$\ddot{I}{•}^n$ trajectory bootstrap

Abstract

We perform an extensive bootstrap study of Hermitian and non-Hermitian theories based on the novel analytic continuation of $⟨\ddot{I}{•}^n⟩$ or $⟨(i\ddot{I}•{)}^n⟩$ in $n$. We first use the quantum harmonic oscillator to illustrate various aspects of the $\ddot{I}{•}^n$ 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 $V(\ddot{I}•)=\ddot{I}{•}^2+\ddot{I}{•}^m$ and the $\mathcal{P}\mathcal{T}$ invariant potential $V(\ddot{I}•)=-(i\ddot{I}•{)}^m$ 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 $⟨\ddot{I}{•}^n⟩$ or $⟨(i\ddot{I}•{)}^n⟩$ are consistent with those from the wave function approach. In the $\mathcal{P}\mathcal{T}$ invariant case, the existence of $⟨(i\ddot{I}•{)}^n⟩$ with non-integer $n$ allows us to bootstrap the non-Hermitian theories with non-integer powers, such as fractional and irrational $m$.

v4: 22 pages, 11 figures, typos corrected, discussions improved; v3: 21 pages, 9 figures, abstract and introduction rewritten, many minor improvements made, hopefully more reader-friendly; v2: 24 pages, 9 figures, irrational example added, typos corrected, discussion improved, references added