Abstract
We perform an extensive bootstrap study of Hermitian and non-Hermitian theories based on the novel analytic continuation of φⁿ or (iφ)ⁿ in n. We first use the quantum harmonic oscillator to illustrate various aspects of the φⁿ trajectory bootstrap method, such as the large n expansion, matching conditions, exact quantization condition, and high energy asymptotic behavior. Then we derive highly accurate solutions for the anharmonic oscillators with the parity invariant potential V(φ)=φ²+φ^m and the PT invariant potential V(φ)=-(iφ)^m for a large range of integral m, showing the high efficiency and general applicability of this new bootstrap approach. For the Hermitian quartic and non-Hermitian cubic oscillators, we further verify that the non-integer n results for φⁿ or (iφ)ⁿ are consistent with those from the wave function approach. In the PT invariant case, the existence of (iφ)ⁿ with non-integer n allows us to bootstrap the non-Hermitian theories with non-integer powers, such as fractional and irrational m.