TBA equations and exact WKB analysis in deformed supersymmetric quantum mechanics
Katsushi Ito (Department of Physics, Tokyo Institute of Technology, Tokyo, 152-8551, Japan); Hongfei Shu (Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China, School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan, 450001, China, Beijing Institute of Mathematical Sciences and Applications, Beijing, 101408, China)
We study the spectral problem in deformed supersymmetric quantum mechanics with polynomial superpotential by using the exact WKB method and the TBA equations. We apply the ODE/IM correspondence to the Schrödinger equation with an effective potential deformed by integrating out the fermions, which admits a continuous deformation parameter. We find that the TBA equations are described by the ℤ4-extended ones. For cubic superpotential corresponding to the symmetric double-well potential, the TBA system splits into the two D 3-type TBA equations. We investigate in detail this example based on the TBA equations and their analytic continuation as well as the massless limit. We find that the energy spectrum obtained from the exact quantization condition is in good agreement with the diagonalization approach of the Hamiltonian.
