Some nodal properties of the quantum harmonic oscillator and other Schrödinger operators in ℝ²

Abstract

For the spherical Laplacian on the sphere and for the Dirichlet Laplacian in the square, Antonie Stern claimed in her PhD thesis (1924) the existence of an infinite sequence of eigenvalues whose corresponding eigenspaces contain an eigenfunction with exactly two nodal domains. These results were given complete proofs respectively by Hans Lewy in 1977, and the authors in 2014 (see also Gauthier-Shalom-Przybytkowski, 2006). In this paper, we obtain similar results for the two dimensional isotropic quantum harmonic oscillator. In the opposite direction, we construct an infinite sequence of regular eigenfunctions with as many nodal domains as allowed by Courant's theorem, up to a factor 1 4 . A classical question for a 2-dimensional bounded domain is to estimate the length of the nodal set of a Dirichlet eigenfunction in terms of the square root of the energy. In the last section, we consider some Schrödinger operators −∆ + V in R 2 and we provide bounds for the length of the nodal set of an eigenfunction with energy λ in the classically permitted region {V (x) < λ}.Arrange the eigenvalues in increasing order, λ 1 (q) < λ 2 (q) < · · · . A classical theorem of C. Sturm [22] states that an eigenfunction u of (1.1) associated with λ k (q) has exactly (k − 1) zeros in ]a, b[ or, equivalently, that the zeros of u divide ]a, b[ into k sub-intervals.In higher dimensions, one can consider the eigenvalue problem for the Laplace-Beltrami operator −∆ g on a compact connected Riemannian manifold (M, g), with Dirichlet condition in case M has a boundary ∂M , − ∆u = λu in M, u| ∂M = 0.( 1.2) Arrange the eigenvalues in non-decreasing order, with multiplicities,Denote by M 0 the interior of M , M 0 := M \ ∂M . Given an eigenfunction u of −∆ g , denote by N (u) := {x ∈ M 0 | u(x) = 0} (1.3) the nodal set of u, and by µ(u) := # { connected components of M 0 \ N (u)} (1.4) the number of nodal domains of u i.e., the number of connected components of the complement of N (u).Courant's theorem [12] states that if −∆ g u = λ k (M, g)u, then µ(u) ≤ k.In this paper, we investigate three natural questions about Courant's theorem in the framework of the 2D isotropic quantum harmonic oscillator.Question 1. In view of Sturm's theorem, it is natural to ask whether Courant's upper bound is sharp, and to look for lower bounds for the number of nodal domains, depending on the geometry of (M, g) and the eigenvalue. Note that for orthogonality reasons, for any k ≥ 2 and any eigenfunction associated with λ k (M, g), we have µ(u) ≥ 2.We shall say that λ k (M, g) is Courant-sharp if there exists an eigenfunction u, such that −∆ g u = λ k (M, g)u and µ(u) = k. Clearly, λ 1 (M, g) and λ 2 (M, g) are always Courantsharp eigenvalues. Note that if λ 3 (M, g) = λ 2 (M, g), then λ 3 (M, g) is not Courant-sharp.The first results concerning Question 1 were stated by Antonie Stern in her 1924 PhD thesis [36] written under the supervision of R. Courant.

Sharp endpoint estimates for eigenfunctions restricted to submanifolds of codimension 2

“…The Hermite operator −∆ + |x| 2 in R n shares some similar features with the spherical Laplacian, such as periodic Hamilton flow and many highly concentrated eigenfunctions, and the problem of obtaining L p eigenfunction bounds has received considerable interest in the context of Bochner-Riesz means [41,40,42,28,23,14,13,12,15], as well as unique continuation problems [16,17,30]. To understand the nodal sets of the Hermite eigenfunctions in R n , the sizes of nodal sets in small balls have been studied, see Bérard-Helffer [4,3], Hanin-Zelditch-Zhou [19,20], Beck-Hanin [2] and Jin [27]. In this paper, we investigate the concentration of the Hermite eigenfunctions in R n by establishing sharp L p bounds over balls.…”

Level Spacings and Nodal Sets at Infinity for Radial Perturbations of the Harmonic Oscillator

“…See (4) for the assumptions we place on V. Although much is known about nodal sets of eigenfunctions of the Laplacian on a compact manifold, comparatively little has been proved about nodal sets of eigenfunctions of Schrödinger operators − 2 2 ∆ + V (x), even on R d . When V (x) → +∞ as |x| → ∞, such operators have a discrete spectrum and a complete eigenbasis for L 2 (R d , dx).…”

“…They show that the nodal set of an eigenfunctions on the sphere at infinity looks locally like the nodal set of a Hermite polynomial. There is also the paper of Canzani-Toth [6] about the persistence of forbidden hypersurfaces in nodal sets of Schrödinger eigenfunctions on a compact manifold and the articles of Bérard-Helfer [2,3,4] on nodal domains for eigenfunctions of the harmonic oscillator and similar operators (mainly in the allowed region). Finally, we mention the articles of Hanin-Zelditch-Zhou [10,11], which study the typical size of the nodal set in F E and near the caustic ∂A E = {|x| 2 = 2E} for random fixed energy eigenfunctions of HO .…”