Exact WKB analysis for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="script">P</mml:mi><mml:mi mathvariant="script">T</mml:mi></mml:mrow></mml:math>-symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture
Syo Kamata
Abstract
We consider exact Wentzel-Kramers-Brillouin analysis to 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 defined by the potential, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>V</e:mi><e:mo stretchy="false">(</e:mo><e:mi>x</e:mi><e:mo stretchy="false">)</e:mo><e:mo>=</e:mo><e:msup><e:mi>ω</e:mi><e:mn>2</e:mn></e:msup><e:msup><e:mi>x</e:mi><e:mn>2</e:mn></e:msup><e:mo>+</e:mo><e:mi>g</e:mi><e:msup><e:mi>x</e:mi><e:mn>2</e:mn></e:msup><e:mo stretchy="false">(</e:mo><e:mi>i</e:mi><e:mi>x</e:mi><e:msup><e:mo stretchy="false">)</e:mo><e:mrow><e:mi>ϵ</e:mi><e:mo>=</e:mo><e:mn>2</e:mn></e:mrow></e:msup></e:math> with <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>ω</k:mi><k:mo>∈</k:mo><k:msub><k:mi mathvariant="double-struck">R</k:mi><k:mrow><k:mo>≥</k:mo><k:mn>0</k:mn></k:mrow></k:msub></k:math>, <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>g</n:mi><n:mo>∈</n:mo><n:msub><n:mi mathvariant="double-struck">R</n:mi><n:mrow><n:mo>></n:mo><n:mn>0</n:mn></n:mrow></n:msub></n:math>. We in particular aim to verify a conjecture proposed by Ai-Bender-Sarkar (ABS), that pertains to a relation between <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>D</q:mi></q:math>-dimensional <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi mathvariant="script">P</s:mi><s:mi mathvariant="script">T</s:mi></s:math>-symmetric theories and analytic continuation (AC) of Hermitian theories concerning the energy spectrum or Euclidean partition function. For the purpose, we construct energy quantization conditions by exact Wentzel-Kramers-Brillouin analysis and write down their transseries solution by solving the conditions. By performing alien calculus to the energy solutions, we verify validity of the ABS conjecture and seek a possibility of its alternative form by Borel resummation theory if it is violated. Our results claim that the validity of the ABS conjecture drastically changes depending on whether <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>ω</w:mi><w:mo>></w:mo><w:mn>0</w:mn></w:math> or <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>ω</y:mi><y:mo>=</y:mo><y:mn>0</y:mn></y:math>: If <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>ω</ab:mi><ab:mo>></ab:mo><ab:mn>0</ab:mn></ab:math>, then the ABS conjecture is violated when exceeding the semiclassical level of the first nonperturbative order, but its alternative form is constructable by Borel resummation theory. The <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi mathvariant="script">P</cb:mi><cb:mi mathvariant="script">T</cb:mi></cb:math> and the AC energies are related to each other by a one-parameter Stokes automorphism, and a median resummed form, which corresponds to a formal exact solution, of the AC energy (resp. <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi mathvariant="script">P</gb:mi><gb:mi mathvariant="script">T</gb:mi></gb:math> energy) is directly obtained by acting Borel resummation to a transseries solution of the <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:mi mathvariant="script">P</kb:mi><kb:mi mathvariant="script">T</kb:mi></kb:math> energy (resp. AC energy). If <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:mi>ω</ob:mi><ob:mo>=</ob:mo><ob:mn>0</ob:mn></ob:math>, then, with respect to the inverse energy level-expansion, not only perturbative/nonperturbative structures of the <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:mi mathvariant="script">P</qb:mi><qb:mi mathvariant="script">T</qb:mi></qb:math> and the AC energies but also their perturbative parts do not match with each other. These energies are independent solutions, and no alternative form of the ABS conjecture can be reformulated by Borel resummation theory. Published by the American Physical Society 2024