Finding gaps in a spectrum

TL;DR

This paper introduces an algebraic method to identify spectral gaps in differential operators, demonstrating high precision in energy level separation for the quartic oscillator, with potential for broader application.

Contribution

The paper presents a novel algebraic algorithm for finding spectral gaps in 1D differential operators, especially effective for complex double-well potentials.

Findings

Accurately separates energy levels in the quartic oscillator

Effective in tunnelling regimes with double-well configurations

Potential to generalize to other 1D problems

Abstract

We propose a method for finding gaps in the spectrum of a differential operator. When applied to the one-dimensional Hamiltonian of the quartic oscillator, a simple algebraic algorithm is proposed that, step by step, separates with a remarkable precision all the energies even for a double-well configuration in a tunnelling regime. Our strategy may be refined and generalised to a large class of 1d-problems.