<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>ϕ</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:math> trajectory bootstrap

We perform an extensive bootstrap study of Hermitian and non-Hermitian theories based on the novel analytic continuation of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mo stretchy="false">⟨</a:mo><a:msup><a:mrow><a:mi>ϕ</a:mi></a:mrow><a:mrow><a:mi>n</a:mi></a:mrow></a:msup><a:mo stretchy="false">⟩</a:mo></a:mrow></a:math> or <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:mo stretchy="false">⟨</e:mo><e:mo stretchy="false">(</e:mo><e:mi>i</e:mi><e:mi>ϕ</e:mi><e:msup><e:mrow><e:mo stretchy="false">)</e:mo></e:mrow><e:mrow><e:mi>n</e:mi></e:mrow></e:msup><e:mo stretchy="false">⟩</e:mo></e:mrow></e:math> in <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>n</k:mi></k:math>. We first use the quantum harmonic oscillator to illustrate various aspects of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msup><m:mi>ϕ</m:mi><m:mi>n</m:mi></m:msup></m:math> trajectory bootstrap method, such as the large <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>n</o:mi></o:math> 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 <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>V</q:mi><q:mo stretchy="false">(</q:mo><q:mi>ϕ</q:mi><q:mo stretchy="false">)</q:mo><q:mo>=</q:mo><q:msup><q:mi>ϕ</q:mi><q:mn>2</q:mn></q:msup><q:mo>+</q:mo><q:msup><q:mi>ϕ</q:mi><q:mi>m</q:mi></q:msup></q:math> and the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi mathvariant="script">P</u:mi><u:mi mathvariant="script">T</u:mi></u:math> invariant potential <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>V</y:mi><y:mo stretchy="false">(</y:mo><y:mi>ϕ</y:mi><y:mo stretchy="false">)</y:mo><y:mo>=</y:mo><y:mo>−</y:mo><y:mo stretchy="false">(</y:mo><y:mi>i</y:mi><y:mi>ϕ</y:mi><y:msup><y:mo stretchy="false">)</y:mo><y:mi>m</y:mi></y:msup></y:math> for a large range of integral <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>m</eb:mi></eb:math>, 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 <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi>n</gb:mi></gb:math> results for <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mo stretchy="false">⟨</ib:mo><ib:msup><ib:mi>ϕ</ib:mi><ib:mi>n</ib:mi></ib:msup><ib:mo stretchy="false">⟩</ib:mo></ib:math> or <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:mo stretchy="false">⟨</mb:mo><mb:mo stretchy="false">(</mb:mo><mb:mi>i</mb:mi><mb:mi>ϕ</mb:mi><mb:msup><mb:mo stretchy="false">)</mb:mo><mb:mi>n</mb:mi></mb:msup><mb:mo stretchy="false">⟩</mb:mo></mb:math> are consistent with those from the wave function approach. In the <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:mi mathvariant="script">P</sb:mi><sb:mi mathvariant="script">T</sb:mi></sb:math> invariant case, the existence of <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:mo stretchy="false">⟨</wb:mo><wb:mo stretchy="false">(</wb:mo><wb:mi>i</wb:mi><wb:mi>ϕ</wb:mi><wb:msup><wb:mo stretchy="false">)</wb:mo><wb:mi>n</wb:mi></wb:msup><wb:mo stretchy="false">⟩</wb:mo></wb:math> with noninteger <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"><cc:mi>n</cc:mi></cc:math> allows us to bootstrap the non-Hermitian theories with noninteger powers, such as fractional and irrational <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:mi>m</ec:mi></ec:math>.