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

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.