Dirichlet Eigenfunctions of the Square Membrane: Courant’s Property, and A. Stern’s and Å. Pleijel’s Analyses

On the number of Courant-sharp Dirichlet eigenvalues

“…This compares well with the bound 7.1·10 6 obtained in [1]. For the unit square C 2 it is known ( [13], [2]) that only the first, second and fourth Dirichlet eigenvalues are Courant-sharp. Hence C(C 2 ) = 3, and the largest Courant-sharp eigenvalue for C 2 is equal to 8π 2 .…”

Non-universality of the Nazarov–Sodin constant

“…Note that the result of Proposition 3.2 is in the same spirit as known constructions of (deterministic) eigenfunctions of arbitrarily high energy with few or bounded number of nodal components that arise in eigenspaces with a spectral measure given by the Cilleruelo measure (see the recent manuscript [1]). Proposition 3.2 can be proved by either considering an explicit construction of a random field f with the given spectral measure ν 0 and noting that for this model there are a.s. no compact nodal components, or, alternatively, by a local computation, e.g.…”

On the Number of Nodal Domains of Toral Eigenfunctions

“…Concerning a lower bound for N fa , there exist Laplace eigenfunctions of arbitrarily large energy E with only 2 nodal domains (at least on the square); hence there is no nontrivial lower bound for their number (this result goes back to Stern [12] although we refer the reader to [1,Theorem 4.1] and the discussion that follows it; the analogous result on the sphere is given by Lewy [8,Theorems 1 and 2]). Moreover, there exist [7,Proposition 3.2] sequences {E} ⊆ S of energy levels with only o(E) nodal domains as E → ∞ for "most" coefficients (a ξ ) ξ∈E .…”

“…Proposition 3.4 (Ψ does not contribute). Let 4 > 0 and Φ be given, and suppose that Ψ is a random function satisfying Proposition 3.3 (i) and (ii) for a sufficiently small 1 (depending on R, 4 and μ). Then there exist K 1 (μ), δ 1 (K, μ) and E 2 (K, δ) such that…”

“…Suppose that there exist τ 1 , τ 2 > 0 such that for all x ∈ Q we have either |h 1 (y)| > τ 1 or |∇h 1 (y)| > τ 2 . Let h 2 ∈ C(Q) with sup y∈Q |h 2 (y)| < τ1 . Then each component Γ of the zero set Z(h 1 ) with d(Γ, ∂Q) > τ1 τ2 generates a componentΓ of the zero set Z(h 1 + h 2 ) withΓ ⊆ Γ + τ 1 τ 2 .…”