ϕn trajectory bootstrap

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Abstract
We perform an extensive bootstrap study of Hermitian and non-Hermitian theories based on the novel analytic continuation of ⟨ ϕ n ⟩ or ⟨ ( i ϕ ) n ⟩ in n . We first use the quantum harmonic oscillator to illustrate various aspects of the ϕ 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 ( ϕ ) = ϕ 2 + ϕ m and the P T invariant potential V ( ϕ ) = − ( i ϕ ) 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 noninteger n results for ⟨ ϕ n ⟩ or ⟨ ( i ϕ ) n ⟩ are consistent with those from the wave function approach. In the P T invariant case, the existence of ⟨ ( i ϕ ) n ⟩ with noninteger n allows us to bootstrap the non-Hermitian theories with noninteger powers, such as fractional and irrational m .
Funding Information
  • National Natural Science Foundation of China (12205386)
  • Guangzhou Municipal Science and Technology Project (2023A04J0006)

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