We study the bootstrap method in harmonic oscillators in one-dimensional quantum mechanics. We find that the problem reduces to the Diracs ladder operator problem and is exactly solvable. Thus, harmonic oscillators allow us to see how the bootstrap method works explicitly.
Recently, novel numerical computation on quantum mechanics by using a bootstrap was proposed by Han, Hartnoll, and Kruthoff. We consider whether this method works in systems with a $theta$-term, where the standard Monte-Carlo computation may fail due
to the sign problem. As a starting point, we study quantum mechanics of a charged particle on a circle in which a constant gauge potential is a counterpart of a $theta$-term. We find that it is hard to determine physical quantities as functions of $theta$ such as $E(theta)$, except at $theta=0$ and $pi$. On the other hand, the correlations among observables for energy eigenstates are correctly reproduced for any $theta$. Our results suggest that the bootstrap method may work not perfectly but sufficiently well, even if a $theta$-term exists in the system.
