WEYL’S LAW

MATT STEVENSON

Abstract. These are notes for a talk given in the Student Analysis Seminar at the University of Michigan. The n Laplacian on a bounded domain in R has a discrete set of Dirichlet eigenvalues, accumulating only at ∞. Let N(λ) be the number of eigenvalues less than λ, then Weyl’s law asserts that the first term of the asymptotic expansion of N(λ) depends only on the dimension and the volume of the domain. Here we sketch the 1912 proof of Weyl, following the expositions of [1], [5], [6].

1. Introduction Let Ω ⊂ Rn be a bounded domain, then we are interested in solving the eigenvalue problem ∆ϕ = λϕ on Ω (with either Dirichlet or Neumann boundary conditions). The asserts that there exists an 2 orthonormal basis {un} of L (Ω) consisting of (Dirichlet or Neumann) eigenfunctions, where the corresponding eigenvalues λ1 ≤ ... ≤ λn ≤ ... → +∞ accumulate only at +∞ and each have finite multiplicity. Let N(λ) := #{λj ≤ λ} be the eigenvalue counting function, then it is natural to consider the asymptotic behaviour of N(λ) as λ → +∞. Consider the following simple examples. Example. Let Ω = [0, a], then the Dirichlet eigenvalue problem becomes the following:

( d2 dx2 ϕ(x) = λϕ(x), ϕ(0) = ϕ(a) = 0.

kπ  Solving the , the Dirichlet eigenfunctions are ϕk(x) = sin a x with corresponding kπ 2 Dirichlet eigenvalues λk = a , for k ∈ N. Consider the eigenvalue counting function: ( ) ( √ ) kπ 2 a λ a √ N(λ) = # {k ∈ : λ < λ} = # k ∈ : < λ = max k ∈ : k < ∼ λ. N k N a N π π

Example. Let Ω = [0, a] × [0, b], then using separation of variables we can deduce that the Dirichlet eigenfunc- jπ  kπ  jπ 2 kπ 2 tions of the Laplacian are ϕj,k(x, y) = sin a x sin b y with eigenvalues λj,k = a + b , for j, k ∈ N. The eigenvalue counting function is then given by ( ) jπ 2 kπ 2 N(λ) = #{(j, k) ∈ × : λ < λ} = # (j, k) ∈ × : + < λ . N N j,k N N a b

Let E = {(x, y) ∈ 2 :( √ x )2 + ( √ y )2 ≤ 1, x ≥ 0, y ≥ 0} be the area in the first quadrant cut out by the λ R λa/π λb/π ellipse, then N(λ) is the number of integer lattices points in Eλ. To each pair (j, k) ∈ N × N such that λj,k < λ, we can associate the unit-area square [j − 1, j] × [k − 1, k]. Then, √ √ X π λa λb area(Ω) N(λ) = area([j − 1, j] × [k − 1, k]) ≤ area(E ) = = λ. λ 4 π π 4π (j,k)∈N×N : λj,k<λ

Date: October 31, 2014. 1 2 MATT STEVENSON

0 To get a lower bound on N(λ), translate Eλ down by 1 unit and to the left by 1 unit to get the set Eλ. As before, we compute areas to find that √ √ √ √ π λa λb λa λb area(Ω) perimeter(Ω)√ N(λ) ≥ − − = λ − λ. 4 π π π π 2π 2π area(Ω) Therefore, we may conclude that the eigenvalue counting function is asymptotically given by N(λ) ∼ 4π λ. These results can be generalized to give the first term of the asymptotic expansion of the eigenvalue counting function in terms of the dimension and the volume of the domain. Lorentz first conjectured this result at a conference at G¨ottingenin 1910, whence Hilbert predicted it would not be solved in his lifetime. Weyl, as a graduate student, solved it 4 months later. The modern formulation of this result is the following. Theorem. (Weyl, 1911) Let N(λ) be the Dirichlet eigenvalue counting function on a bounded domain Ω, then

n/2 N(λ) ∼ cnvol(Ω)λ

ωn n where cn = (2π)n is a constant depending only on the dimension n and ωn is the volume of the unit ball in R . Remark. The proof of Weyl’s law for a 2-dimensional rectangle generalizes to n-dimensional rectangles.

2. Proof of Weyl’s Law for Bounded Domains When analyzing the solutions to PDEs, it is more natural to work in a rather than an Lp-space. Recall that H1(Ω) is the closed subspace of L2(Ω) consisting of weakly-differentiable functions; it carries an inner product Z hu, viH1 := ∇u · ∇v + uv, Ω which makes H1(Ω) into a . Remark that the space D(Ω) of compactly-supported smooth functions 1 1 1 on Ω is contained in H (Ω); their completion with respect to the H -norm is denoted H0 (Ω), and it is the closed subspace of H1(Ω) consisting of trace-zero functions. These two spaces are the appropriate ones in which to frame our eigenvalue problem. Let u ∈ C2(Ω)∩C(Ω) be a Dirichlet eigenfunction of −∆ with eigenvalue λ, then for any v ∈ D(Ω) integration by parts gives that Z Z Z Z ∇u · ∇v = u∂ν v + (−∆u)v = λuv. Ω ∂Ω Ω Ω 1 We say that u ∈ H0 (Ω) is a weak solution of the Dirichlet eigenvalue problem if it satisfies the above equality 1 1 for any v ∈ H0 Ω. Similarly, u ∈ H (Ω) is a weak solution of the Neumann eigenvalue problem if it satisfies 1 1 1 the above equality for all v ∈ H (Ω). Let V denote either H (Ω) or H0 (Ω) and let {λn} denote Neumann or Dirichlet eigenvalues, as appropriate. The spectral theorem applied to the resolvent of the Laplacian asserts that there exists an orthonormal basis 2 {un} of L (Ω) consisting of eigenfunctions of −∆, where the eigenvalues {λn} each have finite multiplicity and accumulate only at +∞. Moreover, the eigenfunctions {un} form an orthonormal basis of V with respect to the H1-inner product. There is a variational characterization of these eigenvalues, called the Minimax Principle.

Lemma. (Minimax Principle, Version 1) Let V and {λn} be as above. Then,

λn = inf ρ(u) = sup ρ(u), u∈H n−1 u∈span(u1,...,un)

k∇uk2 ⊥ L2 where Hn−1 = V ∩ span(u1, . . . , un−1) and ρ(u) := kuk2 is the Rayleigh quotient. L2 WEYL’S LAW 3

Proof. For any u ∈ V , write ∞ ∞ 2 X X 2 kukL2 = hu, ui = h hu, uniun, ui = |hun, ui| , n=1 n=1 using the fact that hun, uni = 1. Then, the norm of the gradient can be expressed as Z Z ∞ ∞ 2 X X 2 k∇ukL2 = ∇u · ∇u ≈ −∆u · u = h−∆ hu, uniun, ui = λn|hu, uni| , Ω Ω n=1 n=1 where the second equality is not entirely justified, as we do not know a priori that u is twice weakly-differentiable. If u is now taken in Hn−1, then hu, uki = 0 for k = 0, . . . , n − 1. Consequently, ∞ ∞ ∞ 2 X 2 X 2 X 2 2 k∇ukL2 = λk|hu, uki| ≥ λn |hu, uki| = λn |hu, uki| = λnkukL2 , k=n k=n k=1 where the second-to-last equality follows by adding back the zero terms. Rearranging, for any u ∈ Hn−1, we have the inequality ρ(u) ≥ λn, and hence the inequality holds for the infimum over u ∈ Hn−1. Furthermore, un ∈ Hn−1 and ρ(un) = λn, so the infimum is attained by a function of Hn−1, so we have equality.  Lemma. (Minimax Principle, Version 2)

λn = inf sup ρ(u), X∈Φn(V ) u∈X where Φn(V ) = {n-dimensional linear subspaces X ⊂ V }.

Proof. As span(u1, . . . , un) ∈ Φn(V ), version 1 of the Minimax Principle gives the inequality

λn = sup ρ(u) ≥ inf sup ρ(u). X∈Φ (V ) u∈span(u1,...,un) n u∈X

Conversely, for any X ∈ Φn(V ), there is nonzero v ∈ X ∩ Hn−1 (by dimension count); hence,

ρ(v) ≥ inf ρ(u) = λn =⇒ sup ρ(v) ≥ λn =⇒ inf sup ρ(v) ≥ λn. u∈Hn−1 v∈X X∈Φn(V ) v∈X 

Lemma. Let µn be the Dirichlet eigenvalues on Ω and let νn be the Neumann eigenvalues on Ω, then νn ≤ µn.

1 1 1 1 Proof. As H0 (Ω) ⊂ H (Ω), it follows that Φn(H0 (Ω)) ⊂ Φn(H (Ω)). Then, version 2 of the Minimax Principle says that νn is given by minimizing the same expression as for µn over a larger space, and hence νn ≤ µn. 

Lemma. (Domain Monotonicity) If Ω1 ⊂ Ω2, then µn(Ω2) ≤ µn(Ω1).

1 1 1 1 Proof. Remark that H0 (Ω1) ⊂ H0 (Ω), so Φn(H0 (Ω1)) ⊂ Φn(H (Ω)). Again applying version 2 of the Minimax Principle, we conclude that µn(Ω2) ≤ µn(Ω1).  Proof. (Of Weyl’s Law) The proof is in 2 steps: first, we assume that the domain is a union of finitely-many cubes, and we know Weyl’s law holds on each cube by our earlier considerations. Secondly, we approximate the domain by from the interior and from the exterior by unions of finitely-many cubes.

Step 1: Assume the bounded domain Ω is a finite union of almost-disjoint cubes {Qi}i. Let {µk(Qi)}k,i and {νk(Qi)}k,i be the collections of all Dirichlet and Neumann eigenvalues on all of the cubes, respectively. Order these sets to get all of the Dirichlet eigenvalues {µ˜k}k and all of the Neumann eigenvalues {ν˜k}k, i.e. so thatµ ˜1 ≤ µ˜2 ≤ ... andν ˜1 ≤ ν˜2 ≤ .... 4 MATT STEVENSON

1 2 1 1 2 1 Define Hf0 (Ω) = {u ∈ L (Ω): u|Qi ∈ H0 (Qi) for all i} and Hf (Ω) = {u ∈ L (Ω): u|Qi ∈ H (Qi) for all i}, then there is the chain of inclusions 1 1 1 ˜1 Hf0 (Ω) ⊂ H0 (Ω) ⊂ H (Ω) ⊂ H (Ω).

As in the Minimax Principles, there is a variational characterization of theµ ˜k’s andν ˜k’s as

µ˜k = inf sup ρ(u), ˜1 X∈Φk(H0 ) u∈X and ν˜k = inf sup ρ(u). ˜1 e X∈Φk(H ) u∈X P 2  2 Here,ρ ˜(u) = k∇uk 2 /kuk 2 is a modified Rayleigh quotient. Remark thatρ ˜ 1 = ρ. Again, as i L (Qi) L (Ω) H (Ω) we are minimizing over larger and larger sets, we conclude that

νek ≤ νk ≤ µk ≤ µfk =⇒ Nµ˜(λ) ≤ Nµ(λ) ≤ Nν (λ) ≤ Nν˜(λ). n/2 n The earlier considerations give Weyl’s law on the rectangles: Nµ,i(λ) = cnvol(Qi)λ + o(λ 2 ), so we can write X n/2 X n n/2 n Nµ˜(λ) = Nµ,i(λ) = cnλ vol(Qi) + o(λ 2 ) = cnvol(Ω)λ + o(λ 2 ). i i n/2 n Similarly, we can write Nν˜(λ) = cnvol(Ω)λ + o(λ 2 ). Combining these two results, we obtain Weyl’s law for the union of finitely-many rectangles: n/2 n n/2 n cnvol(Ω)λ + o(λ 2 ) ≤ Nµ(λ) ≤ Nν (λ) ≤ cnvol(Ω)λ + o(λ 2 ). n Step 2: Let Ω ⊂ R be any bounded domain. Given any  > 0, there exists Ω1, Ω2 finite unions of rectangles such that Ω1 ⊂ Ω ⊂ Ω2 and vol(Ω2\Ω1) < . Domain monotonicity implies that µn(Ω2) ≤ µn(Ω) ≤ µn(Ω1), and hence Nµ,Ω1 (λ) ≤ Nµ,Ω(λ) ≤ Nµ,Ω2 (λ). Consider the following: N (λ) N (λ) lim sup µ,Ω ≤ lim sup µ,Ω2 = c vol(Ω ) ≤ c (vol(Ω) + ) n/2 n/2 n 2 n λ→∞ λ λ→∞ λ Similarly,

Nµ,Ω(λ) Nµ,Ω1 (λ) lim inf ≥ lim inf = cnvol(Ω1) ≥ cn(vol(Ω) − ). λ→∞ λn/2 λ→∞ λn/2 Nµ,Ω(λ) As  > 0 is arbitrary, it follows that limλ→∞ λn/2 = cnvol(Ω).  3. Generalizations Conjecture 1. (Weyl Conjecture) Let Ω ⊂ Rn be a bounded domain with smooth boundary, then n/2 n−1 n−1 N(λ) = cnvol(Ω)λ + cn−1area(∂Ω)λ 2 + o(λ 2 ), where cn, cn−1 are constants depending only on the dimension n. This conjecture was shown by Victor Ivrii in 1982, subject to the conjecture that the set of periodic billiards has measure zero in a bounded domain with smooth boundary.

References [1] Y. Canzani, Analysis on via the Laplacian. Course notes for Harvard Math 253, (2013). [2] I. Chavel, Eigenvalues in Riemannian Geometry. Volume 115, Academic Press, 2nd Edition (1987). [3] H. Iwaniec, Spectral Methods of Automorphic Forms. AMS Volume 53, (2000). [4] W. Muller¨ , Weyl’s Law in the Theory of Automorphic Forms. Groups and Analysis, The Legacy of Hermann Weyl, Cambridge Univ. Press., (2008), pp. 133 -163. [5] W. Strauss, Partial Differential Equations: An Introduction. Wiley, 2nd Edition (2007). [6] G. Tsogtgerel, Spectral Properties of the Laplacian on Bounded Domains. Course Notes for McGill Math 580 (2013).