The $\phi^n$ trajectory bootstrap
- URL: http://arxiv.org/abs/2402.05778v2
- Date: Mon, 11 Mar 2024 14:48:24 GMT
- Title: The $\phi^n$ trajectory bootstrap
- Authors: Wenliang Li
- Abstract summary: Green's functions $G_n=langlephinrangle$ admit analytic continuations to complex $n$.
bootstrap problems can be resolved by the principle of minimal singularity.
- Score: 1.8855270809505869
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: The Green's functions $G_n=\langle\phi^n\rangle$ and their self-consistent
equations admit analytic continuations to complex $n$. The indeterminacy of
bootstrap problems can be resolved by the principle of minimal singularity. We
use the harmonic oscillator to illustrate various aspects of the bootstrap
analysis, such as the large $n$ expansion, matching conditions, exact
quantization condition, and high energy asymptotic behavior. For the Hermitian
quartic and non-Hermitian cubic oscillators, we revisit the $\phi^n$
trajectories at non-integer $n$ by the standard wave function formulation. The
results are in agreement with the minimally singular solutions. Using the
matching procedure, we obtain accurate solutions for anharmonic oscillators
with higher powers. In particular, the existence of $G_n$ with non-integer $n$
allows us to bootstrap the $\mathcal{PT}$ invariant oscillators with
non-integer powers.
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