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

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