Solving anharmonic oscillator with null states: Hamiltonian bootstrap and Dyson-Schwinger equations

As basic quantum mechanical models, anharmonic oscillators are recently revisited by bootstrap methods. An effective approach is to make use of the positivity constraints in Hermitian theories. There exists an alternative avenue based on the null state condition, which applies to both Hermitian and non-Hermitian theories. In this work, we carry out an analytic bootstrap study of the quartic oscillator based on the small coupling expansion. In the Hamiltonian formalism, we obtain the anharmonic generalization of Dirac's ladder operators. Furthermore, the Schrodinger equation can be interpreted as a null state condition generated by an anharmonic ladder operator. This provides an explicit example in which dynamics is incorporated into the principle of nullness. In the Lagrangian formalism, we show that the existence of null states can effectively eliminate the indeterminacy of the Dyson-Schwinger equations and systematically determine $n$-point Green's functions.