Bochner-Riesz profile of anharmonic oscillator L = -d(2)/dx(2) + vertical bar x vertical bar

Peng Chen, Waldemar Hebisch, Adam Sikora*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    Abstract

    We investigate spectral multipliers, Bochner–Riesz means and the convergence of eigenfunction expansion corresponding to the Schrödinger operator with anharmonic potential L=−[formula presented]+|x|. We show that the Bochner–Riesz profile of the operator L completely coincides with such profile of the harmonic oscillator H=−[formula presented]+x2. It is especially surprising because the Bochner–Riesz profile of the one-dimensional standard Laplace operator is known to be essentially different and the case of operators H and 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 an entirely new perspective has to be developed to obtain the critical exponent for Bochner–Riesz means convergence.

    Original languageEnglish
    Pages (from-to)3186-3241
    Number of pages56
    JournalJournal of Functional Analysis
    Volume271
    Issue number11
    DOIs
    Publication statusPublished - 1 Dec 2016

    Keywords

    • Airy function
    • Bochner–Riesz means
    • Schrödinger operators
    • Spectral multipliers

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