Eigenfunctions and Nodal Sets

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Also , /?jv satisfies the osculating equation at the origin Since we are using normal coordinates, the osculating equation is the usual Laplace equation in Euclidean space, le,

Thus , Pf^ is a spherical harmonic of degree N

If TV = 1, /7^v(x) IS a linear polynomial and this shows that df{0)¥'0, then the nodal set around 0 is a very nice piece of C°° manifold

When N>\, the situation is more complicated We shall extend the method of T С Kuo [5] in Lemma 2 4 to prove 1ЫХ/{х) = р^^{Ф(х)) where Ф is a C^ diifeo- morphism between two small neighborhoods of OeR" and Ф(0) = 0 Thus, the nodal set of/around the origin is C^ diifeomorphic to the nodal set of a spherical harmonic around the origin However, there is not much information about nodal sets of sphereical harmonics The following simple observation will be useful

LEMMA 2 3 Suppose that p^ is a spherical harmonic of degree N N>\ Then, the nodal set of p^ around the origin has a singularity at 0

Proof Notice that if S" ^ is the sphere of radius 1 in R" then pj^\sn i is an eigen- function of 5""^ Since N>\,pj^\sn i is not the 1-set eigenfunction Pn\s^ i must have zeros on S"~^ and the homogeneity of Pf^ shows that if xeS"~^ with pj^{x) = 0 then /7jv (/x) = 0 for allr >0 The only case where the nodal sets ofp^ around the origin is a smooth manifold is when the nodal set of/7jv|s" i ii^s on a great circle of S" ^ Since great circles are nodal sets of 1-st eigenfunctions on S"~^ and Л^> I, the assertion of the lemma is immediately seen to be true

We now prove the theorem by induction on the dimension n

If «=1, It IS trivial

Suppose that it is true for « — 1

We now prove it for n

We shall show that the nodal set of/around the origin is C^ diifeomorphic to the nodal set of a spherical harmonic pj^ of degree N around the origin in R" ever, the nodal set of pj^ around the origin is equal to {tx t>0, p^\sn i(x) = 0} Remember that/^^js« i ^s an eigenfunction on the (/i—l)-dim sphere S"~^ Our ductive assumption then applies and shows that Theorem 2 2 is true for the nodal set of/7jv Now recall that we have the relation/(х) = /?^(Ф(х)), where Ф is a C^ dif- feomorphism keeping the origin fixed Suppose that p^ ^ (0)\я = Mq around the origin, where я is a closed set of lower dimension and Mq is an (n— l)-dimC°° fold ТЬеп/-1(0)\Ф"Ч7г) = ф-^(Мо) Thus ф-^^о) IS a C^ manifold We now want to show further that Ф'^(Мо) is C°° Indeed, let уеФ~^{Мо) Then/(7) = 0, and Ф{у)еМс) Apply our previous argument to a small neighborhood of y, we have