  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.
