Exact quantization conditions and full transseries structures for PT symmetric anharmonic oscillators

We study exact Wentzel–Kramers–Brillouin analysis (EWKB) for a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="script">P</a:mi><a:mi mathvariant="script">T</a:mi></a:math> symmetric quantum mechanics (QM) defined by the potential <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:msub><e:mrow><e:mi>V</e:mi></e:mrow><e:mrow><e:mi mathvariant="script">P</e:mi><e:mi mathvariant="script">T</e:mi></e:mrow></e:msub><e:mo stretchy="false">(</e:mo><e:mi>x</e:mi><e:mo stretchy="false">)</e:mo><e:mo>=</e:mo><e:msup><e:mrow><e:mi>ω</e:mi></e:mrow><e:mrow><e:mn>2</e:mn></e:mrow></e:msup><e:msup><e:mrow><e:mi>x</e:mi></e:mrow><e:mrow><e:mn>2</e:mn></e:mrow></e:msup><e:mo>+</e:mo><e:mi>g</e:mi><e:msup><e:mrow><e:mi>x</e:mi></e:mrow><e:mrow><e:mn>2</e:mn><e:mi>K</e:mi></e:mrow></e:msup><e:mo stretchy="false">(</e:mo><e:mi>i</e:mi><e:mi>x</e:mi><e:msup><e:mrow><e:mo stretchy="false">)</e:mo></e:mrow><e:mrow><e:mi>ϵ</e:mi></e:mrow></e:msup></e:mrow></e:math> with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>ω</m:mi><m:mo>∈</m:mo><m:msub><m:mi mathvariant="double-struck">R</m:mi><m:mrow><m:mo>≥</m:mo><m:mn>0</m:mn></m:mrow></m:msub></m:math>, <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>g</p:mi><p:mo>∈</p:mo><p:msub><p:mi mathvariant="double-struck">R</p:mi><p:mrow><p:mo>></p:mo><p:mn>0</p:mn></p:mrow></p:msub></p:math> and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>K</s:mi><s:mo>,</s:mo><s:mi>ϵ</s:mi><s:mo>∈</s:mo><s:mi mathvariant="double-struck">N</s:mi></s:math> to clarify its perturbative/nonperturbative structure. In our analysis, we mainly consider the massless cases, i.e., <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mi>ω</v:mi><v:mo>=</v:mo><v:mn>0</v:mn></v:math>, and derive the exact quantization conditions (QCs) for arbitrary <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mo stretchy="false">(</x:mo><x:mi>K</x:mi><x:mo>,</x:mo><x:mi>ϵ</x:mi><x:mo stretchy="false">)</x:mo></x:math> including all perturbative/nonperturbative corrections. From the exact QCs, we clarify full transseries structure of the energy spectra with respect to the inverse energy level expansion, and then formulate the Gutzwiller trace formula, the spectral summation form, and the Euclidean path integral. For the massive cases, i.e., <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:mi>ω</bb:mi><bb:mo>></bb:mo><bb:mn>0</bb:mn></bb:math>, we show the fact that, by requiring the existence of the solution of the exact QCs, the path of analytic continuation in EWKB is uniquely determined for a given <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:mi>N</db:mi><db:mo>=</db:mo><db:mn>2</db:mn><db:mi>K</db:mi><db:mo>+</db:mo><db:mi>ϵ</db:mi></db:math>, and in consequence the exact QCs, the energy spectra, and the three formulas are all perturbative. Similarities to Hermitian QMs and resurgence are also discussed as additional remarks. Published by the American Physical Society 2024