Supersymmetry of 𝒫𝒯-symmetric tridiagonal Hamiltonians

Modern Physics Letters A ◽

10.1142/s0217732321502424 ◽

2021 ◽

Author(s):

Mohammad Walid AlMasri

Keyword(s):

Energy Spectrum ◽

Orthogonal Polynomials ◽

Complex Potentials ◽

Complex Conjugate ◽

Matrix Elements ◽

Morse Oscillator ◽

Supersymmetric Partner ◽

Natural Home ◽

Symmetric Complex ◽

Physical Quantities

We extend the study of supersymmetric tridiagonal Hamiltonians to the case of non-Hermitian Hamiltonians with real or complex conjugate eigenvalues. We find the relation between matrix elements of the non-Hermitian Hamiltonian [Formula: see text] and its supersymmetric partner [Formula: see text] in a given basis. Moreover, the orthogonal polynomials in the eigenstate expansion problem attached to [Formula: see text] can be recovered from those polynomials arising from the same problem for [Formula: see text] with the help of kernel polynomials. Besides its generality, the developed formalism in this work is a natural home for using the numerically powerful Gauss quadrature techniques in probing the nature of some physical quantities such as the energy spectrum of [Formula: see text]-symmetric complex potentials. Finally, we solve the shifted [Formula: see text]-symmetric Morse oscillator exactly in the tridiagonal representation.

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The Underlying Order Induced by Orthogonality and the Quantum Speed Limit

Quantum Reports ◽

10.3390/quantum3030024 ◽

2021 ◽

Vol 3 (3) ◽

pp. 376-388

Author(s):

Francisco J. Sevilla ◽

Andrea Valdés-Hernández ◽

Alan J. Barrios

Keyword(s):

Energy Spectrum ◽

Energy Distribution ◽

Finite Time ◽

Complete Characterization ◽

Comprehensive Analysis ◽

Orthogonality Condition ◽

Speed Limit ◽

Physical Quantities

We perform a comprehensive analysis of the set of parameters {ri} that provide the energy distribution of pure qutrits that evolve towards a distinguishable state at a finite time τ, when evolving under an arbitrary and time-independent Hamiltonian. The orthogonality condition is exactly solved, revealing a non-trivial interrelation between τ and the energy spectrum and allowing the classification of {ri} into families organized in a 2-simplex, δ2. Furthermore, the states determined by {ri} are likewise analyzed according to their quantum-speed limit. Namely, we construct a map that distinguishes those ris in δ2 correspondent to states whose orthogonality time is limited by the Mandelstam–Tamm bound from those restricted by the Margolus–Levitin one. Our results offer a complete characterization of the physical quantities that become relevant in both the preparation and study of the dynamics of three-level states evolving towards orthogonality.

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Off-diagonal matrix elements for hydrogen-like states: an exact correspondence relationship in terms of orthogonal polynomials and the WKB approximation

Journal of Physics B Atomic Molecular and Optical Physics ◽

10.1088/0953-4075/28/3/007 ◽

1995 ◽

Vol 28 (3) ◽

pp. 341-355 ◽

Cited By ~ 2

Author(s):

H Marxer

Keyword(s):

Orthogonal Polynomials ◽

Diagonal Matrix ◽

Wkb Approximation ◽

Matrix Elements

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Effects of a non-Hermitian potential on the Landau quantization

International Journal of Modern Physics A ◽

10.1142/s0217751x19500726 ◽

2019 ◽

Vol 34 (12) ◽

pp. 1950072 ◽

Cited By ~ 2

Author(s):

B. F. Ramos ◽

I. A. Pedrosa ◽

K. Bakke

Keyword(s):

Magnetic Field ◽

Energy Spectrum ◽

Schrödinger Equation ◽

Complex Potential ◽

Schrodinger Equation ◽

Uniform Magnetic Field ◽

Spinless Particle ◽

Symmetric Complex

In this work, we solve the time-independent Schrödinger equation for a Landau system modulated by a non-Hermitian Hamiltonian. The system consists of a spinless particle in a uniform magnetic field submitted to action of a non-[Formula: see text] symmetric complex potential. Although the Hamiltonian is neither Hermitian nor [Formula: see text]-symmetric, we find that the Landau problem under study exhibits an entirely real energy spectrum.

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