Bochner-Riesz profile of anharmonic oscillator ${\mathcal L}=-\frac{d^2}{dx^2}+|x|$

TL;DR

This paper studies spectral multipliers and Bochner-Riesz means for the anharmonic oscillator with potential |x|, revealing that its profile matches the harmonic oscillator and differs from the standard Laplacian, using novel proof techniques.

Contribution

It demonstrates that the Bochner-Riesz profile of the anharmonic oscillator matches that of the harmonic oscillator, introducing new methods beyond restriction estimates.

Findings

Bochner-Riesz profile of ${ mf L}$ coincides with harmonic oscillator

Profile differs from standard Laplace operator

New proof techniques developed for critical exponent

Abstract

We investigate spectral multipliers, Bochner-Riesz means and convergence of eigenfunction expansion corresponding to the Schr\"odinger operator with anharmonic potential ${\mathcal L}=-\frac{d^2}{dx^2}+|x|$. We show that the Bochner-Riesz profile of the operator ${\mathcal L}$ completely coincides with such profile of the harmonic oscillator ${\mathcal H}=-\frac{d^2}{dx^2}+x^2$. It is especially surprising because the Bochner-Riesz profile for the one-dimensional standard Laplace operator is known to be essentially different and the case of operators ${\mathcal H}$ and ${\mathcal L}$ resembles more the profile of multidimensional Laplace operators. Another surprising element of the main obtained result is the fact that the proof is not based on restriction type estimates and instead entirely new perspective have to be developed to obtain the critical exponent for Bochner-Riesz means convergence.