Equivalent Hermitian operator from supersymmetric quantum mechanics

Abstract

Diagonalizable pseudo-Hermitian Hamiltonians with real and discrete spectra, which are superpartners of Hermitian Hamiltonians, must be $\eta$-pseudo-Hermitian with Hermitian, positive-definite and non-singular $\eta$ operators. We show that despite the fact that an $\eta$ operator produced by a supersymmetric transformation, corresponding to the exact supersymmetry, is singular, it can be used to find the eigenfunctions of a Hermitian operator equivalent to the given pseudo-Hermitian Hamiltonian. Once the eigenfunctions of the Hermitian operator are found the operator may be reconstructed with the help of the spectral decomposition.

9 pages; revised formula (14), published version with erratum

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Cited by in corpus (4)

• Perfect transmission scattering as a PT-symmetric spectral problem
• PT-symmetric models in curved manifolds
• Supersymmetric $\eta$ operators
• Charge oscillations in a simple model of interacting magnetic orbits