The Leray measure of nodal sets for random eigenfunctions on the torus

Abstract

Abstract. We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in d ≥ 2 dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with growing multiplicity N → ∞.The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction is constant, equal to 1/ √ 2π. Our main result is that the variance of Leray measure is asymptotically 1/4πN , as N → ∞, at least in dimensions d = 2 and d ≥ 5.

Quantum chaos for point scatterers on flat tori

“…for instance [13]) which only takes into account the physically relevant energies in the summation 6) where α = 0 is the physical coupling constant. We require the following lemma which gives an asymptotic for the sum over the energies outside the interval [λ − λ δ , λ + λ δ ].…”

“…We introduce the mean normalized spacingŝ 6) and it is easy to see that δ ϕ j x ∼ δ j x as x → ∞ (recall that δ ϕ j are the spacings of the new eigenvalues which interlace with the norms n j ).…”

On the distribution of the nodal sets of random spherical harmonics

“…This justifies devoting a separate section (namely, section 4.2) to the treatment of B c . In analogy to the situation of [10] (cf. section 6.1) and [11] (cf.…”

“…and c m is a constant given by (10). As in case of the Leray nodal measure, we divide the integration range into the nonsingular set SB and its complement SB c (see section 4.2).…”

“…The Leray measure L(f T m ) for the random eigenfunctions on the torus T n was considered by Oravecz, Rudnick and Wigman [10]. The expectation is given by…”