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Mustapha Maamache

Publications and source records attributed to Mustapha Maamache.

16 recordsLinked to original sources
arXiv2026

A Non-Hermitian Relativistic Oscillator Exactly Mapped to a Hermitian Schrodinger Problem

We introduce the Dirac Inverted Oscillator, a new exactly solvable relativistic quantum system obtained through a Hermitian modification of the generalized momentum operator in the Dirac equation. In contrast to the conventional Dirac oscillator, where the generalized momentum is intrinsically non-Hermitian while the Hamiltonian remains Hermitian, the present construction reverses these Hermiticity properties: the generalized momentum becomes Hermitian, whereas the corresponding Dirac Hamiltonian is intrinsically non-Hermitian. This inversion leads to a new class of relativistic Hamiltonians within the framework of non-Hermitian quantum mechanics. Starting from the stationary Dirac equation, we derive the corresponding second-order wave equation and establish the generalized symmetry properties of the model, including pseudo-Hermiticity and pseudo- symmetry. We then prove that the Dirac Inverted Oscillator is exactly related to the conventional Dirac oscillator through a Hermitian but non-unitary similarity transformation generated by a dilation operator. This transformation is bijective and invertible, mapping the original non-Hermitian relativistic problem onto an equivalent Hermitian Schrodinger eigenvalue problem while preserving its complete spectral structure. The transformed Hermitian problem is solved analytically, allowing the exact relativistic spectrum and the corresponding eigenfunctions of the original non-Hermitian Hamiltonian to be determined explicitly. The present work therefore establishes an exact correspondence between a non-Hermitian relativistic Dirac system and a Hermitian Schrodinger problem, providing a new exactly solvable model and illustrating the effectiveness of Hermitian non-unitary similarity transformations for constructing analytically solvable relativistic Hamiltonians beyond the conventional Hermitian framework.

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arXiv2026

Quantum Dynamics of a Particle in a Linear Potential: Invariant Operator Approach and Discrete Spectrum Solutions

We investigate the quantum dynamics of a particle subjected to a linear potential using the Lewis--Riesenfeld invariant operator method. Starting from the time-dependent Schrödinger equation associated with a constant external force, we construct the most general Hermitian quadratic invariant and derive the corresponding coupled differential equations for its time-dependent coefficients. By means of an appropriate sequence of unitary transformations, the invariant operator is reduced to the form of a harmonic oscillator Hamiltonian. This reduction enables a clear classification of the system according to the sign of the conserved quantity ω2. Particular attention is devoted to the physically relevant case ω2 >0, which yields a discrete eigenspectrum. Explicit analytical expressions for the invariant coefficients, the displacement parameters, and the transformed wave functions are obtained. The resulting formalism provides an exact quantum description of a particle under a constant force and establishes a direct connection between invariant theory and harmonic oscillator quantization.

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arXiv2026

Quantum and Thermal Properties of the Klein-Gordon Inverted Harmonic Oscillator with Physical Applications

We develop a systematic framework for the quantum and thermal properties of a Klein-Gordon scalar field subject to an inverted harmonic potential $-{1\over2} m^2ω^2 x^2$. Starting from a non-Hermitian momentum substitution $P \to P - mωx$, we employ a symplectic phase-space rotation $V = \exp\!\left[-\tfracπ{8}(xp+px)\right]$ to map the system onto an analytically tractable effective harmonic oscillator evaluated at $xe^{iπ/4}$. This allows us to define a well-regulated partition function $Z(β,ω,m)$ and derive closed-form expressions for the free energy, entropy, and thermal correlation functions. We then apply this framework to three physical settings: (i) scalar field fluctuations during cosmological inflation, (ii) quantum fields near black-hole horizons, and (iii) order-parameter dynamics near second-order phase transitions in condensed matter. Our results unify previously scattered results in the literature and provide new predictions for the finite-temperature spectral density and entanglement entropy of unstable quantum systems.

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arXiv2025

Geometrical Amplitude factors in the the adiabatic evolution

In a quantum system initially in the n-th eigenstate, an adiabatic evolution of the Hamiltonian ensures that the system remains in the corresponding instantaneous eigenstate while acquiring a phase factor. This phase has two components: one resulting from standard time evolution and another associated with the dependence of the eigenstate on the varying Hamiltonian, known as the Berry phase. In this work, we explore the concept of geometric amplitudes in the context of a Hermitian Hamiltonian with imaginary eigenvalues. We introduce the notion of geometric amplitude and provide a novel derivation of this concept. Our study reveals that a system undergoing cyclic evolution under adiabatic conditions acquires an additional amplitude factor of purely geometric origin. To illustrate this idea, we apply it to a concrete case: a generalized harmonic oscillator.

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arXiv2024

A new symmetry theory for non-Hermitian Hamiltonians

The η pseudo PT symmetry theory, denoted by the symbol η, explores the conditions under which non-Hermitian Hamiltonians can possess real spectra despite the violation of PT symmetry, that is the adjoint of H, denoted H^{†} is expressed as H^{†}=PTHPT. This theory introduces a new symmetry operator, η=PTη, which acts on the Hilbert space. The η pseudo PT symmetry condition requires the Hamiltonian to commute with the η operator, leading to real eigenvalues. We discuss some general implications of our results for the coupled non hermitian harmonic oscillator.

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arXiv2023

On the η pseudo PT symmetry theory for non-Hermitian Hamiltonians: time-dependent systems

In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo PT symmetry , i.e. the non-Hermitian Hamiltonian H is related to its adjoint H^{†} via the relation, H^{†}=PTHPT . We propose a derivation of pseudo PT symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new metric η(t)=PTη(t) that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1,1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new metric and discuss a concrete physical applications of our results.

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arXiv2022

Anti-PT-symmetric harmonic oscillator and its relation to the inverted harmonic oscillator

We treat the quantum dynamics of a harmonic oscillator as well as its inverted counterpart in the Schrödinger picture. Generally in the most papers of the literature, the inverted harmonic oscillator is formally obtained from the harmonic oscillator by the replacement of ω to iω, this leads to unbounded eigenvectors. This explicitly demonstrates that there are some unclear points involved in redefining the variables in the harmonic oscillator inversion. To remedy this situation, we introduce a scaling operator (Dyson transformation) by connecting the inverted harmonic oscillator to an anti-PT-symmetric harmonic oscillator, we obtain the standard quasi-Hermiticity relation which would ensure the time invariance of the eigenfunction's norm. We give a complete description for the eigenproblem. We show that the wavefunctions for this system are normalized in the sense of the pseudo-scalar product. A Gaussian wave packet of the inverted oscillator is investigated by using the ladder operators method. This wave packet is found to be associated with the generalized coherent state that can be crucially utilized for investigating the mean values of the space and momentum operators. We find that these mean values reproduce the classical motion.

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arXiv2022

Inverted oscillator: pseudo hermiticity and coherent states

It is known that the standard and the inverted harmonic oscillator are different. Replacing thus of ω by iω in the regular oscillator is necessary going to give the inverted oscillator H^{r}. This replacement would lead to anti- PT-symmetric harmonic oscillator Hamiltonian (iH^{os}). The pseudo-hermiticity relation has been used here to relate the anti-PT-symmetric harmonic Hamiltonian to the inverted oscillator. By using a simple algebra, we introduce the ladder operators describing the inverted harmonic oscillator to reproduce the analytical solutions.We construct the inverted coherent states which minimize the quantum mechanical uncertainty between the position and the momentum.

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arXiv2019

Pseudo-invariant approach for a particle in a complex time-dependent linear potential

The Lewis and Riesenfeld method has been investigated, by Ramos et al in Ref.[1], for quantum systems governed by time-dependent PT symmetric Hamiltonians and particularly where the quantum system is a particle submitted to action of a complex time-dependent linear potential. We discuss the method they used and propose an alternative one which leads to physically acceptable uncertainty product and to complex x and p expectation values but describe the classical motion. We used, for this situation, a linear pseudo hermitian invariant operator which allow us to solve analytically the time-dependent Schrödinger equation for this problem and to construct a Gaussian wave packet solution. The normalization condition for the invariant eigenfunctions with the Dirac delta function is correctly obtained, contrary to what is stated in Ref.[1].

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arXiv2017

Non-Unitary evolution of quantum time-dependent non-Hermitian systems

We provide a new perspective on non-Hermitian evolution in quantum mechanics by emphasizing the same method as in the Hermitian quantum evolution. We first give a precise description of the non unitary evolution, and collecting the basic results around it and postulating the norm preserving. This cautionary postulate imposing that the time evolution of a non Hermitian quantum system preserves the inner products between the associated states must not be read naively. We also give an example showing that the solutions of time-dependent non Hermitian Hamiltonian systems given by a linear combination of SU(1,1) and SU(2) are obtained thanks to time-dependent non-unitary transformation.

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arXiv2017

Time evolution of quantum systems with time-dependent non-Hermitian Hamiltonian and the pseudo Hermitian invariant operator

We study the time evolution of quantum systems with a time-dependent non-Hermitian Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators.With a time-dependent metric, the pseudo-Hermitian invariant operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the pseudo-Hermitian invariant operator.

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arXiv2017

Quantum Evolution of the Time-Dependent Non-Hermitian Hamiltonians: Real Phases

Explicitly time-dependent pseudo-Hermitian (TDPH) invariants theory systems, with a time-dependent (TD) metric, is developed for a time-dependent non Hermitian (TDNH) quantum systems. We derive a simple relation between the eigenstates of this pseudo-Hermitien (PH) invariant and the solutions of the Schrodinger equation. A physical system is treated in detail: the TD Swanson model, where an explicitly TDPH invariant is derived for this system, the eigenvalues and eigenstates of the invariant are calculated explicitly.

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arXiv2010

Supersymmetric Extension of Non-Hermitian su(2) Hamiltonian and Supercoherent States

A new class of non-Hermitian Hamiltonians with real spectrum, which are written as a real linear combination of su(2) generators in the form $ H=ωJ_{3}+αJ_{-}+βJ_{+}$, $α\neq β$, is analyzed. The metrics which allows the transition to the equivalent Hermitian Hamiltonian is established. A pseudo-Hermitian supersymmetic extension of such Hamiltonians is performed. They correspond to the pseudo-Hermitian supersymmetric systems of the boson-phermion oscillators. We extend the supercoherent states formalism to such supersymmetic systems via the pseudo-unitary supersymmetric displacement operator method. The constructed family of these supercoherent states consists of two dual subfamilies that form a bi-overcomplete and bi-normal system in the boson-phermion Fock space. The states of each subfamily are eigenvectors of the boson annihilation operator and of one of the two phermion lowering operators.

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arXiv2010

An alternative approach to exact wave functions for time-dependent coupled oscillator model of charged particle in variable magnetic field

A general treatment of the quantal time-dependent coupled oscillators in presence of the variable magnetic field is presented. The treatment is based on the use of an alternative canonical transformations, time-dependent unitary transformations and the invariant methods. Exact wave functions for Schrödinger equations of this system are constructed.We applied our theory to a particular case and, co,sequently, showed that our results recovers to the perviously known one.

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arXiv2010

Time-dependent coupled oscillator model for charged particle motion in the presence of a time varyingmagnetic field

The dynamics of time-dependent coupled oscillator model for the charged particle motion subjected to a time-dependent external magnetic field is investigated. We used canonical transformation approach for the classical treatment of the system, whereas unitary transformation approach is used when managing the system in the framework of quantum mechanics. For both approaches, the original system is transformed to a much more simple system that is the sum of two independent harmonic oscillators which have time-dependent frequencies. We therefore easily identified the wave functions in the transformed system with the help of invariant operator of the system. The full wave functions in the original system is derived from the inverse unitary transformation of the wave functions associated to the transformed system.

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