A new non-Hermitian E2-quasi-exactly solvable model

@article{Fring2015ANN,
title={A new non-Hermitian E2-quasi-exactly solvable model},
author={Andreas Fring},
journal={Physics Letters A},
year={2015},
volume={379},
pages={873-876}
}
• A. Fring
• Published 8 December 2014
• Physics, Mathematics
• Physics Letters A
We construct a previously unknown E2-quasi-exactly solvable non-Hermitian model whose eigenfunctions involve weakly orthogonal polynomials obeying three-term recurrence relations that factorize beyond the quantization level. The model becomes Hermitian when one of its two parameters is fixed to a specific value. We analyze the double scaling limit of this model leading to the complex Mathieu equation. The norms, Stieltjes measures and moment functionals are evaluated for some concrete values of… Expand
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References

SHOWING 1-10 OF 27 REFERENCES
E2-quasi-exact solvability for non-Hermitian models
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.Expand
Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Physics, Mathematics
• 1998
The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of $\mathrm{PT}$ symmetry, one obtains newExpand
A PT-invariant potential with complex QES eigenvalues
• Physics, Mathematics
• 2000
Abstract We show that the quasi-exactly solvable eigenvalues of the Schrodinger equation for the PT-invariant potential V ( x )=−( ζ cosh2 x − iM ) 2 are complex conjugate pairs in case the parameterExpand
Quasi‐exactly solvable systems and orthogonal polynomials
• Physics, Mathematics
• 1995
This paper shows that there is a correspondence between quasi‐exactly solvable models in quantum mechanics and sets of orthogonal polynomials {Pn}. The quantum‐mechanical wave function is theExpand
Spontaneous PT-Symmetry Breaking for Systems of Noncommutative Euclidean Lie Algebraic Type
• Physics, Mathematics
• 2014
We propose a noncommutative version of the Euclidean Lie algebra E2. Several types of non-Hermitian Hamiltonian systems expressed in terms of generic combinations of the generators of this algebraExpand
New quasi-exactly solvable sextic polynomial potentials
• Mathematics, Physics
• 2005
A Hamiltonian is said to be quasi-exactly solvable (QES) if some of the energy levels and the corresponding eigenfunctions can be calculated exactly and in a closed form. An entirely new class of QESExpand
Quasi-Hermitian operators in quantum mechanics and the variational principle
• Physics
• 1992
We establish a general criterion for a set of non-Hermitian operators to constitute a consistent quantum mechanical system, which allows for the normal quantum-mechanical interpretation. ThisExpand
Quasi-exactly-solvable problems andsl(2) algebra
Recently discovered quasi-exactly-solvable problems of quantum mechanics are shown to be related to the existence of the finite-dimensional representations of the groupSL(2,Q), whereQ=R, C. It isExpand
Non-Hermitian systems of Euclidean Lie algebraic type with real energy spectra
• Physics, Mathematics
• 2014
We study several classes of non-Hermitian Hamiltonian systems, which can be expressed in terms of bilinear combinations of Euclidean Lie algebraic generators. The classes are distinguished byExpand
Pseudo-Hermitian Representation of Quantum Mechanics
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give aExpand