[Paper Review] Quantum Limits for Harmonic Oscillator

Elie Studnia|arXiv (Cornell University)|May 19, 2019

Spectral Theory in Mathematical Physics2 references5 citations

TL;DR

This paper establishes that any probability measure invariant under the Hamiltonian flow of the harmonic oscillator on the energy sphere $S^{2d-1}$ can be realized as a weak limit of $L^2$-normalized eigenfunctions of the quantum harmonic oscillator. Using a group-theoretic construction via the ortho-symplectic group and microlocal analysis, the authors prove that such measures—particularly elementary (orbit-supported) and convex combinations—are dense in the space of invariant measures, extending quantum ergodicity results to the harmonic oscillator setting.

ABSTRACT

In this note we consider high energy eigenfunctions of the harmonic oscillator in $\mathbb{R}^d$ and prove that any invariant measure on the energy surface can be written as a weak limit of eigenfunctions.

Motivation & Objective

• To extend quantum ergodicity results from the Laplacian on compact manifolds to the harmonic oscillator in $\mathbb{R}^d$.
• To characterize the set of all possible weak limits (Wigner measures) of $L^2$-normalized eigenfunctions of the harmonic oscillator.
• To show that every probability measure on $S^{2d-1}$ invariant under the harmonic oscillator flow arises as a weak limit of eigenfunctions.
• To establish a complete characterization of the quantum limits for the harmonic oscillator using symplectic geometry and group actions.

Proposed method

• Use semiclassical analysis to control pseudodifferential operators $a^w(x,hD)$ with symbols in $\mathscr{C}^\infty_c(\mathbb{R}^{2d})$.
• Prove microlocal concentration of eigenfunctions near the energy sphere $S^{2d-1}$ via a perturbation argument involving a complex absorbing potential.
• Construct eigenfunctions concentrating on individual closed orbits (great circles) using the action of the ortho-symplectic group $\mathrm{OSp}(d,\mathbb{R})$.
• Show that cross-terms between eigenfunctions associated with different orbits are negligible in the weak limit using spectral decay estimates.
• Prove that convex combinations of elementary measures (uniform on orbits) are dense in the space of invariant probability measures on $S^{2d-1}$ via the Krein-Milman theorem.
• Establish a metric topology on the space of linear functionals induced by eigenfunctions and their limits, ensuring weak-* convergence is metrizable and compact.

Experimental results

Research questions

• RQ1 Can every $\mathrm{Ham}$-invariant probability measure on the energy sphere $S^{2d-1}$ of the harmonic oscillator arise as a weak limit of eigenfunctions of the quantum harmonic oscillator?
• RQ2 What group action can be used to generate eigenfunctions concentrating on arbitrary closed orbits of the harmonic oscillator flow?
• RQ3 Are convex combinations of orbit-supported measures dense in the space of all invariant measures on $S^{2d-1}$?
• RQ4 Is the set of quantum limits (weak limits of eigenfunctions) closed under the weak-* topology in the space of invariant measures?
• RQ5 Can the weak-* topology on the space of invariant measures be metrized to allow sequential approximation of arbitrary invariant measures by eigenfunction sequences?

Key findings

• Any probability measure $\mu$ on $S^{2d-1}$ invariant under the Hamiltonian flow of $p(x,\xi) = |x|^2 + |\xi|^2$ is a weak-* limit of eigenfunctions of the harmonic oscillator.
• Elementary measures—uniform on a single closed orbit (great circle)—are realized as weak limits of eigenfunctions via the $d=1$ case and group action.
• Convex combinations of elementary measures are dense in the space of all invariant probability measures on $S^{2d-1}$, as shown by the Krein-Milman theorem.
• The set of all quantum limits (weak limits of eigenfunctions) is closed in the weak-* topology, ensuring completeness of the limit set.
• The weak-* topology on the space of linear functionals induced by eigenfunctions and their limits is metrizable, enabling sequential approximation of any invariant measure by eigenfunction sequences.
• The proof relies on a metric structure on the dual of a Banach space of $C^{Md}$ functions, ensuring convergence is equivalent to sequential convergence in a compact metric space.