Spectral Properties of the Massless Relativistic Quartic Oscillator

TL;DR

This paper provides an explicit spectral analysis of a non-local Schrödinger operator combining the square root of the Laplacian with a quartic potential, revealing eigenvalues, eigenfunctions, and spectral properties.

Contribution

It introduces explicit solutions for the spectral problem of a non-local operator with a quartic potential, including formulas for eigenfunctions and spectral estimates.

Findings

Eigenvalues as zeros of special functions related to the fourth order Airy function

Explicit Fourier transform formulas for eigenfunctions

Spectral gap estimates and eigenvalue distribution analysis

Abstract

An explicit solution of the spectral problem of the non-local Schr\"odinger operator obtained as the sum of the square root of the Laplacian and a quartic potential in one dimension is presented. The eigenvalues are obtained as zeroes of special functions related to the fourth order Airy function, and closed formulae for the Fourier transform of the eigenfunctions are derived. These representations allow to derive further spectral propertiessuch as estimates of spectral gaps, heat trace and the asymptotic distribution of eigenvalues, as well as a detailed analysis of the eigenfunctions. A subtle spectral effect is observed which manifests in an exponentially tight approximation of the spectrum by the zeroes of the dominating term in the Fourier representation of the eigenfunctions and its derivative.