Bootstrap Method in Harmonic Oscillator

Recently, an application of the numerical bootstrap method to quantum mechanics was proposed, and it successfully reproduces the eigenstates of various systems. However, it is unclear why this method works. In order to understand this question, we study the bootstrap method in harmonic oscillators. We find that the problem reduces to the Dirac's ladder operator problem and is exactly solvable analytically. Our result suggests that the bootstrap method may be regarded as a numerical version of the Dirac's approach and it may explain why it works in various systems.