Bounded multiplicity for eigenvalues of a circular vibrating clamped plate
aa r X i v : . [ m a t h . SP ] J u l BOUNDED MULTIPLICITY FOR EIGENVALUES OF ACIRCULAR VIBRATING CLAMPED PLATE
YURI LVOVSKY AND DAN MANGOUBI
Abstract.
We prove that no eigenvalue of the clamped disk canhave multiplicity greater than six. Our method of proof is basedon a new recursion formula, linear algebra arguments and a tran-scendency theorem due to Siegel and Shidlovskii. Introduction and background
The vibrating membrane.
Recall the Dirichlet eigenvalue prob-lem on the unit disk, D .(VM) (cid:26) − ∆ u = λu in D ,u = 0 on ∂ D , where ∆ = div ◦ grad is the (analyst’s) Laplacian. The eigenfunctionsand the corresponding eigenvalues are given in terms of Bessel func-tions of the first kind J m and their positive zeros j m,k . Indeed, it isstraightforward to check that(1) u m,k ( r, φ ) := J m ( j m,k r ) e imφ is an eigenfunction of eigenvalue λ = j m,k . Fourier expansion showsthat any eigenfunction is a linear combination of functions u m,k [4, Ch.V § j m,k is eitherone (in case m = 0) or two (in case m = 0). This was coined asBourget’s Hypothesis before Siegel’s Theorem.We recall the line of proof of Bourget’s hypothesis. First, (see [19,Ch. 15.28]) using a well known (length two) recursion formula for Besselfunctions and their second order ODEs it was shown that if j m,k = j m ′ ,k ′ , then either m = m ′ or j m,k is algebraic. In a second much deeperstep it was shown by Siegel [16] (see also [17]) that all positive zeros ofBessel functions are transcendental.1.2. The vibrating clamped plate.
In this paper we are interestedin the vibrating clamped circular plate ([4, Ch. V § following fourth order eigenvalue problem.(VP) ∆ u = λu in D ,u = 0 on ∂ D ,∂ n u = 0 on ∂ D . Similarly to the vibrating circular membrane, it is readily checkedthat u m,k ( r, φ ) = (cid:0) I m ( w m,k ) J m ( w m,k r ) − J m ( w m,k ) I m ( w m,k r ) (cid:1) e imφ is an eigenfunction of eigenvalue λ = w m,k , where w m,k is a zero of thecross product(2) W m := I ′ m J m − I m J ′ m , and where I m is the modified Bessel function.As in Problem (VM), it is natural to ask whether multiplicitiesoccur. There is extensive literature studying the vibrating clampedplate problem in general domains. The main questions studied arethe isoperimetric problem, eigenvalues inequalities, asymptotic dis-tribution of eigenvalues and the positivity of the ground state (seee.g. [1–3, 5–9, 11–13, 18]). It seems that the question of multiplicity ofeigenvalues for the circular plate has not been addressed so far, and it isstill not known whether eigenvalues are of multiplicity at most two (seein this context Theorem 4.1). From Weyl’s law [4, Ch. VI § k -th eigenvalue m ( λ k ) = o ( k ) as k → ∞ . In this paper we follow the line of proof for the bounded multi-plicity of the eigenvalues of the vibrating membrane, and we adapt it todeal with the eigenvalues of the clamped plate problem. The main newingredient is a recursion formula for the sequence of cross products W m .Although this sequence was extensively studied [10] we could not findthis recursion in the existing literature. Further, it turns out that thisrecursion (of length four) has nice grading and non-cancellation prop-erties which allow to adjust the linear algebra and ODE arguments inthe proof for the vibrating membrane case to our case. When combinedwith Siegel-Shidlovskii Theory (see [15]) it yields Theorem 1.1.
Let m , m , m , m be four distinct non-negative inte-gers. There is no x > for which W m ( x ) = W m ( x ) = W m ( x ) = W m ( x ) = 0 . As a main corollary we obtain
Corollary 1.2.
Let λ be an eigenvalue of Problem (VP). Then, λ isof multiplicity at most six. Remark 1.3.
One can check that the ground state of the disk is ofmultiplicity one (see [10]).
OUNDED MULTIPLICITY OF EIGENVALUES 3
Acknowledgements.
We first learned about the clamped plateproblem from Iosif Polterovich. We are very grateful to Iosif for intro-ducing us to this problem, and explaining to us the beauty and subtlepoints of several surrounding questions. We would like to thank EnricoBombieri, who explained to us some ideas of transcendental numbertheory. This manuscript also benefited from several interesting discus-sions with Lev Buhovski, Aleksandr Logunov, Eugenia Malinnikova,Guillaume Roy-Fortin, and Mikhail Sodin. This paper is part of thePhD thesis of the first author. The support of the Israel Science Foun-dation through grants nos. 753/14 and 681/18 is gratefully acknowl-edged. Part of this work was written while the second author was aninvited researcher of the LabEx Math´ematiques Hadamard project inParis-Sud XI and a Chateaubriand France-Israel fellow. The finan-cial supports of the LMH and the french government are gratefullyacknowledged.2.
Classical facts about Bessel functions
Let m be a non-negative integer. The Bessel function J m can bedefined as the entire function satisfying J ′′ m ( z ) + 1 z J ′ m ( z ) − (cid:18) m z − (cid:19) J m ( z ) = 0and normalized by J m ( z ) = 1 m ! (cid:16) z (cid:17) m + o ( z m ) as z → . The modified Bessel function I m is the entire solution of I ′′ m ( z ) + 1 z I ′ m ( z ) − (cid:18) m z + 1 (cid:19) I m ( z ) = 0normalized so that I m ( z ) = 1 m ! (cid:16) z (cid:17) m + o ( z m ) as z → . It is easily verified that I m ( z ) = ( − i ) m J m ( iz ), where i is a squareroot of − Proposition 2.1 ([19, Ch. II.12]) . The Bessel functions satisfy thefollowing rules: (3) J ′ m ( z ) − mz J m ( z ) = − J m +1 J ′ m ( z ) + mz J m ( z ) = J m − I ′ m ( z ) − mz I m ( z ) = I m +1 I ′ m ( z ) + mz I m ( z ) = I m − YURI LVOVSKY AND DAN MANGOUBI
The next positivity statement (which can be readily seen from theTaylor series expansion) will also be useful.
Proposition 2.2.
The function I m is positive in (0 , ∞ ) . A recursion formula for a cross product of Besselfunctions
As explained in the introduction, the eigenvalues of the clampedplate problem are given in terms of zeros of the functions W m definedin (2). In this section we study this sequence and we present a lengthfour rational recursion relation satisfied by it. We prove Theorem 3.1.
The following recursion formula holds. W m +2 ( z ) + W m +4 ( z ) = 4( m + 2)( m + 3) z (cid:0) W m +1 ( z ) − W m +3 ( z ) (cid:1) + − m + 3 m + 1 (cid:0) W m ( z ) + W m +2 ( z ) (cid:1) . For the proof we need some convenient formulas given in the nextlemma, proved at the end of this section.
Lemma 3.2.
The following formulas hold. (a) W m = I m +1 J m + I m J m +1 (b) W m = I m − J m − I m J m − (c) W m − ( z ) + W m +1 ( z ) = mz I m ( z ) J m ( z )(d) W m − − W m +1 = 2( I m J m ) ′ (e) W m + W m +1 = 2 I m J m +1 (f) W m − W m +1 = 2 I m +1 J m Proof of Theorem 3.1.
For convenience we denote the formula to beproved as A = B − C where A is the left hand side and B, C correspondrespectively to the two terms in the right hand side. By Lemma 3.2 wehave A = 4( m + 3) z I m +3 ( z ) J m +3 ( z ) ,B = 4( m + 2)( m + 3) z I m +2 J m +2 ) ′ ( z ) C = 4 m + 3 z I m +1 ( z ) J m +1 ( z ) . Hence, the statement A + C = B is equivalent to I m +1 ( z ) J m +1 ( z ) + I m − ( z ) J m − ( z ) = 2 mz ( I m J m ) ′ ( z ) . The last identity can be easily validated by expressing I m +1 , I m − , J m +1 and J m − in terms of I ′ m , I m , J ′ m and J m with the rules given in Propo-sition 2.1. (cid:3) OUNDED MULTIPLICITY OF EIGENVALUES 5
Proof of Lemma 3.2.
To prove (a) we use the rules in Proposition 2.1to obtain I m +1 ( z ) J m ( z ) + I m ( z ) J m +1 ( z ) = (cid:16) I ′ m ( z ) − mz I m ( z ) (cid:17) J m ( z ) − I m ( z ) (cid:16) J ′ m ( z ) − mz J m ( z ) (cid:17) = I ′ m ( z ) J m ( z ) − I m ( z ) J ′ m ( z ) = W m ( z ) . Formula (b) is proved similarly. To prove (c) we express W m − usingformula (a), while W m +1 using formula (b). Then, we get W m − + W m +1 = I m ( J m − + J m +1 ) + ( I m − − I m +1 ) J m . At the next step we express J m − , J m +1 , I m − and I m +1 in terms of thefunctions J ′ m , J m , I ′ m and I m using Proposition 2.1. The proof of (d) issimilar. To prove (e) and (f) one uses (a) and (b). (cid:3) Forbidden joint zeros
In this section we observe some forbidden patterns of joint zeros inthe sequence W m . Observe that the forbidden patterns in Theorem 4.1are not covered by Theorem 1.1. Theorem 4.1.
The patterns of joint zeros below are forbidden. (a)
The functions W m and W m +1 have no joint positive zeros. (b) The functions W m and W m +2 have no joint positive zeros.Proof. Since I m is a positive function in (0 , ∞ ), we can deduce fromLemma 3.2(a) and (b) that if W m ( x ) = W m +1 ( x ) = 0 then J m ( x ) = J m +1 ( x ) = 0. This is impossible as it implies J m ( x ) = J ′ m ( x ) =0, which is forbidden by the second order ODE satisfied by J m . Toprove (b) we use Lemma 3.2(c) and (d) and the fact that I m is apositive function in (0 , ∞ ) to deduce a similar contradiction. (cid:3) A joint zero is algebraic
In this section we show that the recursion given in Theorem 3.1combined with the fact that the four functions W , W , W , W do nothave a joint positive zero (as follows from Theorem 4.1) implies that ajoint zero of four distinct W m ’s must be algebraic. We emphasize thatthis implication is independent of the specific nature of functions W m (for example, it does not depend on the non-trivial fact that the W m s are linearly independent - see Appendix). Proposition 5.1.
Let F be a linear subspace of meromorphic functionsin C . Let ( f m ) ∞ m =0 be any sequence in F which satisfies the recursionrelation given in Theorem 3.1 and assume that f , f , f and f haveno common positive zero. Let m , m , m , m be distinct non-negativeintegers. Let x > be such that f m ( x ) = f m ( x ) = f m ( x ) = f m ( x ) = 0 . Then x is algebraic. YURI LVOVSKY AND DAN MANGOUBI
The heart of the proof of Proposition 5.1 is a linear independenceproperty implied by the recursion of Theorem 3.1.
Lemma 5.2.
Let V be a four dimensional linear space over the field ofrational functions with rational coefficients Q ( z ) . Let ( F , F , F , F ) be a basis of V , and define a sequence ( F m ) ∞ m =0 in V by the recursionof Theorem 3.1. Let m , m , m , m be distinct non-negative integers.Then, the set of vectors { F m , F m , F m , F m } is linearly independent. The proof of Lemma 5.2 is based on nice grading and non-cancellationproperties of the recursion in Theorem 3.1. We give its proof below theproof of Proposition 5.1.
Proof of Proposition 5.1.
Consider a space V and a sequence ( F m ) ∞ m =0 as in Lemma 5.2. According to Lemma 5.2 we can uniquely express F m j = X j =0 A jk F j , where A = ( A jk ) ∈ M ( Q ( z )) is an invertible matrix. Since the se-quence ( f m ) satisfies the same recursion we conclude that (not neces-sarily uniquely)(4) f m j = A j f + A j f + A j f + A j f . Taking a least common denominator D ∈ Q [ z ] for all A jk s we get Df m j = ˜ A j f + ˜ A j f + ˜ A j f + ˜ A j f , where D and ˜ A jk are polynomials in Q [ z ]. Evaluation of this identityat the point x results in D ( x ) f m ( x ) f m ( x ) f m ( x ) f m ( x ) = ˜ A ( x ) f ( x ) f ( x ) f ( x ) f ( x ) The left hand side is the zero vector by our assumption, while thevector ( f ( x ) , f ( x ) , f ( x ) , f ( x )) ∈ C is not zero by our assump-tion. We conclude that ˜ A ( x ) ∈ M ( Q ) is non-invertible. Hence,Det( ˜ A )( x ) = Det( ˜ A ( x )) = 0, and since A ∈ M ( Q ( z )) is invert-ible, Det( ˜ A ) is a non-zero polynomial in Q [ z ] and we can conclude that x is algebraic. (cid:3) Proof of Lemma 5.2.
Assume that 0 ≤ m < m < m < m anddefine the parameters ( j, k, l, m ) by m = j, m = 1 + j + k, m = 2 + j + k + l, m = 3 + j + k + l + m. Let us refine the statement in the Lemma. Consider the unique anti-symmetric four-linear form defined on V for which ( F , F , F , F ) := 1.We need to show that ( F m , F m , F m , F m ) = 0. Keeping track of theleading term in these determinant-like expressions we prove OUNDED MULTIPLICITY OF EIGENVALUES 7
Claim.
There exist constants B jklm > such that ( F j , F j + k , F j + k + l , F j + k + l + m ) =( − m B jklm z − k +2 ⌊ l/ ⌋ + m ) + P jklm ( z − ) , where P jklm ∈ Q [ z ] is of degree smaller than k + 2 ⌊ l/ ⌋ + m − . The proof of the preceding claim is by induction on j + k + l + m . The base case ( j, k, l, m ) = (0 , , ,
0) is trivial. For the sake ofshortly written expressions we introduce some notations to expressionsappearing as coefficients in the recursion of Theorem 3.1. α jklm := 4(1 + j + k + l + m )(2 + j + k + l + m ) ,β jklm := 2 + j + k + l + mj + k + l + m ,γ jklm := β jklm + 1 . We now unroll the determinant by applying the recursion given inTheorem 3.1.( F j , F j + k ,F j + k + l , F j + k + l + m ) =( F j , F j + k , F j + k + l , F j + k + l + m + F j + k + l + m )+ − ( F j , F j + k , F j + k + l , F j + k + l + m ) = α jklm z − ( F j , F j + k , F j + k + l , F j + k + l + m − F j + k + l + m )+ − β jklm ( F j , F j + k , F j + k + l , F − j + k + l + m + F j + k + l + m )+ − ( F j , F j + k , F j + k + l , F j + k + l + m ) . After a slight rearrangement we obtain( F j , F j + k ,F j + k + l , F j + k + l + m ) = α jklm z − ( F j , F j + k , F j + k + l , F j + k + l +( m − )+ − α jklm z − ( F j , F j + k , F j + k + l , F j + k + l +( m − )+ − β jklm ( F j , F j + k , F j + k + l , F j + k + l +( m − )+ − γ jklm ( F j , F j + k , F j + k + l , F j + k + l +( m − ) . (5)We denote the expression obtained in (5) by X . In order to apply theinduction hypothesis, we distinguish several cases: Case 1: m ≥
4. In this case one gets by the induction hypothesis thatfor some polynomial ˜ P jklm (of controlled degree) X = − ( − m α jklm B jkl ( m − z − k − ⌊ l/ ⌋− m − − +( − m α jklm B jkl ( m − z − k − ⌊ l/ ⌋− m − − + − ( − m β jklm B jkl ( m − z − k − ⌊ l/ ⌋− m − +( − m γ jklm B jkl ( m − z − k − ⌊ l/ ⌋− m − + ˜ P jklm ( z − ) =( − m α jklm B jkl ( m − z − k − ⌊ l/ ⌋− m + P jklm ( z − ) YURI LVOVSKY AND DAN MANGOUBI where P jklm is a polynomial of degree smaller than k + 2 ⌊ l/ ⌋ + m − Case 2: m = 3. In this case the anti-symmetry of the determinant isused to get X = α jklm z − ( F j , F j + k , F j + k + l , F j + k + l +0 )+ − α jklm z − ( F j , F j + k , F j + k + l , F j + k + l +2 )+ − γ jklm ( F j , F j + k , F j + k + l , F j + k + l +1 )By induction we have X = α jklm B jkl z − k − ⌊ l/ ⌋− + − ( − α jklm B jkl z − k − ⌊ l/ ⌋− − + − ( − γ jklm B jkl z − k − ⌊ l/ ⌋− + ˜ P jklm ( z − ) =( − m α jklm B jkl z − k − ⌊ l/ ⌋− m + P jkl ( z − )where P jklm is of degree smaller than k + 2 ⌊ l/ ⌋ + 2. Case 3: m = 2 , l ≥ X = − α jklm z − ( F j , F j + k , F j + k + l , F j + k + l +1 )+ β jklm ( F j , F j + k , F j + k +( l − , F j + k +( l − )+ − γ jklm ( F j , F j + k , F j + k + l , F j + k + l +0 ) . By induction, X = − ( − α jklm B jkl z − k − ⌊ l/ ⌋− − + β jklm B jk ( l − z − k − ⌊ ( l − / ⌋ + − γ jklm B jkl z − k − ⌊ l/ ⌋ + ˜ P jklm ( z − ) =( − m α jklm B jkl z − k − ⌊ l/ ⌋− m + P jkl ( z − )where P jkl is of degree smaller than k + 2 ⌊ l/ ⌋ + 1. Case 4: m = 2 , l = 0. X = − α jklm z − ( F j , F j + k , F j + k +0 , F j + k +0+1 )+ − γ jklm ( F j , F j + k , F j + k +0 , F j + k +0+0 )where by induction X = α jklm B jk z − k − − + − γ jklm B jk z − k + ˜ P jklm ( z − ) =( − m α jklm B jk z − k − ⌊ l/ ⌋− m + P jk ( z − )and P jk is of degree smaller than k + 1. OUNDED MULTIPLICITY OF EIGENVALUES 9
Case 5: m = 1 , l ≥ X = − α jklm z − ( F j , F j + k , F j + k +( l − , F j + k +( l − )+ − α jklm z − ( F j , F j + k , F j + k + l , F j + k + l +0 )+ β jklm ( F j , F j + k , F j + k +( l − , F j + k +( l − )Hence, by hypothesis X = − α jklm B jk ( l − z − k − ⌊ ( l − / ⌋− + − α jklm B jkl z − k − ⌊ l/ ⌋− + − β jklm B jk ( l − z − k − ⌊ ( l − / ⌋− + ˜ P jklm ( z − ) . Now it becomes a bit trickier to tell which the leading termis. If l is even then it is the second one, so we take B jkl = α jklm B jkl . If l is odd then the first two terms contribute tothe leading term and are of the same sign, so we take B jkl = α jklm ( B jk ( l − + B jkl ). In any case, we obtain X = ( − m B jklm z − k − ⌊ l/ ⌋− m + P jkl ( z − ) , where P jklm is of degree smaller than k + 2 ⌊ l/ ⌋ . Case 6: m = 1 , l = 1. X = − α jklm z − ( F j , F j + k , F j + k +0 , F j + k +0+0 )+ − α jklm z − ( F j , F j + k , F j + k +1 , F j + k +1+0 ) . The induction gives X = − α jklm B jk z − k − + − α jklm B jk z − k − + P jklm ( z − ) =( − m α jklm ( B jk + B jk ) z − k − ⌊ l/ ⌋− m + P jk ( z − )where P jk is of degree smaller than k . Case 7: m = 1 , l = 0 , k ≥ X = − α jklm z − ( F j , F j + k , F j + k +0 , F j + k +0+0 )+ − β jklm ( F j , F j +( k − , F j +( k − , F j +( k − )leading to X = − α jklm B jk z − k − + − β jklm z − k − + ˜ P jklm ( z − ) =( − m α jklm B jk z − k − ⌊ l/ ⌋− m + P jk ( z − )where P jk is of degree smaller than k . Case 8: m = 1 , l = 0 , k = 0. X = − α jklm z − ( F j , F j +0 , F j +0+0 , F j +0+0+0 ) This simple expression gives by our hypothesis X = − α jklm B j z − =( − m α jklm B j z − k − ⌊ l/ ⌋− m . Case 9: m = 0 , l ≥ X = − α jklm z − ( F j , F j + k , F j + k +( l − , F j + k +( l − )+ β jklm ( F j , F j + k , F j + k +( l − , F j + k +( l − )+ γ jklm ( F j , F j + k , F j + k +( l − , F j + k +( l − ) . We are led to the tricky expression X = α jklm B jk ( l − z − k − ⌊ ( l − / ⌋− − + β jklm B jk ( l − z − k − ⌊ ( l − / ⌋− + γ jklm B jk ( l − z − k − ⌊ ( l − / ⌋ + ˜ P jkl ( z − ) . If l is even then the leading term is the first one B jkl z − k − ⌊ l/ ⌋ with B jkl = α jkl B jk ( l − . If l is odd, then the three firstterms are of the same degree − k − ⌊ l/ ⌋ . So, we let B jkl = α jkl B jk ( l − + β jkl B jk ( l − + γ jkl B jk ( l − . In any case we ob-tain X = ( − m B jklm z − k − ⌊ l/ ⌋− m + P jkl ( z − )where P jkl is of degree smaller than k + 2 ⌊ l/ ⌋ − Case 10: m = 0 , l = 2. X = − α jklm z − ( F j , F j + k , F j + k +0 , F j + k +0+1 )+ γ jklm ( F j , F j + k , F j + k +1 , F j + k +1+0 ) X = α jklm B jk z − k − − + γ jklm B jk z − k + ˜ P jklm ( z − ) =( − m α jklm B jk z − k − ⌊ l/ ⌋− m + P jk ( z − )where P jk is of degree smaller than k + 1. Case 11: m = 0 , l = 1 , k ≥ X = − β jklm ( F j , F j +( k − , F j +( k − , F j +( k − )+ γ jklm ( F j , F j + k , F j + k +0 , F j + k +0+0 ) . The last expression gives X = β jklm B j ( k − z − k − − + γ jklm B jk z − k + P jklm ( z − ) =( − m ( β jklm B j ( k − + γ jklm B jk ) z − k − ⌊ l/ ⌋− m + P jk ( z − )with P jk of degree smaller than k − OUNDED MULTIPLICITY OF EIGENVALUES 11
Case 12: m = 0 , l = 1 , k = 0. X = γ jklm ( F j , F j +0 , F j +0+0 , F j +0+0+0 ) . This is simply a positive constant (by induction) X = γ jklm B j = ( − m γ jklm B j z − k − ⌊ l/ ⌋− m . Case 13: m = 0 , l = 0 , k ≥ X = α jklm z − ( F j , F j +( k − , F j +( k − , F j +( k − )+ − β jklm ( F j , F j +( k − , F j +( k − , F j +( k − ) . Hence, X = α jklm B j ( k − z − k − − + − β jklm B j ( k − z − k − + ˜ P jklm ( z − ) =( − m α jklm B j ( k − z − k − ⌊ l/ ⌋− m + P jk ( z − )where P jk is of degree smaller than k − Case 14: m = 0 , l = 0 , k = 1. X = α jklm z − ( F j , F j +0 , F j +0+0 , F j +0+0+0 )Thus, X = α jklm B j z − = ( − m α jklm B j z − k − ⌊ l/ ⌋− m . Case 15: m = 0 , l = 0 , k = 0 , j ≥ X = β jklm ( F j − , F j − , F j − , F j − ) . By induction this is a constant X = β jklm B ( j − = ( − m β jklm B ( j − z − k − ⌊ l/ ⌋− m . (cid:3) Some elements from Siegel-Shidlovskii Theory - a zerois transcendental
We recall the notion of a Siegel E -function. Let E be a power series. E ( z ) = ∞ X k =0 a k z k k ! . Definition 6.1 ([17, Ch. II.1]) . E is called an E -function if the fol-lowing two conditions hold(a) a k ∈ Q for all k .(b) If a k = p k /q k , where p k , q k ∈ Z are coprime, then a k = o ( k εk ),and q k = o ( k εk ) as k → ∞ for all ε > We remark that any E -function is entire and E functions consti-tute a ring. The examples we are interested in are the functions J m , J ′ m , I m , I ′ m . It is readily verified that these are all E -functions.Siegel-Shidlovskii theory is concerned with transcendental propertiesof values of E -functions which satisfy a linear ODE system. The fol-lowing theorem is one of the corner stones of the theory. It was provedfor second order ODEs in [16] and [17], and then it was simplified andextended to linear systems by Shidlovskii. Theorem 6.2 ([14; 15, Ch. 3 §
13, Second Fundamental Theorem]) . Let E , . . . , E k be algebraically independent E -functions over the fieldof rational functions C ( z ) . Let E = ( E , . . . , E k ) satisfy a linear ODEsystem of the form (6) E ′ = AE , where A ∈ M k ( C ( z )) . Let α be algebraic and distinct from the poles of A ij . Then, the set of numbers { E j ( α ) } kj =1 is algebraically independentover Q . The assumptions in Theorem 6.2 are verified in the case relevant tothis paper by an earlier theorem of Siegel.
Theorem 6.3 ([16], see also [15, Ch. 9, §
5, Lemma 6]) . The four E -functions J m , J ′ m , I m , I ′ m are algebraically independent over C ( z ) . As a corollary we have
Corollary 6.4.
Let x >
0. If W m ( x ) = 0 then x is transcendental. Proof of Corollary 6.4.
The vector of functions E = ( J m , J ′ m , I m , I ′ m )satisfies an ODE of the form (6) with A ( z ) = − m z − z m z − z Let α be a positive algebraic number. By Theorems 6.3 and 6.2 thefour values J m ( α ) , J ′ m ( α ) , I m ( α ) , I ′ m ( α ) are algebraically independent.In particular, W m ( α ) as a polynomial in these numbers is not 0. (cid:3) Proof of the main Theorem 1.1
The main theorem now follows easily.
Proof of Theorem 1.1.
Let x > W m , W m , W m and W m . The functions W , W , W and W have no common zero byTheorem 4.1. Hence, we can apply proposition 5.1 to conclude thatthe positive number x must be algebraic. On the other hand, byCorollary 6.4 it must be transcendental. This is a contradiction. (cid:3) OUNDED MULTIPLICITY OF EIGENVALUES 13
Remark 7.1.
The full power of Theorem 4.1 was not used in the proofof Theorem 1.1. The weaker statement that W , W , W and W haveno common zero follows also by computing a fourth order ODE for W and showing that ( W , W , W , W ) is obtained from ( W , W ′ , W ′′ , W ′′′ )by an invertible transformation. We leave the details to the reader.8. Appendix-Shortest recursion possible
We explain how our arguments for the proof of Theorem 1.1 alsoshow that any four distinct W m ’s are algebraically independent overthe field of rational functions C ( z ). In particular, it follows that thelinear recursion in Theorem 3.1 cannot be shortened while keepingrational coefficients. Claim 8.1 (base case) . The four functions W , W , W , W are alge-braically independent over the field C ( z ) .Proof. We may express the function W m as a linear combination of thefour functions I J , I ′ J , I J ′ and I ′ J ′ over the field Q ( z ) simply byexpanding the defining formula (2) by means of the classical recursionsin Proposition 2.1. One calculates(7) W W W W = − − − − − z − z z − z z I J I ′ J I J ′ I ′ J ′ . By Theorem 6.3 the four functions I J , I ′ J , I J ′ , I ′ J ′ are alge-braically independent over the field C ( z ). Since the linear system (7)is invertible and due to the simple fact that the set of non-zero poly-nomials is preserved by linear transformations we obtain that W , W , W , W are algebraically independent over C ( z ) too. (cid:3) Theorem 8.2.
Let m , m , m , m be four distinct non-negative inte-gers. Then, W m , W m , W m and W m are algebraically independentover the field C ( z ) .Proof. By equation (4) we can express W m j = A j W + A j W + A j W j + A j W , with A ∈ GL ( C ( z )). By Claim 8.1 it follows that W m , W m , W m and W m are algebraically independent. (cid:3) References [1] M. S. Ashbaugh and R. D. Benguria,
On Rayleigh’s conjecture for the clampedplate and its generalization to three dimensions , Duke Math. J. (1995),no. 1, 1–17.[2] M. S. Ashbaugh and R. S. Laugesen, Fundamental tones and buckling loadsof clamped plates , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) (1996), no. 2,383–402. [3] Q.-M. Cheng and G. Wei, A lower bound for eigenvalues of a clamped plateproblem , Calc. Var. Partial Differential Equations (2011), no. 3-4, 579–590.[4] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I , Inter-science Publishers, Inc., New York, N.Y., 1953.[5] R. J. Duffin,
On a question of Hadamard concerning super-biharmonic func-tions , J. Math. Physics (1949), 253–258.[6] P. R. Garabedian, A partial differential equation arising in conformal mapping ,Pacific J. Math. (1951), 485–524.[7] J. Hadamard, M´emoire sur le probl`eme d’analyse relatif `a l’´equilibre des plaques´elastiques encastr´ees. , M´em. Sav. ´etrang. (2) (1908), no. 4 (French).[8] P. J. H. Hedenmalm, A computation of Green functions for the weighted bi-harmonic operators ∆ | z | − α ∆ , with α > −
1, Duke Math. J. (1994), no. 1,51–78.[9] H. A. Levine and M. H. Protter, Unrestricted lower bounds for eigenvaluesfor classes of elliptic equations and systems of equations with applications toproblems in elasticity , Math. Methods Appl. Sci. (1985), no. 2, 210–222.[10] L. Lorch, Monotonicity of the zeros of a cross product of Bessel functions ,Methods Appl. Anal. (1994), no. 1, 75–80.[11] N. S. Nadirashvili, Rayleigh’s conjecture on the principal frequency of theclamped plate , Arch. Rational Mech. Anal. (1995), no. 1, 1–10.[12] L. E. Payne, G. P´olya, and H. F. Weinberger,
On the ratio of consecutiveeigenvalues , J. Math. and Phys. (1956), 289–298.[13] ˚A. Pleijel, On the eigenvalues and eigenfunctions of elastic plates , Comm. PureAppl. Math. (1950), 1–10.[14] A. B. ˇSidlovski˘ı, A criterion for algebraic independence of the values of aclass of entire functions , Izv. Akad. Nauk SSSR. Ser. Mat. (1959), 35–66(Russian).[15] A. B. Shidlovskii, Transcendental numbers , De Gruyter Studies in Mathemat-ics, vol. 12, Walter de Gruyter & Co., Berlin, 1989. Translated from the Russianby Neal Koblitz; With a foreword by W. Dale Brownawell.[16] C. L. Siegel, ¨Uber einige Anwendungen diophantischer Approximationen , Abh.Preuß. Akad. der Wissensch., Phys.-Math. K1 (1929), no. 1, 58 pp.[17] C. L. Siegel, Transcendental Numbers , Annals of Mathematics Studies, no. 16,Princeton University Press, Princeton, N. J., 1949.[18] G. Talenti,
On the first eigenvalue of the clamped plate , Ann. Mat. Pura Appl.(4) (1981), 265–280 (1982).[19] G. N. Watson,
A treatise on the theory of Bessel functions , Cambridge Mathe-matical Library, Cambridge University Press, Cambridge, 1995. Reprint of thesecond (1944) edition.
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