High-energy eigenfunctions of the Laplacian on the torus and the sphere with nodal sets of complicated topology
Alberto Enciso, Daniel Peralta-Salas, Francisco Torres de Lizaur
aa r X i v : . [ m a t h . A P ] O c t HIGH-ENERGY EIGENFUNCTIONS OF THE LAPLACIAN ONTHE TORUS AND THE SPHERE WITH NODAL SETS OFCOMPLICATED TOPOLOGY
A. ENCISO, D. PERALTA-SALAS, AND F. TORRES DE LIZAUR
Abstract.
Let Σ be an oriented compact hypersurface in the round sphere S n or in the flat torus T n , n >
3. In the case of the torus, Σ is further assumed tobe contained in a contractible subset of T n . We show that for any sufficientlylarge enough odd integer N there exists an eigenfunctions ψ of the Laplacianon S n or T n satisfying ∆ ψ = − λψ (with λ = N ( N + n −
1) or N on S n or T n , respectively), and with a connected component of the nodal set of ψ givenby Σ, up to an ambient diffeomorphism. Introduction
Let M be a closed manifold of dimension n > g . The Laplace eigenfunctions of M satisfy the equation∆ u k = − λ k u k , where 0 = λ < λ λ . . . are the eigenvalues of the Laplacian. The zero set u − k (0) is called the nodal set of the eigenfunction.The study of the nodal sets of the eigenfunctions of the Laplacian in a compactRiemannian manifold is a classical topic in geometric analysis with a number ofimportant open problems [18, 19]. When the Riemannian metric is not fixed, thenodal set is quite flexible. Indeed, it has been recently shown that [9], given aseparating hypersurface Σ in M , there is a metric g on the manifold for whichthe nodal set u − (0) of the first eigenfunction is precisely Σ. This result has beenextended to the class of metrics conformal to a metric g prescribed a priori [10],and to higher codimension submanifolds arising as the joint nodal set of severaleigenfunctions corresponding to a degenerate eigenvalue [5].For a fixed Riemannian metric, the problem is much more rigid than when onecan freely choose a metric adapted to the geometry of the hypersurface that oneaims to recover from the nodal set of the eigenfunctions. In this case, the techniquesdeveloped in [9, 5, 10] do not work. Nevertheless, since the Hausdorff measure of thenodal sets of the eigenfunctions grows as the eigenvalue tends to infinity [13, 14], oneexpects that the nodal set may become topologically complicated for high-energyeigenfunctions.Our goal in this paper is to establish the existence of high-energy eigenfunctionsof the Laplacian on the round sphere S n and the flat torus T n with nodal setsdiffeomorphic to a given submanifold. All along this paper, S n denotes the unitsphere in R n +1 and T n is the standard flat n-torus, ( R / π Z ) n . More precisely, our main theorem shows that for a sequence of high enougheigenvalues, there exist m eigenfunctions of the Laplacian on S n or T n with ajoint nodal set diffeomorphic to a given codimension m submanifold Σ. For theconstruction we need to assume that the normal bundle of Σ is trivial. This meansthat a small tubular neighborhood of the submanifold Σ must be diffeomorphicto Σ × R m . In the statement, structural stability means that any small enoughperturbation of the corresponding eigenfunction (in the C k norm with k >
1) stillhas a union of connected components of the nodal set that is diffeomorphic to thesubmanifold Σ under consideration. Throughout, diffeomorphisms are of class C ∞ and submanifolds are C ∞ and without boundary. Theorem 1.1.
Let Σ be a finite union of (disjoint, possibly knotted or linked)codimension m > compact submanifolds of S n or T n , n > , with trivial nor-mal bundle. In the case of the torus, we further assume that Σ is contained in acontractible subset. If m = 1 , we also assume that Σ is connected. Then for anylarge enough odd integer N there are m eigenfunctions ψ , . . . , ψ m of the Laplacianwith eigenvalue λ = N ( N + n − (in S n ) or λ = N (in T n ), and a diffeomor-phism Φ such that Φ(Σ) is the union of connected components of the joint nodal set ψ − (0) ∩ · · · ∩ ψ − m (0) . Furthermore, Φ(Σ) is structurally stable.
An important observation is that the proof of this theorem yields a reasonablycomplete understanding of the behavior of the diffeomorphism Φ, which is, in par-ticular, connected with the identity. Oversimplifying a little, the effect of Φ is touniformly rescale a contractible subset of the manifold that contains Σ to have adiameter of order 1 /N . In particular, the control that we have over the diffeomor-phism Φ allows us to prove an analog of this result for quotients of the sphere byfinite groups of isometries (lens spaces). Notice that Φ(Σ) is not guaranteed tocontain all the components of the nodal set of the eigenfunction.The proof of the main theorem involves an interplay between rigid and flexibleproperties of high-energy eigenfunctions of the Laplacian. Indeed, rigidity appearsbecause high-energy eigenfunctions in any Riemannian n -manifold behave, locallyin sets of diameter 1 / √ λ , as monochromatic waves in R n do in balls of diameter 1.We recall that a monochromatic wave is any solution to the Helmholtz equation∆ φ + φ = 0. The catch here is that, in general, one cannot check whether a givenmonochromatic wave in R n actually corresponds to a high-energy eigenfunction onthe compact manifold.To prove the converse implication, what we call an inverse localization theorem(see Sections 2 and 3), it is key to exploit some flexibility that arises in the problemas a consequence of the fact that large eigenvalues of the Laplacian in the torus orin the sphere have increasingly high multiplicities (for this reason the proof does notwork in a general Riemannian manifold). The inverse localization is a powerful toolto ensure that any monochromatic wave in a compact set of R n can be reproducedin a small ball of the manifold by a high-energy eigenfunction. This allows us totransfer any structurally stable nodal set that can be realized in Euclidean spaceto high-energy eigenfunctions on S n and T n . The inverse localization was firstintroduced in [11] to construct high-energy Beltrami fields on the torus and thesphere with topologically complicated vortex structures, and was also exploitedin [6, 7] to solve a problem of M. Berry [2] on knotted nodal lines of high-energy ODAL SETS OF HIGH-ENERGY EIGENFUNCTIONS 3 eigenfunctions of the harmonic oscillator and the hydrogen atom, and in [17] toanalyze the nodal sets of the eigenfunctions of the Dirac operator.One should notice that the techniques introduced in [8] to prove the existence ofsolutions to second-order elliptic PDEs in R n (including the monochromatic waves)with a prescribed nodal set Σ do not work for compact manifolds. The reason isthat the proof is based on the construction of a local solution in a neighborhoodof Σ, which is then approximated by a global solution in R n using a Runge-typeglobal approximation theorem. For compact manifolds the complement of the setΣ is precompact, so we cannot apply the global approximation theorem obtainedin [8]. In fact, as is well known, this is not just a technical issue, but a fundamentalobstruction in any approximation theorem of this sort. This invalidates the wholestrategy followed in [8] and makes it apparent that new tools are needed to provethe existence of Laplace eigenfunctions with geometrically complex nodal sets incompact manifolds.We finish this introduction with two corollaries. It is known that an orientedcodimension one or two submanifold in S n or T n has trivial normal bundle [15],therefore the main theorem implies the following: Corollary 1.2.
Let Σ be an oriented, compact, connected hypersurface in S n or T n , n > . In the case of the torus, we further assume that Σ is contained in acontractible subset. Then for any large enough odd integer N there is an eigen-function ψ of the Laplacian with eigenvalue λ = N ( N + n − (in S n ) or λ = N (in T n ), and a diffeomorphism Φ such that Φ(Σ) is a structurally stable connectedcomponent of the nodal set ψ − (0) . Corollary 1.3.
Let Σ be a finite union of (disjoint, possibly knotted or linked)codimension two compact submanifolds in S n or T n , n > . In the case of the torus,we further assume that Σ is contained in a contractible subset. Then for any largeenough odd integer N there is a complex-valued eigenfunction ψ of the Laplacianwith eigenvalue λ = N ( N + n − (in S n ) or λ = N (in T n ), and a diffeomorphism Φ such that Φ(Σ) is a union of structurally stable connected components of the nodalset ψ − (0) . The paper is organized as follows. In Sections 2 and 3 we prove an inverselocalization theorem for the eigenfunctions of the Laplacian on S n and T n , respec-tively. Theorem 1.1 is then proved in Section 4. Finally, in Section 5, we prove arefinement of the inverse localization Theorem on S n that allows us to approximateseveral given monochromatic waves by a single eigenfunction of the Laplacian indifferent small regions of S n .2. An inverse localization theorem on the sphere
In this section we prove an inverse localization theorem for eigenfunctions of theLaplacian on S n for n >
2. We recall that the eigenvalues of the Laplacian on the n -sphere are of the form N ( N + n − N is a nonnegative integer, and thecorresponding multiplicity is given by d ( N, n ) := (cid:18) N + n − N (cid:19) N + n − N + n − . A. ENCISO, D. PERALTA-SALAS, AND F. TORRES DE LIZAUR
For the precise statement of the theorem, let us fix an arbitrary point p ∈ S n and take a patch of normal geodesic coordinates Ψ : B → B centered at p . Hereand in what follows, B ρ (resp. B ρ ) denotes the ball in R n (resp. the geodesic ballin S n ) centered at the origin (resp. at p ) and of radius ρ , and we shall drop thesubscript when ρ = 1. For the ease of notation, we will use the R m -valued functions φ := ( φ , · · · , φ m ) and ψ := ( ψ , · · · , ψ m ), and the action of the Laplacian on suchfunctions is understood componentwise. Theorem 2.1.
Let φ be an R m -valued monochromatic wave in R n , satisfying ∆ φ + φ = 0 . Fix a positive integer r and a positive constant δ ′ . For any large enoughinteger N , there is an R m -valued eigenfunction ψ of the Laplacian on S n witheigenvalue N ( N + n − such that (cid:13)(cid:13)(cid:13)(cid:13) φ − ψ ◦ Ψ − (cid:16) · N (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) C r ( B ) δ ′ . To prove Theorem 2.1, we will proceed in two successive approximation steps.First, we will approximate the function φ in B by an R m -valued function ϕ thatcan be written as a finite sum of terms of the form c j | x − x j | n − J n − ( | x − x j | )with c j ∈ R m and x j ∈ R n , j = 1 , ..., N ′ , for N ′ large enough (Proposition 2.2below). Notice that any function of this form is a monochromatic wave. In thesecond step, we show that there is a collection of m eigenfunctions ( ψ , ..., ψ m ) =: ψ in S n with eigenvalue N ( N + n −
1) such that, when considered in a ball of radius N − , they approximate ϕ := ( ϕ , ..., ϕ m ) in the unit ball, provided that N is largeenough. Proposition 2.2.
Given any δ > , there is a constant R > and finitely manyconstant vectors { c j } N ′ j =1 ⊂ R m and points { x j } N ′ j =1 ⊂ B R such that the function ϕ := N ′ X j =1 c j | x − x j | n − J n − ( | x − x j | ) approximates the function φ in the unit ball as k φ − ϕ k C r ( B ) < δ . Proof.
It is more convenient to work with complex-valued functions, so we set e φ := φ + iφ . First, we notice that, since e φ is also a solution of the Helmholtzequation, it can be written in the ball B as an expansion(2.1) e φ = ∞ X l =0 d ( l,n − X k =1 b lk j l ( r ) Y lk ( ω ) , where r := | x | ∈ R + and ω := x/r ∈ S n − are spherical coordinates in R n , Y lk is a basis of spherical harmonics of eigenvalue l ( l + n − j l are n -dimensionalhyperspherical Bessel functions and b lk ∈ C m are constant coefficients. ODAL SETS OF HIGH-ENERGY EIGENFUNCTIONS 5
The series in (2.1) is convergent in the L sense, so for any δ ′ >
0, we cantruncate the sum at some integer L (2.2) φ := L X l =0 d ( l,n − X k =1 b lk j l ( r ) Y lk ( ω )so that it approximates e φ as(2.3) k φ − e φ k L ( B ) < δ ′ . The C m -valued function φ decays as | φ ( x ) | C/ | x | n − for large enough | x | (because of the decay properties of the spherical Bessel functions). Hence, Her-glotz’s theorem (see e.g. [12, Theorem 7.1.27]) ensures that we can write(2.4) φ ( x ) = Z S n − f ( ξ ) e ix · ξ dσ ( ξ ) , where dσ is the area measure on S n − := { ξ ∈ R n : | ξ | = 1 } and f is a C m -valuedfunction in L ( S n − ).We now choose a smooth C m -valued function f approximating f as k f − f k L ( S n − ) < δ ′ , which is always possible since smooth functions are dense in L ( S n − ). The functiondefined as the inverse Fourier transform of f ,(2.5) φ ( x ) := Z S n − f ( ξ ) e ix · ξ dσ ( ξ ) , approximates φ uniformly: by the Cauchy–Schwarz inequality, we get | φ ( x ) − φ ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12) Z S n − ( f ( ξ ) − f ( ξ )) e ix · ξ dσ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) C k f − f k L ( S n − ) < Cδ ′ (2.6)for any x ∈ R n .Our next objective is to approximate the function f by a trigonometric poly-nomial: for any given δ ′ , we will find a constant R >
0, finitely many points { x j } N ′ j =1 ⊂ B R and constants { c j } N ′ j =1 ⊂ C m such that the smooth function in R n f ( ξ ) := 1(2 π ) n N ′ X j =1 c j e − ix j · ξ , when restricted to the unit sphere, approximates f in the C norm,(2.7) k f − f k C ( S n − ) < δ ′ . In order to do so, we begin by extending f to a smooth function g : R n → C m with compact support, g ( ξ ) := χ ( | ξ | ) f (cid:18) ξ | ξ | (cid:19) , where χ ( s ) is a real-valued smooth bump function, being 1 when, for example, | s − | < , and vanishing for | s − | > . The Fourier transform b g of g is Schwartz, A. ENCISO, D. PERALTA-SALAS, AND F. TORRES DE LIZAUR so it is easy to see that, outside some ball B R , the L norm of b g is very small, Z R n \ B R | b g ( x ) | dx < δ ′ , and therefore we get a very good approximation of g by just considering its Fourierrepresentation with frequencies within the ball B R , that is,(2.8) sup ξ ∈ R n (cid:12)(cid:12)(cid:12)(cid:12) g ( ξ ) − Z B R b g ( x ) e − ix · ξ dx (cid:12)(cid:12)(cid:12)(cid:12) < δ ′ / . Next, let us show that we can approximate the integral Z B R b g ( x ) e − ix · ξ dx by the sum(2.9) f ( ξ ) := 1(2 π ) n N ′ X j =1 c j e − ix j · ξ with constants c j ∈ C m and points x j ∈ B R , so that we have the bound(2.10) sup ξ ∈ S n − (cid:12)(cid:12)(cid:12)(cid:12) Z B R b g ( x ) e − ix · ξ dx − f ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) < δ ′ / . Indeed, consider a covering of the ball B R by closed sets { U j } N ′ j =1 , with piecewisesmooth boundaries, pairwise disjoint interiors, and diameters not exceeding δ ′′ .Since the function e − ix · ξ b g ( x ) is smooth, we have that for each x, y ∈ U j sup ξ ∈ S n − (cid:12)(cid:12)b g ( x ) e − ix · ξ − b g ( y ) e − iy · ξ | < Cδ ′′ , with the constant C depending on b g (and therefore on δ ′ ) but not on δ ′′ . If x j isany point in U j and we set c j := (2 π ) n b g ( x j ) | U j | in (2.9), we getsup ξ ∈ S n − (cid:12)(cid:12)(cid:12)(cid:12) Z B R b g ( x ) e − ix · ξ dx − f ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) N ′ X j =1 Z U j sup ξ ∈ S n − (cid:12)(cid:12)b g ( x ) e − ix · ξ − b g ( x j ) e − ix j · ξ (cid:12)(cid:12) dx Cδ ′′ , with C depending on δ ′ and R but not on δ ′′ nor N ′ . By taking δ ′′ so that Cδ ′′ <δ ′ /
2, the estimate (2.10) follows.Now, in view of (2.8) and (2.10), one has k f − g k C ( S n − ) < δ ′ , so the estimate (2.7) follows upon noticing that the function f is the restrictionto S n − of the function g .To conclude, set e ϕ ( x ) := Z S n − f ( ξ ) e ix · ξ dσ ( ξ ) = N ′ X j =1 c j (2 π ) n Z S n − e i ( x − x j ) · ξ dσ ( ξ ) == N ′ X j =1 c j | x − x j | n − J n − ( | x − x j | ) , ODAL SETS OF HIGH-ENERGY EIGENFUNCTIONS 7 then from Equation (2.7) we infer that k e ϕ − φ k C ( R n ) Z S n − | f ( ξ ) − f ( ξ ) | dσ ( ξ ) < Cδ ′ , and from Equations (2.3) and (2.6) we get the L estimate(2.11) k e φ − e ϕ k L ( B ) C k e ϕ − φ k C ( R n ) + C k φ − φ k C ( R n ) ++ k φ − e φ k L ( B ) < Cδ ′ . Furthermore, both e ϕ and e φ are C m -valued functions satisfying the Helmholtz equa-tion in R n (note that the Fourier transform of e ϕ is supported on S n − ), so bystandard elliptic regularity estimates we have k e φ − e ϕ k C r ( B ) C k e φ − e ϕ k L ( B ) < Cδ ′ . This in particular implies that k φ − Re e ϕ k C r ( B ) < Cδ ′ , and taking δ ′ small enough so that Cδ ′ < δ , resetting c j := Re c j , and defining ϕ := Re e ϕ , the proposition follows. (cid:3) The second step consists in showing that, for any large enough integer N , wecan find an R m -valued eigenfunction ψ of the Laplacian on S n with eigenvalue N ( N + n −
1) that approximates, in the ball B /N , when appropriately rescaled,the function ϕ in the unit ball. The proof is based on asymptotic expansions ofultraspherical polynomials, and uses the representation of ϕ as a sum of shiftedBessel functions which we obtained in the previous proposition as a key ingredient.It is then straightforward that Theorem 2.1 follows from Propositions 2.2 and 2.3,provided that N is large enough and δ is chosen so that 2 δ < δ ′ . Proposition 2.3.
Given a constant δ > , for any large enough positive inte-ger N there is an R m -valued eigenfunction ψ of the Laplacian on S n with eigen-value N ( N + n − satisfying (cid:13)(cid:13)(cid:13)(cid:13) ϕ − ψ ◦ Ψ − (cid:16) · k (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) C r ( B ) < δ . Proof.
Consider the ultraspherical polynomial of dimension n + 1 and degree N , C nN ( t ), which is defined as(2.12) C nN ( t ) := Γ( N + 1)Γ( n )Γ( N + n ) P ( n − , n − N ( t ) , where Γ( t ) is the gamma function and P ( α, β ) N ( t ) are the Jacobi polynomials (seee.g [16, Chapter IV, Section 4.7]). We have included a normalizing factor so that C nN (1) = 1 for all N .Let p, q be two points in S n , considered as vectors in R n +1 with | p | = | q | = 1.The addition theorem for ultraspherical polynomials ensures that C nN ( p · q ) (where p · q denotes the scalar product in R n +1 of the vectors p and q ) can be written as(2.13) C nN ( p · q ) = 2 π n +12 Γ( n +12 ) 1 d ( N, n ) d ( N,n ) X k =1 Y Nk ( p ) Y Nk ( q ) , A. ENCISO, D. PERALTA-SALAS, AND F. TORRES DE LIZAUR with { Y Nk } d ( N,n ) k =1 being an arbitrary orthonormal basis of eigenfunctions of theLaplacian on S n (spherical harmonics) with eigenvalue N ( N + n − ϕ is written as the finite sum ϕ ( x ) = N ′ X j =1 c j | x − x j | n − J n − ( | x − x j | ) , with coefficients c j ∈ R m and points x j ∈ B R . With these c j and x j we define, forany point p ∈ S n , the function ψ ( p ) := N ′ X j =1 c j n − Γ( n ) C nN ( p · p j ) , where p j := Ψ − ( x j N ). As long as N > R , p j is well defined. In view of Equa-tion (2.13) it is clear that ψ is an R m -valued eigenfunction of the Laplacian on S n with eigenvalue N ( N + n − ψ . To beginwith, note that if we consider points p := Ψ − ( xN ) with N > R and x ∈ B R , wehave(2.14) p · p j = cos (cid:0) dist S n ( p, p j ) (cid:1) = cos (cid:18) | x − x j | + O ( N − ) N (cid:19) , as N → ∞ . The last equality comes from Ψ : B → B being a patch of nor-mal geodesic coordinates (by dist S n ( p, p j ) we mean the distance between p and p j considered on the sphere S n ). From now on we set(2.15) e ψ ( x ) := ψ ◦ Ψ − (cid:18) xN (cid:19) . When N is large, one hasΓ( N + 1)Γ( N + n ) = N − n + O ( N − n ) , so from Equation (2.14) we infer C nN ( p · p j ) = (cid:18) Γ (cid:16) n (cid:17) N − n + O ( N − n ) (cid:19) P ( n − , n − N (cid:18) cos (cid:18) | x − x j | + O ( N − ) N (cid:19)(cid:19) . (2.16)By virtue of Darboux’s formula for the Jacobi polynomials [16, Theorem 8.1.1], wehave the estimate1 N n − P ( n − , n − N (cid:16) cos tN (cid:17) = 2 n − J n − ( t ) t n − + O ( N − ) , uniformly in compact sets (e.g., for | t | R ). Hence, in view of Equation (2.16),the function e ψ can be written as e ψ ( x ) = N ′ X j =1 c j n − Γ( n ) C nN (cid:16) cos (cid:16) | x − x j | + O ( N − ) N (cid:17)(cid:17) = N ′ X j =1 c j | x − x j | n − J n − ( | x − x j | ) + O ( N − ) , ODAL SETS OF HIGH-ENERGY EIGENFUNCTIONS 9 for N big enough and x, x j ∈ B R . From this we get the uniform bound(2.17) k ϕ − e ψ k C ( B ) < δ ′ for any δ ′ > N large enough.It remains to promote this bound to a C r estimate. For this, note that, sincethe eigenfunction ψ has eigenvalue N ( N + n − e ψ verifieson B the equation ∆ e ψ + e ψ = 1 N A e ψ , with A e ψ := − ( n − e ψ + G ∂ e ψ + G ∂ e ψ , where ∂ k e ψ is a matrix whose entries are k -th order derivatives of e ψ , and G k ( x, N )are smooth matrix-valued functions with uniformly bounded derivatives, i.e.,(2.18) sup x ∈ B | ∂ αx G k ( x, N ) | C α , with constants C α independent of N .Since ϕ satisfies the Helmholtz equation ∆ ϕ + ϕ = 0 , the difference ϕ − e ψ satisfies∆( ϕ − e ψ ) + ( ϕ − e ψ ) = 1 N A e ψ , and, considering the estimates (2.17) and (2.18), by standard elliptic estimates weget k ϕ − e ψ k C r,α ( B ) < C k ϕ − e ψ k C ( B ) + CN k A e ψ k C r − ,α ( B ) < Cδ ′ + CN k ϕ − e ψ k C r,α ( B ) + CN k ϕ k C r,α ( B ) , so we conclude that, for N big enough and δ ′ small enough, k ϕ − e ψ k C r ( B ) Cδ ′ + C k ϕ k C r,α N < δ .
The proposition then follows. (cid:3) An inverse localization theorem on the torus
In this section we prove an inverse localization theorem for eigenfunctions of theLaplacian on T n for n >
3. We recall that the eigenvalues of the Laplacian on the n -torus are the integers of the form λ = | k | for some k ∈ Z n . In particular, the spectrum of the Laplacian in T n contains theset of the squares of integers.As in the previous section, we fix an arbitrary point p ∈ T n and take a patchof normal geodesic coordinates Ψ : B → B centered at p . Theorem 3.1.
Let φ be an R m -valued function in R n , satisfying ∆ φ + φ = 0 . Fixa positive integer r and a positive constant δ ′ . For any large enough odd integer N ,there is an R m -valued eigenfunction ψ of the Laplacian on T n with eigenvalue N such that (cid:13)(cid:13)(cid:13)(cid:13) φ − ψ ◦ Ψ − (cid:16) · N (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) C r ( B ) δ ′ . Proof.
Arguing as in the proof of Proposition 2.2 we can readily show that for any δ >
0, there exists an R m -valued function φ on R n that approximates the function φ in the ball B as(3.1) k φ − φ k C ( B ) < δ , and that can be represented as the Fourier transform of a distribution supportedon the unit sphere of the form φ ( x ) = Z S n − f ( ξ ) e iξ · x dσ ( ξ ) . Again S n − denotes the unit sphere { ξ ∈ R n : | ξ | = 1 } and f is a smooth C m -valuedfunction on S n − satisfying f ( ξ ) = ¯ f ( − ξ ).Let us now cover the sphere S n − by finitely many closed sets { U k } N ′ k =1 withpiecewise smooth boundaries and pairwise disjoint interiors such that the diameterof each set is at most ǫ . We can then repeat the argument used in the proof ofProposition 2.2 to infer that, if ξ k is any point in U k and we set c k := f ( ξ k ) | U k | , the function e ψ ( x ) := N ′ X k =1 c k e iξ k · x approximates the function φ uniformly with an error proportional to ǫ : k e ψ − φ k C ( B ) < Cǫ . The constant C depends on δ but not on ǫ nor N ′ , so one can choose the maximaldiameter ǫ small enough so that(3.2) k e ψ − φ k C ( B ) < δ . In turn, the uniform estimate k e ψ − φ k C ( B ) k e ψ − φ k C ( B ) + k φ − φ k C ( B ) < δ can be readily promoted to the C r bound(3.3) k e ψ − φ k C r ( B ) < Cδ . This follows from standard elliptic estimates as both e ψ (whose Fourier transformis supported on S n − ) and φ satisfy the Helmholtz equation:∆ e ψ + e ψ = 0 , ∆ φ + φ = 0 . Furthermore, replacing e ψ by its real part if necessary, we can safely assume thatthe function e ψ is R m -valued.Let us now observe that for any large enough odd integer N one can choosethe points ξ k ∈ U k ⊂ S n − so that they have rational components (i.e., ξ k ∈ Q n )and the rescalings N ξ k are integer vectors (i.e., N ξ k ∈ Z n ). This is because for n >
3, rational points ξ ∈ S n − ∩ Q n of height N (and so with N ξ ∈ Z n ) areuniformly distributed on the unit sphere as N → ∞ through odd values [4] (in fact,the requirement for N to be odd can be dropped for n > ξ k as above, we are now ready to prove the inverse localization theoremin the torus. Without loss of generality, we will take the origin as the base point ODAL SETS OF HIGH-ENERGY EIGENFUNCTIONS 11 p , so that we can identify the ball B with B through the canonical 2 π -periodiccoordinates on the torus. In particular, the diffeomorphism Ψ : B → B that appearsin the statement of the theorem can be understood to be the identity.Since N ξ k ∈ Z n , it follows that the function ψ ( x ) := N ′ X k =1 c k e iNξ k · x is 2 π -periodic (that is, invariant under the translation x → x + 2 π a for any vector a ∈ Z n ). Therefore it defines a well-defined function on the torus, which we willstill denote by ψ .Since the Fourier transform of ψ is now supported on the sphere of radius N , ψ is an eigenfunction of the Laplacian on the torus T n with eigenvalue N ,∆ ψ + N ψ = 0 . The theorem then follows provided that δ is chosen small enough for Cδ < δ ′ . (cid:3) We conclude this section noticing that the statement of Theorem 3.1 does nothold for T . The reason is that rational points ξ ∈ S ∩ Q with N ξ ∈ Z are nolonger uniformly distributed on the unit circle (not even dense) as N → ∞ throughany sequence of odd values, counterexamples can be found in [3]. Nevertheless, aslightly different statement can be proved using [3]: Theorem 3.2.
Let φ be an R m -valued function in R , satisfying ∆ φ + φ = 0 .Fix a positive integer r and a positive constant δ ′ . Then there exists a sequence ofintegers { N l } ∞ l =1 ր ∞ , and R m -valued eigenfunctions ψ l of the Laplacian on T with eigenvalues N l such that (cid:13)(cid:13)(cid:13)(cid:13) φ − ψ l ◦ Ψ − (cid:16) · N l (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) C r ( B ) δ ′ for l large enough. Proof of the main theorem
For the ease of notation, we shall write M n to denote either T n or S n . LetΦ ′ be a diffeomorphism of M n mapping the codimension m submanifold Σ intothe ball B / ⊂ M n , and the ball B / into itself. In S n , the existence of such adiffeomorphism is trivial, while in the case of T n it follows from the assumptionthat Σ is contained in a contractible set.Consider the submanifold Σ ′ in B / ⊂ R n defined as Φ ′ (Σ) in the path of normalgeodesic coordinates: Σ ′ := (Ψ ◦ Φ ′ )(Σ) . It is shown in [8, Theorem 1.3] if m > m = 1, that thereis an R m -valued monochromatic wave φ = ( φ , · · · , φ m ), satisfying ∆ φ + φ = 0 in R n , and a diffeomorphism Φ (close to the identity, and different from the identityonly on B / ) such that Φ (Σ ′ ) ⊂ B / is a union of connected components of thejoint nodal set φ − (0) ∩ ... ∩ φ − m (0). In addition, the construction in [8] ensuresthat the regularity condition rk ( ∇ φ , · · · , ∇ φ m ) = m holds at any point of Φ (Σ ′ ),so it is a structurally stable nodal set of φ by Thom’s isotopy theorem [1]. Now, the inverse localization theorem (Theorem 2.1 in the case of S n and Theo-rem 3.1 for T n ) allows us to find, for any large enough odd integer N , an R m -valuedfunction ψ = ( ψ , · · · , ψ m ) in M n satisfying ∆ ψ = − λψ (with λ := N ( N + n − λ := N in the sphere or the torus, respectively) and such that ψ ◦ Ψ − ( · N )approximates φ in the C r ( B ) norm as much as we want.The structural stability property ensures the existence of a second diffeomor-phism Φ : R n → R n close to the identity, and different from the identity onlyon B / , such that Φ (Φ (Σ ′ )) is a union of connected components of the jointnodal set of the R m -valued function ψ ◦ Ψ − ( · N ). Therefore, the correspondingsubmanifold Φ(Σ) := Ψ − (cid:16) N Φ (Φ ((Ψ ◦ Φ ′ )(Σ))) (cid:17) is a union of connected components of the nodal set of ψ . The map Φ : M n → B N thus defined is easily extended to a diffeomorphism of the whole manifold M n .Finally, we have by the construction that Φ(Σ) is structurally stable, and henceTheorem 1.1 follows.5. Final remark: inverse localization on the sphere in multipleregions
Theorem 2.1 in Section 2 can be refined to include inverse localization at differentpoints of the sphere. This way, we get an eigenfunction of the Laplacian thatapproximates several given solutions of the Helmholtz equation in different regions.The fast decay of ultraspherical polynomials of high degree outside the domainswhere they behave as shifted Bessel functions is behind this multiple localization.Notice that, in contrast, trigonometric polynomials do not exhibit this decay, hencethe lack of an analog of the following result in the case of the torus. All along thissection we assume that n > { p α } N ′ α =1 be a set of points in S n , with N ′ an arbitrarily large (but fixedthroughout) integer. We denote by Ψ α : B ρ ( p α ) → B ρ the corresponding geodesicpatches on balls of radius ρ centered at the points p α . We fix a radius ρ such thatno two balls intersect, for example by setting ρ := 12 min α = β dist S n ( p α , p β ) . We further choose the points { p α } N ′ α =1 so that no pair of points are antipodal in S n ⊂ R n +1 , i.e. p α = − p β for all α , β . The reason is that the eigenfunctions of theLaplacian on the sphere with eigenvalue N ( N + n −
1) have parity ( − N : ψ ( p α ) = ( − N ψ ( − p α )(they are the restriction to the sphere of homogenous harmonic polynomials ofdegree N ); so that prescribing the behavior of an eigenfunction in a ball aroundthe point p α automatically determines its behavior in the antipodal ball. Proposition 5.1.
Let { φ α } N ′ α =1 be a set of N ′ R m -valued monochromatic waves in R n , m n , satisfying ∆ φ α + φ α = 0 . Fix a positive integer r and a positiveconstant δ . For any large enough integer N , there is an R m -valued eigenfunction ODAL SETS OF HIGH-ENERGY EIGENFUNCTIONS 13 ψ of the Laplacian on S n with eigenvalue N ( N + n − such that (cid:13)(cid:13)(cid:13)(cid:13) φ α − ψ ◦ Ψ − α (cid:16) · N (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) C r ( B ) < δ for all α N ′ .Proof. We use the notation introduced in the proof of Proposition 2.3 withoutfurther mention. Applying Theorem 2.1 to each φ α we obtain, for high enough N , R m -valued eigenfunctions of the Laplacian { ψ α } N ′ α =1 satisfying the bound (cid:13)(cid:13)(cid:13)(cid:13) φ α − ψ α ◦ Ψ − α (cid:16) · N (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) C r ( B ) < δ ′ . For each α , the R m -valued eigenfunction ψ α ( p ) is a linear combination (with coef-ficients in R m ) of ultraspherical polynomials C nN ( p · q j ), where { q j } is a finite setof points such that dist S n ( p α , q j ) is proportional to N − , for all j . Recall that theultraspherical polynomials satisfy the asymptotic formula C nN ( p · q ) = Γ( n ) N n − P ( n − , n − N (cos(dist S n ( p, q ))) + O ( N − n ) , so considering the fact that the Jacobi polynomials behave as (see [16, Theorem7.32.2]) N − n P ( n − , n − N (cos t ) = O ( N − ) t , uniformly for N − < t < π − N − , we can conclude that the functions C nN ( p · q j )are uniformly bounded as | C nN ( p · q j ) | C ρ N for any point p satisfyingmin j dist S n ( p, q j ) > ρ and min j dist S n ( p, − q j ) > ρ , and where C ρ is a constant depending only on ρ . The same decay is thus alsoexhibited by the eigenfunction ψ α , k ψ α k C ( S n \ ( B ( p α ,ρ ) ∪ B ( − p α ,ρ )) CN since it is just a normalized linear combination of ultraspherical polynomials (herethe constant C depends on ρ and on the particular coefficients in the expansion of ψ α , that is, on φ α and δ ′ ).Now, if we define the R m -valued eigenfunction ψ := N ′ X α =1 ψ α and we choose N large enough, the statement of the proposition follows for r = 0.By standard elliptic estimates, the C bound can be easily promoted to a C r bound,so we are done. (cid:3) References
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Instituto de Ciencias Matem´aticas, Consejo Superior de Investigaciones Cient´ıficas,28049 Madrid, Spain
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