such that (cid:98) P k ( (cid:126)η + ) (cid:98) P k ( (cid:126)η − ) < 0. For t positivesmall enough, the function S n ( (cid:126)c + t (cid:126)η ± ) has the sign of (cid:98) P k ( (cid:126)c + t (cid:126)η ± ), and this contradictsthe fact that the function S is positive in Ω I n .We have just proved that, in a neighborhood of zero, (cid:98) P k vanishes if and only if ( ξ − ξ )( ξ − ξ ) vanishes. The polynomials (cid:98) P k and ( ξ − ξ )( ξ − ξ ) are both harmonic andhomogeneous, and they have the same zero set in some neighborhood of zero. Accordingto Lemma 4.9, they divide each other, so that there exists a nonzero constant ρ such that (cid:98) P k = ρ ( ξ − ξ )( ξ − ξ ). 18 .3.2 Example 2 In this example, we choose (cid:126)c = (¯ c , ¯ c , ¯ c , c , c ), with ¯ c < c < c . Call (cid:98) P k the polynomialgiven by (45). Lemma 6.8. The polynomial (cid:98) P k has the following properties. For any permutation σ ∈ s ( ξ , ξ , ξ ) , of the first three variables, (49) (cid:98) P k ( ξ , ξ , ξ , ξ , ξ ) = ε ( σ ) (cid:98) P k ( ξ σ (1) , ξ σ (2) , ξ σ (3) , ξ , ξ ) , (cid:98) P k = 0 ⇔ ( ξ − ξ )( ξ − ξ )( ξ − ξ ) = 0 , (cid:98) P k ( (cid:126)ξ ) = ρ P ( ξ , ξ , ξ ) , where ρ is a nonzero constant. This means that (cid:98) P k has degree , and that (50) S ( (cid:126)c + (cid:126)ξ ) = ρ P ( ξ , ξ , ξ ) (cid:0) ω ( (cid:126)c, (cid:126)ξ ) (cid:1) , where the function ω ( (cid:126)c, (cid:126)ξ ) tends to zero when (cid:126)ξ tends to zero, see Notation 4.3.Proof. Similar to the previous proof. Let (cid:126)c ∈ ∂ Ω I n be a boundary point, i.e. a point of the form (cid:126)c = (¯ c , . . . , ¯ c , . . . , ¯ c p , . . . , ¯ c p ),where p is a positive integer, where ¯ c < ¯ c < · · · < ¯ c p , are points in I, and where (cid:126)c issuch that ¯ c j is repeated k j times, with k + · · · + k p = n .We write (cid:126)x = (cid:126)c + (cid:126)ξ , with (cid:126)ξ close to 0. The function S n is an eigenfunction of the operator − ∆ + Q , and vanishes at the point (cid:126)c ∈ I n . By Bers’s theorem [8], there exists a harmonic homogeneous polynomial (cid:98) P k , of degree k , such that(51) S n ( (cid:126)c + (cid:126)ξ ) = (cid:98) P k ( (cid:126)ξ ) + ω k +1 ( (cid:126)c, (cid:126)ξ ) , where the function ω k +1 ( (cid:126)c, (cid:126)ξ ) is a function of (cid:126)ξ , depending on (cid:126)c , such that ω k +1 ( (cid:126)c, t(cid:126)ξ ) = O ( t k +1 ). Note that, for the time being, we have no a priori information on the degree k .We relabel the coordinates of (cid:126)ξ , according to (7) – (9), and we write this vector as(52) (cid:126)ξ = (cid:0) ξ (1) , . . . , ξ ( p ) (cid:1) , where ξ ( j ) = ( ξ j, , . . . , ξ j,k j ).The permutation group s k j acts by permuting the entries of ξ ( j ) . Given σ j ∈ s k j , ≤ j ≤ p , we denote by σ = ( σ , . . . , σ p ) ∈ s k × · · · × s k p the permutation in s n which permutesthe entries of ξ ( j ) by σ j .For the same vector (cid:126)c , we look at the local behavior of the Vandermonde polynomial P n ,and we rewrite (19) as(53) P n ( (cid:126)c + (cid:126)ξ ) = ρ ( (cid:126)c ) P k (cid:0) ξ (1) (cid:1) · · · P k p (cid:0) ξ ( p ) (cid:1) (cid:16) ω ( (cid:126)c, (cid:126)ξ ) (cid:17) . emma 6.9. The polynomial (cid:98) P k given by (51) has the following properties.1. For any permutation σ = ( σ , . . . , σ p ) ∈ s k × · · · × s k p ⊂ s n , (54) (cid:98) P k ( σ · (cid:126)ξ ) = ε ( σ ) (cid:98) P k ( (cid:126)ξ ) . 2. The zero set of (cid:98) P k is characterized by (55) (cid:98) P k ( (cid:126)ξ ) = 0 ⇔ p (cid:89) j =1 P k j (cid:0) ξ ( j ) (cid:1) = 0 . 3. There exists a nonzero constant ρ ( (cid:126)c ) such that (56) (cid:98) P k ( (cid:126)ξ ) = ρ ( (cid:126)c ) P k ( ξ (1) ) . . . P k p ( ξ ( p ) ) . This means that (cid:98) P k has degree k = (cid:80) j k j ( k j − , and that (57) S n ( (cid:126)c + (cid:126)ξ ) = ρ ( (cid:126)c ) P k ( ξ (1) ) . . . P k p ( ξ ( p ) ) (cid:0) ω ( (cid:126)c, (cid:126)ξ ) (cid:1) , where the function ω ( (cid:126)c, (cid:126)ξ ) tends to zero when (cid:126)ξ tends to zero, see Notation 4.3.Proof. Assertion 1. From the form of (cid:126)c , and the definition of σ = ( σ , . . . , σ p ), we havethe relations, ε ( σ ) S n ( (cid:126)c + t(cid:126)ξ ) = S n ( σ · ( (cid:126)c + t(cid:126)ξ )) = S n ( (cid:126)c + tσ · (cid:126)ξ ) . It follows that (cid:98) P k ( tσ · (cid:126)ξ ) + ω k +1 ( (cid:126)c, tσ · (cid:126)ξ ) = ε ( σ ) (cid:0) (cid:98) P k ( t(cid:126)ξ ) + ω k +1 ( (cid:126)c, t(cid:126)ξ ) (cid:1) . The assertion follows by dividing by t and letting t tend to zero. Assertion 2. The first assertion implies that the polynomial (cid:98) P k vanishes whenever thepolynomial (cid:81) pj =1 P k j (cid:0) ξ ( j ) (cid:1) vanishes. Part ( ⇐ ) of the second assertion follows.Assume that there exists some (cid:126)η = ( η (1) , . . . , η ( p ) ) such that (cid:98) P k ( (cid:126)η ) = 0 and p (cid:89) j =1 P k j (cid:0) η ( j ) (cid:1) (cid:54) = 0 . Since (cid:98) P k is harmonic, nonconstant, and vanishes at (cid:126)η , it must change sign, and thereexist (cid:126)η ± such that (cid:98) P k ( (cid:126)η + ) (cid:98) P k ( (cid:126)η − ) < 0. Using the first assertion and the properties of theVandermonde polynomials, we see that one can choose (cid:126)η ± ∈ Ω n , with Ω n as in (31). Itfollows that for t small enough, the vectors (cid:126)c + t(cid:126)η ± are in Ω I n , defined in (41). For thesevectors, one has S n ( (cid:126)c + t(cid:126)η ± ) = (cid:98) P k ( t(cid:126)η ± ) + ω k +1 ( (cid:126)c, t(cid:126)η ± ) . This equality contradicts the fact that S n is positive in Ω I n . Assertion 3. Notice that the polynomials (cid:98) P k ( ξ ) and (cid:81) pj =1 P k j (cid:0) ξ ( j ) (cid:1) are both harmonicand homogeneous, with the same zero set in a neighborhood of 0. We can then applyLemma 4.9, which implies that they divide each other, so that these polynomials mustbe proportional. The lemma is proved.As a consequence of the preceding lemma, we have, Corollary 6.10. Let (cid:126)c ∈ ∂ Ω I n be as above. with the notation (14) , we have the relations, (58) D k ( ∂ x (1) ) · · · D k p ( ∂ x ( p ) ) S n ( (cid:126)x ) (cid:12)(cid:12)(cid:12) (cid:126)x = (cid:126)c = D k ( ∂ ξ (1) ) · · · D k p ( ∂ ξ ( p ) ) S n ( (cid:126)c + (cid:126)ξ ) (cid:12)(cid:12)(cid:12) (cid:126)ξ =0 (cid:54) = 0 . .4 Strong upper bound We can now prove Assertion 3a in Theorem 1.1, using Gelfand’s strategy, as explained inSection 5. Proposition 6.11. Let (cid:126)b ∈ R n \{ } . Call ¯ c < · · · < ¯ c p the zeros of the linear combination S (cid:126)b of the first n eigenfunctions of problem (37) . Call k j the order of vanishing of S (cid:126)b at ¯ c j . Call (cid:126)c the vector (¯ c , . . . , ¯ c , . . . , ¯ c p , . . . , ¯ c p ) , where c j , ≤ j ≤ p is repeated k j times.Then,1. k + · · · + k p ≤ ( n − ,2. If k + · · · + k p = ( n − , then there exists a nonzero constant C such that S (cid:126)b = C S (cid:126)s ( (cid:126)c ) , where the linear combination S (cid:126)s ( (cid:126)c ) is given by developing the determinant (59) (cid:12)(cid:12)(cid:12) (cid:126)h ( c ) . . . (cid:126)h ( k − ( c ) . . . (cid:126)h ( c p ) . . . (cid:126)h ( k p − ( c p ) (cid:126)h ( x ) (cid:12)(cid:12)(cid:12) , and where (cid:126)h ( m ) ( a ) is the vector (cid:0) h ( m )1 ( a ) , . . . , h ( m ) n ( a ) (cid:1) of the m th derivatives of the h j ’s evaluated at the point a .Proof. Assertion 1. Assume that k + · · · + k p ≥ n . This implies that the coefficients b , . . . , b n , satisfy the system of n equations,( b , . . . , b n ) (cid:16) (cid:126)h ( c ) . . . (cid:126)h ( k − ( c ) . . . (cid:126)h ( c p ) . . . (cid:126)h ( k p − ( c p ) (cid:17) = 0where the left hand side is the product of the row matrix ( b , . . . , b n ) by the n × n matrix (cid:16) (cid:126)h ( c ) . . . (cid:126)h ( k − ( c ) . . . (cid:126)h ( c p ) . . . (cid:126)h ( k p − ( c p ) (cid:17) . Using (58), we see that the determinant of the latter matrix is nonzero. This implies that (cid:126)b = 0, a contradiction. Assertion 2. Using (58) again (with n − n ), we see that the coefficient of h n ( x )in the linear combination S (cid:126)s ( (cid:126)c ) is nonzero, so that S (cid:126)s ( (cid:126)c ) is not identically zero. It followsthat the family of ( n − 1) vectors F := (cid:110) (cid:126)h ( c ) , . . . , (cid:126)h ( k − ( c ) , . . . , (cid:126)h ( c p ) , . . . , (cid:126)h ( k p − ( c p ) (cid:111) is free. Both functions S (cid:126)b and S (cid:126)s ( (cid:126)c ) vanish at order k j at ¯ c j , for 1 ≤ j ≤ p . This means thatthe vectors (cid:126)b and (cid:126)s ( (cid:126)c ) are both orthogonal to F , which implies that they are proportional.The proposition is proved. Remark 6.12. In this paper, we have considered a Dirichlet Sturm-Liouville problemwith smooth coefficients. In less regular cases, one can still improve Statement 1.6 byintroducing the number N (cid:126)b of nodes of S (cid:126)b (zeros at which the function changes sign),and the number A (cid:126)b of anti-nodes (zeros at which the function retains its sign). Then, N (cid:126)b + 2 A (cid:126)b ≤ n − 1. This result is stated in [12, p. 275], and proved in [11, Chap. III.5] inthe more general framework of Chebyshev systems of continuous functions.21 eferences [1] V. Arnold. Topology of real algebraic curves (works of I.G. Petrovsky and theirdevelopment)[in Russian]. Usp. Mat. Nauk 28:5 (1973) 260–262. Translated by O. Viroin V. Arnold, collected works, Vol. 2, Springer 2014, pp. 251–254. 3[2] V. Arnold. 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