Laplace-Beltrami spectrum of ellipsoids that are close to spheres and analytic perturbation theory
LLAPLACE-BELTRAMI SPECTRUM OF ELLIPSOIDS THAT ARE CLOSE TOSPHERES AND ANALYTIC PERTURBATION THEORY
SURESH ESWARATHASAN AND THEODORE KOLOKOLNIKOV
Abstract.
We study the spectrum of the Laplace Beltrami operator on ellipsoids. For ellipsoidsthat are close to the sphere, we use analytic perturbation theory to estimate the eigenvalues upto two orders. We show that for biaxial ellipsoids sufficiently close to the sphere, the first L eigenvalues have multiplicity at most two, and characterize those that are simple. For the triaxialellipsoids sufficiently close to the sphere that are not biaxial, we show that at least the first sixteeneigenvalues are all simple.We also give the results of various numerical experiments, including comparisons to our resultsfrom the analytic perturbation theory, and approximations for the eigenvalues of ellipsoids thatdegenerate into infinite cylinders or two-dimensional disks. We propose a conjecture on the exactnumber of nodal domains of near-sphere ellipsoids. Introduction
This article aims to compute the Laplace-Beltrami spectrum (and its multiplicities) of a class of 2-dimensional ellipsoids in R through analytic perturbation theory [18], more specifically eigenvalueperturbations [11]. We describe our main results first before providing motivations and a discussionof related results.1.1. Main results.
Let us give the definition for the main object of study in our article:
Definition Let a, b, c ą
0. We denote by E a,b,c Ă R the ellipsoid given by " p x, y, z q P R | x a ` y b ` z c “ * . In other words, E a,b,c is an ellipsoid with axes a, b , and c .Let ´ ∆ g be the positive Laplace-Beltrami operator on E a,b,c and consider the correspondingeigenvalue problem ´ ∆ g ϕ Λ “ Λ ϕ Λ . Recall that on the sphere S “ E , , , the eigenvalues areΛ “ l p l ` q having multiplicity 2 l `
1, with l P Z ě . We are now in a position to state main results: Theorem Let L P N and α, β P R with at least one being non-zero. Consider the biaxial ellipsoid E a,a,b where a “ ` εα, b “ ` εβ where ε P R ` and g ε the metric from R restricted to E a,a,b .Then there exists ε p α, β, L q such that for all ε ă ε and Λ P spec p´ ∆ g q X r , L p L ` qs , we have Λ “ l p l ` q ` ε Λ ` O p ε q for l “ , , , . . . L and m “ ´ l, . . . , l with Λ being given by the explicit formula Λ “ ´ αl p l ` q ` p α ´ β q l p l ` qp l ` q p l ´ q ` l ´ m ` l ´ ˘ . (3) Moreover, each Λ has multiplicity two except for those whose expansion has m “ , which in thiscase corresponds to multiplicity one. a r X i v : . [ m a t h . SP ] F e b SURESH ESWARATHASAN AND THEODORE KOLOKOLNIKOV
Theorem Let l P N and α, β, γ P R be given with at least one being non-zero. Consider thetriaxial ellipsoid E a,b,c where a “ ` αε, b “ ` βε and c “ ` γε and g ε the metric from R restricted to E a,b,c .Then there exists ε p α, β, γ, l q such that for all ă ε ă ε and Λ P spec p´ ∆ g q X r l p l ` q ´ l, l p l ` q ` l s , we have Λ “ l p l ` q ` Λ ε ` O p ε q (5) where Λ is an eigenvalue of a p l ` q ˆ p l ` q matrix and whose entries yield explicit formulasin l, α, β, and γ . Thus, given L P N , there exists ε p α, β, γ, L q such that the expansion (5) holdsfor all Λ P spec p´ ∆ g q X r , L p L ` qs . Lastly, for l “ , , in particular, there exists ε such that spec p´ ∆ g q X r l p l ` q ´ l, l p l ` q ` l s contains only simple eigenvalues for all ε ă ε . For these explicit formulas pertaining to Λ , see Proposition 35 in Section 4.1.2. Some motivations.
From the point of view of classical mechanics, ellipsoids E a,b,c formone of the oldest known examples of integrable systems, themselves holding a venerable placein the subject. Their quantum analogues have been intensively studied in the last forty years,with numerous contributions arising from a beautiful mixture of symplectic geometry and WKBapproximations, the connection being exploited by microlocal/semiclassical analysis. We surveysome related and microlocally-oriented results in the next section.We emphasize that the spectrum of geometric spaces with large symmetry groups has beenexplicitly computed [23] with a partial list being compact rank-one symmetric spaces (CROSSes),certain projective spaces, Steifel manifolds, and Grassmannians. However, it appears that not muchis known for manifolds lacking large symmetry groups like biaxial and triaxial ellipsoids let alonetheir multiplicities. While microlocal analysis has addressed the approximation of the eigenvaluesin the semiclassical limit, we have not found any literature providing bounds on multiplicity.The study of Laplace’s equation on R using ellipsoidal-type coordinate systems is well-developedand is centered around the analysis of the Lam´e equation. In fact, the word “ellipsoidal harmonics”has been attached to a variety of families of functions including for eigenfunctions on E a,b,c . Seethe treatise of Dassios [7] for a survey and in particular Chapter 4.4 for a brief treatment onproduct-form eigenfunctions on E a,b,c .We make note that the use of analytic perturbation theory allows for both accurate approximations(as confirmed by our numerics in Section 5.1) and multiplicity calculation . Furthermore, it providesus the opportunity to bypass the use of Bohr-Sommerfeld quantization rules and the computation ofsubprincipal symbols, per microlocal analysis, which are two highly powerful but technical concepts.In this vein, the contents of our article appear to be novel.The question of what is spec p´ ∆ g q has played a prominent role in recent years in data analy-sis. For example, in [19] eigenvalues of the Laplace–Beltrami operator were used to extract “fin-gerprints” which characterize surfaces and solid objects. In [2, 4], these eigenvalues (and theircorresponding eigenfunctions) were used for dimensionality reduction and data representation.We close our motivations section by briefly discussing the concept of shape DNA in shape match-ing. By “shape DNA”, we mean the first N elements of spec p´ ∆ g q for p M, g q . Our results canbe encompassed as the computation of this shape DNA for ellipsoids that are close to spheres.Note that shape DNA plays a crucial role in the representation of data sets, itself being usefuk incopyright protection and database retrieval. For more applications, see [19].1.3. Related results from semiclassics.
There are a number of relevant results from the semi-classical analysis literature that require discussion. We begin with the work of Pankratova [16].In this article, the author uses a special ellipsoidal coordinate system and the so-called parabolic-equation method (in the spirit of Babich and Lazutkin) to compute high-frequency asymptotics foreigenvalues arising only from product-form eigenfunctions of ´ ∆ g . MALL PERTURBATIONS OF SPHERES 3
Three works of greatest relevance to Theorems 2 and 35 are those of Sj¨ostrand [22], Colin deVerd`ere-Parisse [6], and Toth [24], each of which we describe in detail. First, the work of Toth[24] not only proves the quantum integrability of E a,b,c (i.e. the existence of a second quantumHamiltonian on E a,b,c that commutes with ∆ g ) but also formulates an interesting conjecture: thejoint spectrum for these quantum Hamiltonians is encoded by a second-order complex ODE withautomorphic boundary conditions.An asymptotic description of eigenvalues for semiclassical Schr¨odinger operators whose potentialssatisfy a non-degeneracy condition is given in the work of Sj¨ostrand [22]. In the case of biaxial2-dimensional ellipsoids, one can reduce to a Mathieu-type operator A p (cid:126) q on the non- S -invariantaxis and then use quantum Birkhoff normal forms to read off formulas for the energies in a fixedwindow (in our setting, this corresponds to the low-energy spectrum with the constraints that m « l ) but to order (cid:126) . This leads us to the more geometric work of Colin de Verdi`ere-Parisse.The articles [5, 6] investigate Bohr-Sommerfeld quantization rules in the presence of singularitiesgenerated by select classes of quantum Hamiltonians. The spectrum of ellipsoids is studied as anapplication by Colin de Verdi`ere-Vu Ngoc and depends strongly on some previous work of Colin deVerdi`ere-Parisse in one dimension. The work [6] determines that the energies of a certain class of 1-dimensional semiclassical Schr¨odinger operators P p (cid:126) q in a fixed window can be explicitly deducedfrom solving for the coefficients a j in the equation e ř j aj (cid:126) j (cid:126) “ h . Followingthis reasoning in the case of biaxial 2-dimensional ellipsoids should allow one to reproduce themultiplicity information given in Theorem 2. To perform this calculation however, it appears oneneeds to push the quantum Birkhoff normal forms to greater precision via the calculation of P p (cid:126) q ’ssubprincipal terms. In some sense, our work proceeds in this direction albeit through the lens ofanalytic perturbation theory. This is one avenue in which our Theorems 2 and 4 are new.In closing this discussion, we point the connections between our method and high-frequencyquasimode constructions in semiclassical analysis. If we write out the Laplace-Beltrami operatoron E a,b,c as ´ (cid:126) ` ∆ g ` εA ` O p ε q ˘ , it becomes clear that we are utilizing an additional smallparameter ε whilst bounding (cid:126) away from zero to write quasimodes on S . In fact, we are computingthe quasifrequencies for ´ ∆ g but not in a high-frequency regime. This naturally results in ourTheorems 2 and 4 not being descriptive of high frequencies but at the upshot of being descriptivefor multiplicities.1.4. Outline of the paper.
Our main tool is the theory of perturbations, a sharply defined setof ideas that is described for instance in the classic applied mathematics text of Hinch [11] andin a more pure, theoretical fashion in the treatise of Kato [12]. While there exists a number ofsources from pure mathematics rigorizing asymptotics, including [12], there appear to be muchfewer sources demonstrating the analyticity of eigenvalues for analytic families of metrics. In thispaper, we utilize a combination of results from Rellich’s perturbation theory notes (unfortuntaely,now discontinued) from the Courant Mathematical Institute [18] and an article of Bando-Urakawa[1] on eigenvalues for certain families of Laplace-Beltrami operators. In fact, for the benefit of easierreading, Section 2 is dedicated to an appreciable reproduction of useful results from [1] along withsome alternate proofs coming from Rellich’s Courant notes.Once we have explained our theoretical tools, particularly in Theorem 6, Sections 4 and 4 arededicated to explicit calculations with the coordinate representations of ´ ∆ g on E a,b,c and theultraspherical harmonic basis on L p S q . Section 3 utilizes the symmetries of E a,b,b to reduce oureigenvalue calculation problem to one for analytic families of ordinary differential equations, allow-ing us to apply the theory of Sturm-Liouville equations as well as make deductions on multiplicities.The triaxial ellipsoid E a,b,c is the most difficult computationally, so Section 4 and the appendix are SURESH ESWARATHASAN AND THEODORE KOLOKOLNIKOV focused in this direction. The lack of rotation-invariance (in other words, an invariant S -actionon E a,b,c ) obstruct most simplifications hence requiring a more in-depth analysis.Our final section, namely Section 5, is focused on verifying the accuracy of our analytic methodsvia numerics. A combination of MATLAB calculations as well as Laplace-Beltrami eigenfunctionapproximations on surfaces, as generated by code of Macdonald-Brandman-Rooth [14], are providedfor comparisons: these demonstrate that our analytic results are in fact accurate up to a designated,yet still high, order. We also provide some simulations that address the shapes of regions whereeigenfunctions are non-zero (which go by the moniker of “nodal domains” in the spectral geometrycommunity) on different ellipsoids E a,b,c .2. Analytic Perturbation Theory
Let M be a compact, n -dimensional, smooth manifold without boundary. Let S p M q be the spaceof all C symmetric covariant 2-tensors on M and M the set of all C Riemannian metrics on M .Following the texts [8, 10], we can put a Frechet norm on S p M q . Using this fact, [1, Proposition1.2] gives a metric ρ on M which will play a role in the statement of our following theorem althoughits precise form is not needed. Theorem (Berger’s Lemma) . For g P M and h i P S p M q fixed for i “ , . . . , N , let g p ε q “ g ` ř Ni “ ε i h i where ε ă ε p M, t h u i q . Let Λ be an eigenvalue of ´ ∆ g of multiplicity l . Thenspec p´ ∆ g p ε q q consists of elements that have an analytic dependence in ε in the following way:Given Λ , there exists ε p M, ε q along with Λ m p ε q P R and ψ m p ε q P C p M q , for m P t , . . . , l u , suchthat (1) Λ m and ψ m depend real-analytically on ε ă ε , uniformly for each m P t , . . . , l u , (2) Λ j p q “ Λ , for m P t , . . . , l u , and (3) t ψ m p ε qu lm “ is orthonormal with respect to the inner product x , y g ε This “lemma” is originally due to Berger [3] however some gaps needed to be resolved andwere filled by Bando-Urakawa [1]. Their own proofs though, albeit terse, heavily relied on variousfacts from the perturbation theory of eigenvalue problems, so we reproduce a sufficient numberof arguments due to Rellich [18] for the following two reasons: 1) sake of completeness and 2) tore-illustrate the beautiful blend of ideas and formulas presented by Rellich.The proof of Theorem 6 actually follows as an immediate consequence of a slightly more generalresult. However, we first give a necessary definition:
Definition A family of metric t g ε u ε Ă M depends real-analytically on ε if there exists a family t g i u i “ Ă S p M q and an ε p M, t g u i q such that ř i “ ε i g i converges to g in the metric topology of M , for all ε ă ε . Theorem (cf. [1, Theorem 1] ]) Let g ε P M be a one-parameter family of metrics dependingreal-analytically on ε ă ε with respect to the metric ρ on M , for some ε p M q ą . Let Λ bean eigenvalue of ´ ∆ g of multiplicity l . Then the spectrum spec p´ ∆ g p ε q q consists of elements thathave an analytic dependence in ε in the following way: for Λ , there exists ε p M, ε , Λ q along with Λ m p ε q P R and ψ m p ε q P C p M q X L p M, g ε q , for m P t , . . . , l u , such that (1) Λ m and ψ m depend real-analytically on ε ă ε (with respect to their corresponding topolo-gies), for each m P t , . . . , l u , (2) Λ j p q “ Λ and ψ m p q is in the ´ ∆ g -eigenspace associated to Λ , for m P t , . . . , l u , and (3) t ψ m p ε qu lm “ is orthonormal with respect to the inner product x , y g ε MALL PERTURBATIONS OF SPHERES 5
Main tools and Rellich’s Theorem.
The proof of this Theorem 8 hinges upon the afore-mentioned robust and clever result of Rellich [18] and an auxilliary lemma about linear differentialoperators whose coefficients have an analytic dependence on a small parameter. First, we startwith a definition for the notion of real-analytic operators:
Definition (Real-analytic families of operators) . For ε ą
0, let A p ε q P L p H s p M q ,H s p M qq , the Banach space of bounded operators from H s p M q to H s p M q . We say A p ε q is real-analytic in ε if there exists a A i P L p H s , H s q with the property that A p ε q “ ř i ε i A i and asequence of constants t a i u i where } A i } ď a i such that } A p ε q} ď ÿ n ε n a n ă 8 . In fact, the definition goes both ways: starting off with the series expansion and finiteness of itsnorm, that L is Banach gives us that A p ε q is in fact an element of L p H s , H s q . We now give atechnical lemma that is useful for analysis in coordinate charts: Lemma
Let U Ă R n be a coordinate chart on M . Let L ε be family of differential operators on M which can locally be expressed in U as L ε “ ÿ | α |ď m a α p ε, x q D αx where each a α has a real-analytic dependence on ε ă ε p U q , uniformly for x P U . Then the familyof bounded operators L ε : H m p M q Ñ L p M q is real-analytic.Proof. We leave the proof as an easy exercise for the interested reader. (cid:3)
It is important to note that in local coordinates p x , . . . , x n q on M , the coefficients of the Laplace-Beltrami operator are simply products of functions which are themselves analytic in ε thanks toour analyticity assumption on g ε (this assumption itself implying analyticity for the coefficients of g ´ , which appear in the local coordinate expressions of ∆ g ε , thanks to the analyticity of det ´ p g q and the adjugate matrix of g .)Finally, we arrive at our main technical results in the theoretical portion of this article: Theorem (Rellich’s Theorem) . Let I be an interval containing 0. Let s ą s ě be integers.Let A ε be a real-analytic family of bounded operators mapping from H s to H s with A “ : A .Assume that (1) each operator A ε , ε P I , is self-adjoint with domain H s but with respect to the innerproduct x , y s . In other words, A ε is a densely defined unbounded operator on H s p M q andhas D p A ε q “ D p A ˚ ε q , (2) A is a positive operator on its domain, and (3) Λ is an eigenvalue of A with multiplicity l that is also isolated in the spectrum, that isthere exists δ p Λ q ą such that spec p A q X r´ δ ` Λ , δ ` Λ s “ t Λ u .Then there exists I Ă I containing 0, l real-analytic families of eigenvalues t Λ m p ε qu lm “ andeigenvectors t ψ m p ε qu lm “ of A ε for ε P I such that ‚ Λ j p q “ Λ , for m P t , . . . , l u , ‚ t ψ m p ε qu lm “ is orthonormal with respect to the inner product x , y s for all ε P I , and more-over, ‚ given any d , d ă δ , there exists I Ă I such that for all ε P I , spec p A ε q X r´ d ` Λ , d ` Λ s “ t Λ m p ε qu lm “ . SURESH ESWARATHASAN AND THEODORE KOLOKOLNIKOV
A similar statement can be found in the classic texts of Kato [12] and Riesz-Nagy [17]. Notealso that a priori, our intervals I and I depend on Λ therefore making this result inherentlynon-uniform across the entirety of spec p A q .In fact, by applying Theorem 11 to each element of the spectrum of A below a fixed threshold L say, and carefully choosing the intervals I we then immediately have the Corollary
Given L ą , there exists ε p L q such that spec p A p ε qq X r , L s consists entirely ofanalytic eigenvalues as described in Theorem 11.Proof of Theorem 8 using Lemma 10 and Theorem 11. The proof is almost immediate after con-sidering the following well-known isometry between L p M, g ε q and L p M, g q which we give in local-coordinate form: U ε p f qp x q “ d det g p x q det g ε p x q f p x q . Thanks to the analyticity of b det g p x q det g ε p x q , it follows immediately that U ε ∆ g ε U ´ ε ` I satisfies thehypotheses of Lemma 10 and is a bounded family of operators from H p M, g ε q to L p M, g q where s “ s “
0. Furthermore, these operators are densely defined on L p M q , self-adjoint, andpositive, thus satisfying the hypotheses of Theorem 11.We conclude the proof by noting that for ˜ ψ m p ε q an eigenvector of U ε ∆ g ε U ´ ε ` I as per theconclusions of Theorem 11, ψ m p ε q : “ U ´ ε ˜ ψ m p ε q gives us the desired eigenvectors corresponding to g ε . Hence, Λ m p ε q “ x ` U ε ∆ g ε U ´ ε ` I ˘ ˜ ψ m p ε q , ˜ ψ m p ε qy g which itself admits an power series expansion therefore verifying the analyticity. The last step isto just shift the spectrum by -1. (cid:3) Proof of Theorem 11: Some technical statements.
The idea behind the proof of Rellich’sTheorem is both natural and computational in nature, however there are a number of movingparts that we must carefully identify in a top-down format. Throughout this section, we considerassumptions (1)-(3) in the statement of Theorem 11.For the sake of simplicity, as our operators of interest are themselves Laplace-Beltrami operatorscorresponding to perturbed metrics g ε , we assume that A ε admits a discrete, non-negative spectrumand that Dom p A ε q “ Dom p A q “ H p M, dV g q . Now, set B ε : “ A ´ A ε and µ p ε q : “ Λ ´ Λ p ε q where A ε ψ p ε q “ Λ p ε q ψ p ε q . This leads us to the following series of simple lemmas: Lemma (Restriction and matrix identity) . Let ϕ Λ be a eigenvector of eigenvalue Λ and ψ p ε q aneigenvector of A ε . We have p A ´ Λ q ψ p ε q “ p B ε ´ µ p ε qq ψ p ε q and hence xp A ´ Λ q ϕ Λ , ψ p ε qy “ x ϕ Λ , p B ε ´ µ p ε qq ψ p ε qy “ by self-adjointness with domain H . Lemma (Pseudo inverses) . Let A be our self-adjoint, unbounded operator on L p M q and δ beas in the hypotheses Theorem 11. There exists R P L p L q X L p H q such that ‚ R Π E Λ “ . ‚ R p A ´ Λ q “ p A ´ Λ q R “ I ´ Π E Λ Proof.
We invoke spectral calculus for unbounded operators and denote by E σ the spectral measurefor A (which is in our case discrete). Then we can set R “ ż Λ ´ δ σ ´ Λ dE σ ` ż Λ ` δ σ ´ Λ dE σ . The boundedness follows from spectral multiplier taking values less than δ ´ on the spectrum. (cid:3) MALL PERTURBATIONS OF SPHERES 7
Notice that for ψ p ε q an eigenfunction of A ε , we have the orthogonal decomposition ψ p ε q “ ψ p t q ` R p B ε ´ µ p ε qq ψ p ε q where ψ p t q P E Λ ; this follows immediately from a combination of Lemma13 and Lemma 14. Iterating this expression, if we have a convergent operator series, allows us toexpress ψ p ε q completely in terms of data coming from E Λ . This notion motivates the main idea ofRellich’s proof as seen through the following natural generalization of this series. Lemma (Neumann series for ψ p ε q ) . Let µ be a free parameter and B ε “ ř i “ ε i A i be analyticin ε in the sense of Definition 9, with each of its terms A i having norm } A i } op ď c where c ą is fixed. Consider ψ P E Λ and set S p ε q “ R ˝ p B ε ´ µ q with } R } “ r .Then there exists ε , µ small enough such that for | ε | ă ε and | µ | ă µ , we have that ψ p ε q “ ÿ n “ S p ε q n p ψ p ε qq (16) exists in L . This element ψ p ε q will be shown to be our desired eigenfunction for A ε . It should be noted thatthe expression for this putative eigenfunction involves only information from a fixed eigenspace for A , namely E Λ . Also, the reason for having only a single number c bounding the norms of ouroperator-valued coefficients is because we only deal with 2nd-order differential operators on M ,therefore yielding only 2nd-order operators with uniformly bounded real coefficients. Proof of Lemma 15.
We only need to verify that ř n “ S p t q n has a small norm for t sufficientlysmall. Hence, we must further bound ř n “ } R ˝ p B ε ´ µ q} n , which in turn leads us to bounding } R ˝ p B ε ´ µ p ε qq} n ď } R } n ¨ }p B ε ´ µ p ε qq} n . If we choose t , µ ą } R ˝ p B ε ´ µ p ε qq} ď r ˜ µ ` ÿ i “ ε i c ¸ ă , we then have a convergent Neumann series. This completes our proof. (cid:3) Remark . In our case of Laplace-Beltrami operators arising as perturbations from that on S ,we can take r “ δ can be taken to be as greater than or equal to 2 always.2.3. Proof of Theorem 11: Weierstrass factorization and final steps.
Let us massage ourseries representation for the putative eigenfunction ψ p ε q in equation (16) a bit further, under theassumption that µ p ε q is sufficiently small. To provide some motivation, let E Λ “ span t ϕ Λ ,m u lm “ and suppose that ψ p ε q is actually an eigenvector A ε . Then we know Π E Λ ψ p ε q “ ř lm “ c m p ε q ϕ Λ ,m for some values c m p ε q ; plugging in this linear combination into the series representation of R p B ε ´ µ q with the identity in Lemma 13, gives l ÿ m “ c m p ε qx ϕ Λ ,m , p B ε ´ µ p ε qq ÿ n “ S n ϕ Λ ,m y “ m “ , . . . , l .This set of equations (18) s commonly referred to as the “first solvability conditions” in asymp-totics. Rellich’s idea was to remove the dependence of µ on ε and treat it as an independentvariable. If we can solve these equations in c m for sufficiently small µ , then the theory of zeroes ofanalytic functions return our desired eigenvalues and eigenvectors.We label v m,m p ε q : “ x ϕ Λ ,m , p B ε ´ µ q ř n “ S n ϕ Λ ,m y ; note that v m,m p ε q “ v m ,m p ε q and hencethe corresponding l ˆ l matrix is Hermitian. We lift these functions into R µ,ε and consider theresulting determinant of F p µ, ε q : “ det ` v m,m p ε q ˘ m,m (19) SURESH ESWARATHASAN AND THEODORE KOLOKOLNIKOV
For | ε | ă ε and | µ | ă µ , we know that F p µ, ε q is an analytic function in µ and ε . Notice that inthe special case of µ “ r µ p ε q “ Λ ´ Λ p ε q , and if | µ p ε q| ă µ , then F p µ p ε q , ε q “ ψ p t q .It is through the Weierstrass Preparation Theorem and that we can show the existence of suchsufficiently small µ p ε q , and therefore Λ p ε q , as encapsulated by the following proposition: Proposition
Consider the analytic function F p µ, ε q defined in (19). We have that F p µ, q “ µ l and therefore F p µ, ε q “ ´ µ l ` p l ´ p ε q µ l ´ ` ¨ ¨ ¨ ` p p ε q µ ` p p ε q ¯ ˆ E p µ, ε q where each p i p ε q is analytic in ε and E p µ, ε q is a non-vanishing analytic function in a rectangle I p µ q .Proof. This is an immediate consequence of the Malgrange-Weierstrass Preparation Theorem withthe derivative conditions applied in µ . For its rigorous statement, see the treatise by Guillemin-Golubitsky [10, Chapter 4]. (cid:3) Proof that Proposition 20 implies Theorem 11.
Take ε and µ small enough to satisfy the hypothe-ses of Proposition 20. Thanks to the corresponding matrix for the determinant F being Hermitian, F is itself real and we have a complete factorization of ´ µ l ` p l ´ p ε q µ l ´ ` ¨ ¨ ¨ ` p p ε q µ ` p p ε q ¯ into monomials of the form µ ´ r µ m p ε q for m “ , . . . , l where | µ ´ r µ m p ε q| P r , µ q and ε P r , ε q .By analyticity of F p µ, ε q , it follows that r µ m p ε q is analytic for all m .Notice that F p ˜ µ m p ε q , ε q “
0, implying there exists non-trivial solution vectors ÝÑ c p ε q P R l where l ÿ m “ c m p t q v m,m p ε q “ m “ , . . . , l . With this, we are ready to show ψ p ε q as defined in Lemma 15, with µ “ ˜ µ m p ε q and ψ p ε q having the coefficients ÝÑ c p ε q , is an eigenfunction of A ε of eigenvalue Λ m p ε q “ Λ ` ˜ µ m p ε q .We have p A ´ Λ q ψ p ε q“ p A ´ Λ q R p B ε ´ ˜ µ m p t qq loooooooooooooomoooooooooooooon orthogonality ψ p ε q“ p B ε ´ ˜ µ m p ε qq ψ p ε q ´ Π E Λ pp B ε ´ ˜ µ m p ε qq ψ p ε qq looooooooooooooooooooooooooooomooooooooooooooooooooooooooooon Lemma 14 “ p B ε ´ ˜ µ m p ε qq ψ p ε q ´ l ÿ m “ ˜ l ÿ m “ c m p ε qx ϕ Λ ,m , p B ε ´ ˜ µ p ε qq ÿ n “ S n ϕ Λ ,m y ¸ ϕ Λ ,m loooooooooooooooooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooooooooooooooooon Lemma 15 “ p B ε ´ ˜ µ m p ε qq ψ p ε q with the last equation following from the implications of F p ˜ µ m p ε q , ε q “ A ε ψ p ε q “ p Λ ` ˜ µ m p ε qq ψ p ε q , and finally yielding the eigenfunction we aimed for. Since A p ε q isself-adjoint and Λ is real, so is µ p ε q . By positivity of A ε , we know that Λ ` ˜ µ p ε q ě | ˜ µ p ε q| ď δ .Orthonormality follows from executing the Gram-Schmit procedure.To prove the final part of the theorem, let Λ “ Λ ` p d ´ d q . Then the operator norm of } p A ` p δ q Π E Λ q ´ Λ } op is bounded below, as A ` p δ q Π E Λ has no spectrum in J , and this MALL PERTURBATIONS OF SPHERES 9 continues to be the case in the interval r´ d ` Λ , d ` Λ s for parameters | t | ă t p Λ , δ q where ε ď ε thanks to analyticity which in turn provides continuity of our operator norms in ε . This completesour proof. (cid:3) Biaxial Ellipsoids in R Coordinate calcuations and reduced equations.
Let ϕ P p , π q and θ P r , π q . A naturalset of ellipsoidal coordinates are p a sin ϕ cos θ, a sin ϕ cos θ, b cos ϕ q . We don’t make any assumptionson the sizes of a or b yet. The induced metric in these coordinates is of the form g “ a cos ϕ ` b sin ϕ, g “ g “ , g “ a sin ϕ. Notice that the functions g and g are, respectively, the squared chord length for the ellipse in R given by x { a ` z { b squared and the squared radius of the S cross section of our ellipsoid E a,b .Therefore the corresponding Laplace-Beltrami operator takes the form∆ g “ p a cos ϕ ` b sin ϕ q ´ B ϕϕ ` p a sin ϕ q ´ B θθ ` h p ϕ qB ϕ , (22)where h p ϕ q “ a det p g q ´ B ϕ p a det p g qp a cos ϕ ` b sin ϕ q ´ q . In the upcoming sections, we set a “ ` αε and b “ ` βε where α, β ‰ ε . For ε “ {
2, the metric g ε for E a,b is analytic and admits a finite polynomial expansion in ε . Hence, Theorem 6 applies and we canperform calculations using analytic series in ε for a possibly smaller threshold ε .Thanks to the natural S action on E a,b , basic representation theory (see for instance Terras’treatise [23]) tells us that L p E a,b , dV q has a basis consisting of separable eigenfunctions of the form u p ϕ q e imθ where m P Z . Plugging this ansatz into the Laplace-Beltrami operator and performing thestandard calculations leads to the following separated equations written in Sturm-Liouville form: ˆ a sin ϕ ? a cos ϕ ` b sin ϕ B ϕ ˆ a sin ϕ ? a cos ϕ ` b sin ϕ B ϕ ˙ ` Λ a sin ϕ ˙ uu “ m “ ´ B θθ ff where f p θ q “ e imθ . Thus, the factor u satisfies the following reduced equation ˜ a sin ϕ a a cos ϕ ` b sin ϕ B ϕ ˜ a sin ϕ a a cos ϕ ` b sin ϕ B ϕ ¸ ` Λ a sin ϕ ¸ u ´ m u “ , (23)exhibiting some dependence on the integral parameter m ; in the upcoming calculations, we incor-porate the parameter a into Λ and abuse notation by calling the new eigenvalue Λ p ε q . We willuse these equations to compute approximations for the purported basis of L p E a,b , dV q using theanalytic perturbation introduced in Section 2.3.2. Deriving the first solvability condition.
The theory of 2nd-order ODEs tells us that theeigenvalues Λ p ε q , for each m , are simple. Thanks to the term m and therefore being able to use ˘ m to generate potentially different eigenfunctions in ϕ , we know that we can write each solutionin ϕ as either u ` m or u ´ m : it is of these u ˘ m that we take the expansion and in turn approximateΛ p ε q , which we see has multiplicity at least 2 for m ‰ m “ r´ , s . Hence we know that given l p l ` q P spec p S q , there exists ε p S , l p l ` qq such that the following expansions are valid:Λ p ε q “ l p l ` q ` ε Λ ` ε Λ ` . . .u ˘ m p ϕ, ε q “ u p ϕ q ` εu p ϕ q ` ε u p ϕ q ` . . . . We make note that we are abusing notation by using Λ to represent the first-order coefficient ofΛ; as we have fixed the eigenvalue Λ, this use is unambiguous. The simplicity of the Sturm-Liouville spectrum is used when writing the order 0 (in ε ) term for u ˘ m . To these expansions, we apply A ε “ ϕ ˜ sin ϕ a a cos ϕ ` b sin ϕ B ϕ ˜ sin ϕ a a cos ϕ ` b sin ϕ B ϕ ¸ ´ m a sin ϕ ¸ and work out the formal series for A ε u ˘ m p ε q “ ´ Λ p ε q u ˘ m p ε q . We expand A ε in ε to obtain A ε “ A ` εA ` O p ε q where the big-O notation means that the “implicit” object is a 2nd-order differential operator, andwith A “ B ϕϕ ` cot ϕ B ϕ ´ m sin ϕ (24) A “ p α ´ β q ` ϕ B ϕϕ ` p ϕ qB ϕ ˘ ´ αA (25)A gathering of the 0th-order and 1st-order terms in ε yield the following two equations: A u “ ´ l p l ` q u , and (26) A u ` l p l ` q u “ ´ A u ´ Λ u . (27)The solution to (26) is given by the spherical harmonics u “ c P ml p cos ϕ q where c is any constant.The operator A is self-adjoint with respect to the measure sin ϕdϕ , whose corresponding innerproduct we denote by x , y . Taking the product of both sides of (27) with respect to u we thenobtain: x u , A u ` l p l ` q u y “ x u , A u ` l p l ` q u y “ , so that ´x u , A u y ´ Λ x u , u y “ “ ´x u , A u yx u , u y . (28)Next we compute, A u “ p α ´ β q (cid:32)`` ´ l p l ` q ϕ ` m ˘ u ` ϕ cos ϕ B ϕ u ˘( ` αl p l ` q u where we used the Legendre equation (26) to rewrite B ϕϕ u “ ´ ´ cot ϕ B ϕ ` m sin ϕ ´ l p l ` q ¯ u . Therefore (28) yields the formulaΛ “ p β ´ α q ş (cid:32)` ´ l p l ` q sin ϕ ` m ˘ u ` p ϕ q cos ϕ B ϕ u ( u sin ϕdϕ ş π u sin ϕdϕ. ´ αl p l ` q . Changing variables cos ϕ “ t, we then obtainΛ “ p β ´ α q ş ´ (cid:32)` ´ l p l ` q ` ´ t ˘ ` m ˘ P p t q ´ ` ´ t ˘ tP p t q P p t q ( dt ş π P p t q dt. ´ αl p l ` q (29)where P p t q “ B l ` m dt p l ` m q “` ´ t l ˘ m ‰ . Integrating by parts, we rewrite: ż ´ ` ´ t ˘ tP P “ ´ ż ´ ` ´ t ˘ P so that (29) becomesΛ “ p β ´ α q ´ l p l ` q ` m ` ˘ ` p l p l ` q ´ q ş ´ t P dt ş π P dt. + ´ αl p l ` q MALL PERTURBATIONS OF SPHERES 11
Finally, the ratio ş ´ t P dt ş ´ P “ l ´ m ` l ´ p l ` qp l ´ q is evaluated in Appendix 6. This yields formula (3).The multiplicity statement in Theorem 2 follows easily from here after taking into account thatfor each l p l ` q , we get a single analytic eigenvalue for m “ m “ ˘ , . . . , ˘ l . Now apply Corollary 12 which implies that the multiplicities cannot behigher than 2. 4. Triaxial ellipsoids in R Coordinate calculations.
We now pursue the calculation of eigenvalues for triaxial ellip-soids. For organizational reasons, let us write ∆ g in coordinates. In R , we take the coordinates p a cos ϕ cos θ, b cos ϕ sin θ, c sin ϕ q where ϕ P p , π q and θ P r , π q . This leads us the followingexpression of ∆ g as follows: ∆ g “ A B ϕϕ ` B B ϕθ ` C B θθ ` E B ϕ ` F B θ where A : “ g D , B : “ ´ g D , C : “ g D ,E : “ D ´ { ˆ´ g D ´ { ¯ ϕ ´ ´ g D ´ { ¯ θ ˙ “ B ϕ g D ´ g D B ϕ D ´ B θ g D ` g D B θ DF : “ D ´ { ˆ´ g D ´ { ¯ θ ´ ´ g D ´ { ¯ ϕ ˙ “ B θ g D ´ g D B θ D ´ B ϕ g D ` g D B ϕ D where D p θ, ϕ q : “ g g ´ g g g p θ, ϕ q : “ cos ϕ ` a cos θ ` b sin θ ˘ ` c sin ϕg p θ, ϕ q : “ sin ϕ ` a sin θ ` b cos θ ˘ g p θ, ϕ q : “ g “ ` b ´ a ˘ sin p ϕ q sin p θ q Deriving the first solvability condition.
We want our triaxial ellipsoid to be a smallperturbation of S , so we set a : “ ` αε, b : “ ` βε, c : “ ` γε Note that our metric coefficients g ij are analytic and non-vanishing in ε , therefore making theinverse metric g ij ’s coefficients analytic as well. We are in a position to apply Theorem 6.Thus, we can carefully massage ∆ g into an analytic series of operators, specifically as∆ g “ A ` εA ` ε A p ε q where A “ B ϕϕ ` ϕ B θθ ` cos ϕ sin ϕ B ϕ and A : “ “ p β ´ α q cos θ cos ϕ ` p γ ´ β q cos ϕ ´ γ ‰ B ϕϕ ` ` p α ´ β q cos θ ´ α ˘ sin ϕ B θθ ` p α ´ β q cos ϕ sin θ cos θ sin ϕ B ϕθ ` p β ´ α q sin θ cos θ sin ϕ B θ ` „´ p β ´ α q p cos p θ qq ´ β ` γ ¯ cos ϕ ` p α ´ β q cos θ ´ α ` β ´ γ cos ϕ sin ϕ B ϕ . Now, we set u to be a solution of ∆ g u “ ´ Λ p ε q u. Expanding similarly as in Section 3 thanks to Theorem 6, we are can again write the following twoanalytic series in ε : u “ u ` εu ` . . . Λ p ε q “ l p l ` q ` ε Λ ` . . . Gathering terms in ε yields the 0th-order equation of A u “ ´ l p l ` q u for l P Z ` and whose solution is given by u “ C v p θ, ϕ q ` l ÿ m “ C m v m p θ, ϕ q ` D m w m p θ, ϕ q , where v m “ cos p mθ q P ml p cos p ϕ qq ; w m “ sin p mθ q P ml p cos p ϕ qq ;the set t w , t w m , v m u lm “ u form a basis for the space of spherical harmonics with eigenvalue l p l ` q with the P ml being the associated Legendre functions given by P ml p t q “ A l,m ` ´ t ˘ m { B m ` l B t m ` l ”` ´ t ˘ l ı . The normalization constants A l,m are chosen so that x v m , v m y “ x w m , w m y “ , (30)where the inner product x , y is with respect to the metric on S , that is x v, w y : “ ż π ż π v p θ, ϕ q w p θ, ϕ q sin ϕdϕdθ. At order ε we have A u ` l p l ` q u “ ´ Λ u ´ A u . (31)Note that A is self-adjoint with respect to the inner product x , y . Multiplying (31) by v k or w k and integrating we then obtain the solvability conditions x v k , u y Λ “ ´ x v k , A u y , k “ . . . l, x w k , u y Λ “ ´ x w k , A u y , k “ . . . l. Thanks to orthonormality, we have that x v k u y “ C k and x w k u y “ D k . Thus, in contrast to the biaxial case, the first solvability equation returns a more complicatedsystem of linear equations to solve. This eigenvalue problem problem for Λ can be read off as M V “ Λ V (32)where M is the p l ` q ˆ p l ` q matrix and V is 2 l ` C m and D m . We now seek to simplify the “matrix elements” x v m , A u y . MALL PERTURBATIONS OF SPHERES 13
Expanding into Fourier modes.
To execute this simplification of the quantities x v m , A u y , we expand in terms of Fourier modes on S in the variable θ . For simplicity of notation, set P p ϕ q : “ P ml p cos p ϕ qq . We find that A p cos p mθ q P q “ g m ´ p ϕ q cos pp m ´ q θ q ` g m p ϕ q cos p mθ q ` g m ` p ϕ q cos pp m ` q θ q (33a)and A p sin p mθ q P q “ g m ´ p ϕ q sin pp m ´ q θ q ` g m p ϕ q sin p mθ q ` g m ` p ϕ q sin pp m ` q θ q (33b)and g m ´ p ϕ q “ β ´ α ˆ l p l ` q sin ϕ ` ` ´ l p l ` q ´ m ˘ ` m p m ´ q sin ϕ ˙ P ` β ´ α ˆ ´ cos ϕ sin ϕ ` p m ´ q cos ϕ sin ϕ ˙ P ϕ , (34a) g m ` p ϕ q “ β ´ α ˆ l p l ` q sin ϕ ` ` ´ l p l ` q ´ m ˘ ` m p m ` q sin ϕ ˙ P ` β ´ α ˆ ´ cos ϕ sin ϕ ` p´ m ´ q cos ϕ sin ϕ ˙ P ϕ , (34b) g m p ϕ q “ ` p α ` β ´ γ q m ` l p l ` q γ ` p α ` β ´ γ q l p l ` q cos ϕ ˘ P `p α ` β ´ γ q sin p ϕ q cos p ϕ q P ϕ . (34c)In above expressions, we have used the standard Legendre ODE (albeit with t “ cos ϕ ), P ϕϕ “´ cos ϕ sin ϕ P ϕ ` ” m sin ϕ ´ l p l ` q ı P in order to eliminate any occurrence of P ϕϕ . In total, each quantity x v m , A u y simplifies to x v m , A u y “ x v m , A v m y C m ` x v m , A v m ` y C m ` ` x v m , A v m ´ y C m ´ x w m , A u y “ x w m , A w m y D m ` x w m , A w m ` y D m ` ` x w m , A w m ´ y D m ´ where we used the convention that C m “ m R t , , . . . , l u and similarly D m “ m Rt , , . . . , l u . In other words, odd-frequency Fourier modes couple with neighbouring odd-frequencymodes, and even-frequency Fourier modes couple with their neighbors. Therefore the eigenvalueproblem (32) decomposes into four distinct subproblems: one for even-indexed C ’s, one for odd-indexed C ’s, and so forth. As a result, Λ is characterized by the following. Proposition
Define l e “ " l, if l is even l ´ , if l is odd ; l o “ " l, if l is odd l ´ , if l is evenand define the tridiagonal matrices M cos ,e “ ´ »———————– x v , A v y x v , A v y . . . x v , A v y x v , A v y x v , A v y . . . ... x v , A v y x v , A v y . . . ... . . . . . . . . . x v l e ´ , A v l e y . . . x v l e , A v l e ´ y x v l e , A v l e y fiffiffiffiffiffiffiffifl (36a) M cos ,o “ ´ »———————– x v , A v y x v , A v y . . . x v , A v y x v , A v y x v , A v y . . . ... x v , A v y x v , A v y . . . ... . . . . . . . . . x v l o ´ , A v l y . . . x v l , A v l ´ y x v l , A v l y fiffiffiffiffiffiffiffifl (36b) M sin ,e “ ´ »———————– x w , A w y x w , A w y . . . x w , A w y x w , A w y x w , A w y . . . ... x w , A w y x w , A w y . . . ... . . . . . . . . . x w l e ´ , A w l e y . . . x w l e , A w l e ´ y x w l e , A w l e y fiffiffiffiffiffiffiffifl (36c) M sin ,o “ ´ »———————– x w , A w y x w , A w y . . . x w , A w y x w , A w y x w , A w y . . . ... x w , A w y x w , A w y . . . ... . . . . . . . . . x w l o ´ , A w l o y . . . x w l , A w l ´ y x w l o , A w l o y fiffiffiffiffiffiffiffifl (36d) Then for Λ “ l p l ` q ` ε Λ ` O ` ε ˘ , as given by Theorem 6, we have that Λ is one of the l ` eigenvalues of the four matrices (36). The entries of the matrices (36) are computable explicitly – see Appendix 6 – with the followingresult: x v m , A v m y “ γl p l ` q ` p α ` β ´ γ q l p l ` qp l ` q p l ´ q ` l ` m ` l ´ ˘ for m ‰ , (37a) x w m , A w m y “ x v m , A v m y , for m ‰ , (37b) x v , A v y “ p α ` β ´ γ q l p l ` q p l ` q p l ´ q ` l p l ` q γ, (37c) x w , A w y “ p β ` α ´ γ q l p l ` q p l ` q p l ´ q ` l p l ` q γ, (37d) x v m ´ , A v m y “ x v m , A v m ´ y “ p β ´ α q l p l ` qp l ´ q p l ` qˆ a p l ´ m ` q p l ´ m ` q p l ` m ´ q p l ` m q ˆ " m ą ? m “ , (37e)and x w m ´ , A w m y “ x w m , A w m ´ y “ x v m ´ , A v m y , for m ě . (37f)4.4. Explicit calculations for l “ , , . We carry out Proposition 35 in the simple cases of l “ , MALL PERTURBATIONS OF SPHERES 15
For l “
1, we arrive at three possible choices for Λ :Λ “ ´ p α ` β ` γ q with u “ v ; (38)Λ “ ´ p α ` β ` γ q with u “ v ; (39)Λ “ ´ p α ` β ` γ q with u “ w . (40)Note that this coincides with the formula (3) for the biaxial case by taking γ “ α : the formula (40)corresponds to mode m “ m “ ˘ . For a genericchoice where all of α, β, γ are distinct, we find that spectrum of ∆ g near 1 ¨ “ ` ε Λ ` O p ε q .For l “ , matrices (36) are of the size 2x2 1x1, 1x1 and 1x1, respectively. The five resultingeigenvalues, respectively, are:Λ “ ´ p α ` β ` γ q ˘ a α ` β ` γ ´ αβ ´ αγ ´ βγ, ´ p α ` β ` γ q ´ p α ` β ` γ q , ´ p α ` β ` γ q . (41)Again, for a generic choice where all of α, β, γ are distinct, we find that spectrum of ∆ g near to2 ¨ “ ` ε Λ ` O p ε q . When γ “ α , eigenvalues (41) become, in theorder as listed,Λ “ ´ α ´ β, ´ α ´ β, ´ α ´ β, ´ α ´ β, ´ α ´ β, and as expected, they coincide with the formula (3), with m “ , ˘ , ˘ , ˘ , ˘
1, respectively.Finally, for l “ , matrices (36) are of the size 2x2, 2x2, 2x2 and 1x1, respectively, and yield thefollowing eigenvalues for the correction Λ : M cos ,e : ´ α ´ β ´ γ ˘ a α ` β ´ αβ ´ αγ ´ βγ ` γ M cos ,o : ´ γ ´ β ´ α ˘ a γ ` β ´ γβ ´ γα ´ βα ` α M sin ,e : ´ α ´ β ´ γM sin ,o : ´ γ ´ α ´ β ˘ a γ ` α ´ γα ´ γβ ´ αβ ` β . As with previous cases, we have verified that all these eigenvalues are distinct whenever α, β, γ aredistinct. Taking γ “ β , they become M cos ,e : ´ α ´ β, ´ α ´ βM cos ,o : ´ α ´ β, ´ α ´ βM sin ,e : ´ a ´ bM sin ,o : ´ a ´ b, ´ a ´ b. These agree with formula (3), with m “ ˘ , ˘ , ˘ , ˘ , ˘ , ˘ , , respectively. ε “ . ε “ . m Λ Λ Λ numeric Λ numeric ´ Λ ε %err Λ numeric Λ numeric ´ Λ ε %err0 2 -2.400 1.7772 -2.228 7.17% 1.8844 -2.312 3.67%6 -6.2857 5.4079 -5.921 5.80% 5.6950 -6.100 2.95%12 -12.266 10.8463 -11.53 6.00% 11.4052 -11.896 3.02%1 2 -0.8 1.9250 -0.750 6.25% 1.9612 -0.776 3.00%6 -5.1429 5.5227 -4.773 7.19% 5.7521 -4.958 3.60%12 -11.2 10.9509 -10.49 6.34% 11.4573 -10.854 3.09%2 6 1.7143 5.8404 -1.596 6.90% 5.9173 -1.654 3.52%12 8 11.2595 -7.405 7.44% 11.6152 -7.696 3.80%3 12 -2.666 11.7527 -2.473 7.24% 11.8716 -2.568 3.68% Table 1.
Comparison between asymptotic value of Λ and the estimated numericalvalue for the biaxial ellipsoid. Here, a “ b “ ` ε (corresponding to α “ , β “
1) and with ε “ . and Λ numeric ´ Λ ε . Numerical experiments and conjectures
Biaxial eigenvalues.
Equation (23) can be used to compute the eigenvalues numerically forany bi-axial ellipsoid. To solve (23) numerically, we discretize the space using the usual centereddifference discretization. The procedure leads to an N ˆ N -matrix eigenvalue problem Av “ Λ v where the eigenfunction v p ϕ q is approximated by v p ϕ j q « v j , ϕ j “ jπN . We impose Neumannboundary conditions at the poles: v p q “ , v p π q “ . In our computations, we have set N “ numeric to 4 significant digits; we verified that doubling the mesh-sizedid not change the answer within that precision.Table 1 provides a comparison between the analytical result for Λ , as given in equation (3),and the numerical approximation using the above procedure. The axial parameters a “ b “ ` ε are considered with ε “ . ε “ .
05. While we did not prove this analytically, thenumerics suggest that the relative error scales linearly with ε , as would be expected assuming thatΛ is analytic in ε. Triaxial eigenvalues.
We used the algorithm described in [14] to compute the eigenvaluesnumerically for a true triaxial ellipsoid. As opposed to the numerical method used for the biaxialeigenvalues, the code used in this case implements the closest-point algorithm developed in [20, 15].To compare with the numerics, we take a “ , b “ ` ε, c “ ´ ε, so that p α, β, γ q “ p , , ´ q . Table2 compares the numerics with the analytic formulas for first nine triaxial eigenvalue subprincipalterms, as obtained in Proposition 35.The code in [14] generates a sparse matrix that corresponds to a discretization of the Laplace-Beltrami operator for a surface. We use the spatial resolution of 0 . , ˆ , ˆ , the control on the error is ratherpoor; we expect no more than 2-3 digits of precision. For this reason, the error does not scalelinearly in ε as would be expected: it would require too many meshpoints to resolve up to O p ε q numerically for this two-dimensional non-symmetric problem: the method relies on the MATLABsparse eigenvalue solver eigs which is not sufficiently accurate for such large matrices. Indeed, thisproblem provides a good test case for the closest-point method algorithm. MALL PERTURBATIONS OF SPHERES 17 ε “ . ε “ . Λ Λ numeric Λ numeric ´ Λ ε err Λ numeric Λ numeric ´ Λ ε err2 -1.6 1.69763 -1.511 0.088 1.84107 -1.589 0.01062 0 2.05566 0.278 0.278 2.01048 0.1047 0.1052 1.6 2.33333 1.666 0.066 2.1603 1.603 0.0036 -3.95897 5.24037 -3.798 0.160 5.59748 -4.025 -0.0666 -3.42857 5.45296 -2.735 0.693 5.68282 -3.171 0.2566 0 6.04863 0.2431 0.243 6.0048 0.0479 0.04796 3.42857 6.82912 4.145 0.717 6.36925 3.692 0.2636 3.95897 6.83013 4.150 0.191 6.3857 3.857 -0.101 Table 2.
Comparison between asymptotic value of Λ and the estimated numericalvalue for the triaxial ellipsoid. Here, p a, b, c q “ p ` ε, b “ ´ ε, q (correspondingto p α “ , β, γ q “ p , ´ , q ) and with ε “ . and Λ numeric ´ Λ ε . We conclude this numerical exploration with a discussion of multiplicity of eigenvalues. For thebiaxial case, due to the monotonicity of the formula (3) with respect to m , there are exactly l double eigenvalues (corresponding to m “ . . . l ) and a one single eigenvalue (corresponding to m “
0) near Λ “ l p l ` q , for a total of 2 l ` α ‰ β ‰ γ , extensivenumerical experiments indicate that for a fixed l, the perturbations Λ as given in Proposition 35are all distinct (we verified this analytically when l ď is an eigenvalue of at most 2x2matrix in that case, and numerically for l up to 5). As a consequence, we repeat the followinggenerally held belief in the spectral-geometry literature: Suppose α ‰ β ‰ γ. Given any L , thereexists ε such that for all ε ă ε , the set t Λ : Λ ď L u contains only simple eigenvalues.5.3. Observations on nodal domains.
Figure 1 shows the first few eigenmodes with l “ , , , , as given by Proposition 35, for several values of α, β, γ. The corresponding eigenfunctions areplotted, as well as their nodal lines.A nodal domain of u is a connected component of the subset Ω p u q “ t x P M : u p x q ‰ u ,the regions of M where u is positive or negative. In this section, we briefly explore the nodaldomain structures (or better yet, shapes) of our ellipsoidal harmonics. Recall that a version ofCourant’s nodal domain theorem says that on a compact, boundaryless manifold, the number ofnodal domains for eigenfunctions produced from the n -th eigenspace (counting multiplicities) isbounded above by n .In the biaxial case near a sphere, there are 2 l ` l p l ` q with l dou-ble eigenvalues and one simple eigenvalue. The corresonding eigenfunctions have P ml p ϕ q cos p mθ q , m “ . . . l and P ml p ϕ q sin p mθ q , m “ . . . l as their leading order terms. These are shown in Figure1 (rows 1 to 4) for l “ . . . . Note that those corresponding to double eigenvalues, with leadingorder terms P ml p ϕ qp C cos p mθ q` D sin p mθ q , all have p l ` ´ m q m nodal domains when 0 ă m ď l ;whereas the simple eigenvalue P l p ϕ q has p l ` q nodal domains. Moreover, formula (3) shows thatthe 2 l ` l p l ` q are monotone in m. This allows for a full characterization forthe number of nodal domains in the biaxial case.Figure 1 suggests that when deforming a biaxial ellipsoid to a triaxial ellipsoid, the nodal linetopology changes only at the “north” and “south” poles, where 2 m nodal lines intersect, and doesso in two very specific ways. For example, when m “ , the pole is desingularized either in thisway: or this way: . The latter transformation does not Figure 1.
The first few eigenfunctions of some ellipsoids, computed from Propo-sition 35. Corresponding nodal lines and Λ are also shown. Parameters are asindicated.affect the number of nodal domains, whereas the former reduces it by either 2 p m ´ q if m ă l , orby m ´
1, if m “ l. Based on extensive numerical observations for l “ . . .
5, we offer the followingconjecture on the number of nodal domains for our near-sphere ellipsoids.
Conjecture
Define the sequence N k , k “ . . . l as follows: N “ l ` N m ´ “ p l ` ´ m q m, N m “ N m ´ , m “ . . . l ; (43) and sequence ˆ N k , k “ . . . l as follows: ˆ N m “ " N m ´ p m ´ q , m “ . . . l ´ l ` , m “ l ; ˆ N m ´ “ N m ´ . (44) For ε sufficiently small such that Theorem 4 holds, arrange the l ` eigenvalues near the thelevel Λ “ l p l ` q in increasing order. Their nodal domain count is as follows: (a) For an oblate ellipsoid ( α “ γ ą β ), the nodal domain count is N , . . . , N l . (b) For a prolate ellipsoid ( α “ γ ă β ), the nodal domain count is N l , . . . , N . (c) For a triaxial ellipsoid ( α ‰ β ‰ γ q , the nodal domain count is ˆ N , ˆ N , . . . ˆ N l . For example take l “ . Then the three sequences in Conjecture 42 are:
MALL PERTURBATIONS OF SPHERES 19 eigevalue index noda l do m a i n s l=2 l=3 l=4 l=5 l=6 l=7 triaxialoblateprolate Figure 2.
The number of nodal domains versus eigenvalue index for the three casesas given in Conjecture 42. Notice the fluctuations against Courant’s bound.(a) Oblate: N , . . . , N l “ , , , , , , , , . (b) Prolate: N l , . . . , N “ , , , , , , , , . (c) Triaxial: ˆ N , . . . , ˆ N l “ , , , , , , , , . Part of the motivation for this conjecture comes from the observed low multiplicities in thespectrum of our ellipsoids. Conjecture 42 states (and the reader can verify) that the sequence(a) corresponds to the number of nodal domains in Figure 1 (row 4) whereas the sequence (c)describes the number of nodal domains for rows 5 and 6. Note that any triaxial ellipsoid near thesphere is conjectured to have the same nodal sequence, regardless of the relative sizes of α, β, γ.
This is reflected in the fact the sequence ˆ N k is symmetric: ˆ N k “ ˆ N l ´ k . We verified this conjecturenumerically, for numerous values of α, β, γ and with l up to 5.In Figure 2 we plot the number of nodal domains as a function of the eigenvalue index for anarbitrary triaxial ellipsoid, with parameter l up to 7 . For large l, the maximum number of nodaldomains asymptotes to „ l (by taking m “ l { l (corresponding to m “ S . Levy [13] constructs high-frequency examples of spherical harmonics that obtain exactlytwo nodal domains. Eremenko-Jakobson-Nadirashvilli [9] construct harmonics that obtain variousprescribed topological configurations in their nodal structure. Both works use perturbation-typearguments.5.4. Numerics for large perturbations of spheres.
Figure 3 shows the numerically computedeigenvalues of a biaxial ellipsoid using the method described in Section 5.1. In this regime, we set a “ b from 0 . . The case of the sphere corresponds to the solid black vertical line b “
1, and as expected, multipleeigenvalues collide at this point with Λ “ l p l ` q . There are numerous eigenvalue crossings far awayfrom the sphere.With a “ b Ñ 8 , the ellipsoid takes a cigar-type shape. In this case, all ofthe eigenfunctions corresponding to mode m appear to asymptote to Λ „ m . Furthermore, theeigenfunctions appear to be “microlocalized” (that is, exhibiting its main oscillations) near thecenter of the cigar, as illustrated in Figure 4. It is an interesting open question to explain this“microlocalization” in this asymptotic regime.
Figure 3.
Λ graphed as a function of the axial parameter b with a “ c “ a ). The first ten eigenvalues for modes m “ , . . . , Figure 4.
Eigenfunctions on cigar-type ellipsoids; in the insert gives the axial pro-file of an eigenfunction with eigenvalue Λ “ . Figure 5.
Eigenfunctions on disk-type ellipsoids. The solid black lines indicate thezero curves of the plotted eigenfunctions.
MALL PERTURBATIONS OF SPHERES 21
With a “ b Ñ , the ellipsoid degenerates into a two-dimensional disk. In thiscase, the problem appears to degenerate into a union of the eigenvalues of a unit disk with eitherDirichlet or Neumann boundary conditions as illustrated in Figure 5. In this limit, the numericssuggest that the eigenvalues approach roots of either J m p? Λ q “ J m p? Λ q “ J m is theBessel function of order m. This behaviour is reminiscent of eigenvalue asymptotics in the presenceof degenerating metrics at least in the case of the “singular” manifold being boundaryless; see forinstance [21] and the references therein.6.
Appendix A: Calculation of (37)
We will use the notation v m p ϕ, θ q “ cos p mθ q P ml p cos ϕ q , w m p ϕ, θ q “ sin p mθ q P ml p cos ϕ q ,P ml p t q : “ A l,m Q ml p t q ,Q ml p t q : “ ` ´ t ˘ m { B m ` l B t m ` l ”` ´ t ˘ l ı (45)where A l,m is chosen so that ż π ż π v m p ϕ, θ q sin ϕdϕdθ “ . A couple of key integrals that appear in the computations are: J “ ż ´ ` ´ t ˘ l dt ; J “ ż ´ t ` ´ t ˘ l dt. (46)All of the quantities will be ultimately expressed in terms of their ratio: J J “ l ` . (47)We start with the following lemma. Lemma ş ´ t p P ml p t qq dt ş ´ ` P ml p t q ˘ dt “ ş ´ t p Q ml p t qq dt ş ´ ` Q ml p t q ˘ dt “ l ´ m ` l ´ p l ` q p l ´ q . (49) Proof.
This follows by successive integration by parts. We start with ş ´ p Q ml p t qq : ż ´ p Q ml p t qq “ ż ´ ` ´ t ˘ m B m ` l B t m ` l ”` ´ t ˘ l ı B m ` l B t m ` l ”` ´ t ˘ l ı dt “ ż ´ ˆ p l q ! p l ´ m q ! t m ` l ` . . . ˙ B m ` l B t m ` l ”` ´ t ˘ l ı dt (50) “ p l q ! p l ` m q ! p l ´ m q ! J (51)where J is as in (46). A similar computation yields ż ´ p Q ml p t qq t dt “ ż ´ ˆ t ` ´ t ˘ m B m ` l B t m ` l ”` ´ t ˘ l ı˙ B m ` l B t m ` l ”` ´ t ˘ l ı dt “ p l q ! p m ` l q ! p l ´ m q ! ˆ l ´ m ` l ´ p l ` q p l ´ q ˙ J , (52)where (47) was used.Equation (49) follows immediately. (cid:4) Derivation of (37a-d).
From (33) we obtain x v m , A v m y “ ş π ` P ml p cos ϕ q ˘ sin ϕdϕ ż π dϕ ˆ " g m p ϕ q P ml p cos ϕ q sin ϕ, m ‰ g p ϕ q P l p cos ϕ q sin ϕ ` g ´ p ϕ q P l p cos ϕ q sin ϕ, m “ x w m , A w m y “ ş π ` P ml p cos ϕ q ˘ sin ϕdϕ ż π dϕ ˆ " g m p ϕ q P ml p cos ϕ q sin ϕ, m ‰ g p ϕ q P l p cos ϕ q sin ϕ ´ g ´ p ϕ q P l p cos ϕ q sin ϕ, m “ g m p ϕ q “ (cid:32) A ` B cos ϕ ( P ml p cos ϕ q ` C sin p ϕ q cos p ϕ q B ϕ P ml p cos ϕ q where C “ p α ` β ´ γ q , A “ Cm ` γl p l ` q , B “ Cl p l ` q . Then ż π g m p ϕ q P ml p cos ϕ q sin ϕdϕ “ ż ´ dt ! p P ml p t qq ` A ` Bt ˘ ´ Ct ` ´ t ˘ P ml p t qB t P ml p t q ) “ ˆ A ` C ˙ ż ´ p P ml p t qq dt ` ˆ B ´ C ˙ ż ´ p P ml p t qq t dt This yields ş π g m p ϕ q P ml p cos ϕ q sin ϕdϕ ş π ` P ml p cos ϕ q ˘ sin ϕdϕ “ A ` C ` ˆ B ´ C ˙ ş ´ p P ml p t qq t dt ş ´ ` P ml p t q ˘ dt “ p α ` β ´ γ q l p l ` qp l ` q p l ´ q ` l ` m ` l ´ ˘ ` l p l ` q γ. (55)where we used Lemma 48.In the case when m “ , we also need to evaluate ş π g ´ p ϕ q P l p cos ϕ q sin ϕdϕ ş π p P l p cos ϕ q q sin ϕdϕ . We compute2 α ´ β ż π g ´ p ϕ q P l p cos ϕ q sin ϕdϕ “ ż ´ ` ` l p l ` q t ˘ ` P l p t q ˘ ´ t ` ´ t ˘ B t ` P l p t q ˘ P l p t q dt “ ż ´ ` P l p t q ˘ dt ` ˆ l p l ` q ´ ˙ ż ´ t ` P l p t q ˘ dt so that, using Lemma 48 we obtain ş π g ´ p ϕ q P l p cos ϕ q sin ϕdϕ ş π ` P l p cos ϕ q ˘ sin ϕdϕ “ p α ´ β q l p l ` q p l ` q p l ´ q . (56)Combining (53, 54, 55, 56) yields (37a-d). Derivation of (37e-f )
MALL PERTURBATIONS OF SPHERES 23
Note that x v m ´ , L v m y “ C m ´ ş π g m ´ p ϕ q P m ´ l p cos p ϕ qq sin ϕdϕ, where C m “ ş π cos p θm q dθ “ " π, m “ π, m ě . We further write ż π g m ´ p ϕ q P m ´ l p cos ϕ q sin ϕdϕ “ β ´ α A l,m A l,m ´ ˆ (cid:32) p l ` q lI ` ` ´ l p l ` q ´ m ˘ I ` m p m ´ q I ` I ( where I “ ż π Q ml p cos ϕ q Q m ´ l p cos ϕ q sin ϕ dϕ “ I “ ż π Q ml p cos ϕ q Q m ´ l p cos ϕ q sin ϕdϕ ; I “ ż π Q ml p cos ϕ q Q m ´ l p cos ϕ q sin ϕdϕ ; I “ ż π cos p ϕ q ` ´ sin ϕ ` p m ´ q ˘ B ϕ Q ml p cos ϕ q Q m ´ l p cos ϕ q dϕ C m A m,l “ ż π p Q ml p cos ϕ qq sin ϕdϕ All these integrals are all evaluated using successive integration parts, until they are expressedin terms ş ´ ` ´ t ˘ l dt and ş ´ t ` ´ t ˘ l dt. Skipping the details, we obtain I “ ż P ml P m ´ l sin ϕ dϕ “ ,I “ ´ p l q ! p m ´ ` l q ! p l ´ m q ! J I “ p l q ! p m ` l q ! p l ´ m q ! ˆ J ´ ˆ m ` p l ´ m q p l ´ m ´ q p l ´ q ˙ p m ` l q p m ` l ´ q J ˙ C m A m,l “ p l q ! p l ` m q ! p l ´ m q ! p l q ! p l ` m q ! p l ´ m q ! J After some algebra we then obtain x v m ´ , L v m y “ p β ´ α q c C m ´ C m l p l ` qp l ´ q p l ` q a p l ´ m ` qp l ´ m ` q p l ` m ´ q p l ´ m q which shows (37e). Formula (37f) is evaluated analogously.7. Acknowledgements
We are grateful to Tony Wong for his help with codes from [14] that were instrumental ingenerating numerics for the triaxial ellipsoid. Thanks also to Nilima Nigam, Iosif Polterovich,and Holger Dullin for their insights. SE and TK were supported by the NSERC Discovery Grantprogram during the writing of this article.
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