E2-quasi-exact solvability for non-Hermitian models

Abstract

We propose the notion of $E_2$-quasi-exact solvability and apply this idea to find explicit solutions to the eigenvalue problem for a non-Hermitian Hamiltonian system depending on two parameters. The model considered reduces to the complex Mathieu Hamiltonian in a double scaling limit, which enables us to compute the exceptional points in the energy spectrum of the latter as a limiting process of the zeros for some algebraic equations. The coefficient functions in the quasi-exact eigenfunctions are univariate polynomials in the energy obeying a three-term recurrence relation. The latter property guarantees the existence of a linear functional such that the polynomials become orthogonal. The polynomials are shown to factorize for all levels above the quantization condition leading to vanishing norms rendering them to be weakly orthogonal. In two concrete examples we compute the explicit expressions for the Stieltjes measure.

21 pages, 2 figures

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Cited by in corpus (9)

• Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians
• Attoscience in Phase Space
• A new non-Hermitian E2-quasi-exactly solvable model
• Broken Hermiticity phase transition in Bose-Hubbard model
• Displaced harmonic oscillator $V\sim \min [(x+d{)}^2,(x-d{)}^2]$ as a benchmark double-well quantum model
• Stability boundaries of a Mathieu equation having PT symmetry
• Quantisations of exactly solvable ghostly models
• Zig-zag-matrix algebras and solvable quasi-Hermitian quantum models
• A unifying E2-quasi-exactly solvable model