Tridiagonality, supersymmetry and non self-adjoint Hamiltonians

Abstract

In this paper we consider some aspects of tridiagonal, non self-adjoint, Hamiltonians and of their supersymmetric counterparts. In particular, the problem of factorization is discussed, and it is shown how the analysis of the eigenstates of these Hamiltonians produce interesting recursion formulas giving rise to biorthogonal families of vectors. Some examples are proposed, and a connection with bi-squeezed states is analyzed.

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• Supersymmetry of $\mathcal{P}\mathcal{T}$- symmetric tridiagonal Hamiltonians
• Markov generators as non-hermitian supersymmetric quantum Hamiltonians: spectral properties via bi-orthogonal basis and Singular Value Decompositions
• Few-grid-point simulations of Big Bang singularity in quantum cosmology
• Resonances and continued-fraction Green's functions in non-Hermitian Bose-Hubbard-like quantum models
• Extended coupled SUSY, pseudo-bosons and weak squeezed states