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We revisit the problem of the deformed oscillator with position-dependent mass [da Costa et al., J. Math. Phys. {\bf 62}, 092101 (2021)] in the classical and quantum formalisms, by introducing the effect of the mass function in both kinetic and potential energies. The resulting Hamiltonian is mapped into a Morse oscillator by means of a point canonical transformation from the usual phase space $(x, p)$ to a deformed one $(x_\gamma, \Pi_\gamma)$. Similar to the Morse potential, the deformed oscillator presents bound trajectories in phase space corresponding to an anharmonic oscillatory motion in classical formalism and, therefore, bound states with a discrete spectrum in quantum formalism. On the other hand, open trajectories in phase space are associated with scattering states and continuous energy spectrum. Employing the factorization method, we investigate the properties of the coherent states, such as the time evolution and their uncertainties. A fast localization, classical and quantum, is reported for the coherent states due to the asymmetrical position-dependent mass. An oscillation of the time evolution of the uncertainty relationship is also observed, whose amplitude increases as the deformation increases., Comment: 24 pages, 11 figures

We find quantum systems having finite distinct discrete energy levels can reflect infinite distinct energy levels under suitable form of position dependent mass systems. A model example has been investigated considering the fractional Harmonic Oscillator [6].In addition to this,we also notice the same behaviour in other quantum models. In all the cases, we find finite distinct discrete levels becoming infinite distinct discrete levels under position dependent mass quantum systems without the change in the potential energy.

Bell’s inequality is investigated in parity-time (PT) symmetric quantum mechanics, using a recently developed form of the inequality by Maccone, with two PT-qubits in the unbroken phase with real energy spectrum. It is shown that the inequality produces a bound that is consistent with the standard quantum mechanics even after using Hilbert space equipped with CPT inner product and therefore, the entanglement has identical structure with standard quantum mechanics. Consequently, the no-signaling principle for a two-qubit system in PT-symmetric quantum theory is preserved.

We formulated the oscillators with position-dependent finite symmetric decreasing and increasing mass. The classical phase portraits of the systems were studied by analytical approach (He’s frequency formalism). We also study the quantum mechanical behaviour of the system and plot the quantum mechanical phase space for necessary comparison with the same obtained classically. The phase portrait in all the cases exhibited closed loop reflecting the stable system but the quantum phase portrait exhibited the inherent signature (cusp or kink) near origin associated with the mass. Although the systems possess periodic motion, the discrete eigenvalues do not possess any similarity with that of the simple harmonic oscillator having m = 1.

The solution of the Schrödinger equation for the two quasi-exactly solvable potentials is presented using the Lie algebra approach. It is shown that all models give rise to the same basic differential equation which is quasi-exactly solvable. The eigenvalues, eigenfunctions and the allowed potential parameters are given for each of the two models in terms of the roots of a set of algebraic quasi-exact solvable methods.

We find real spectral behaviour in negative potentials having PT-symmetry in nature. Apart from linear non-polynomial and polynomial operators, we also consider few nonlinear quantum operators. In the case of polynomial operators, we use appropriate supersymmetry Hamiltonians and verify the iso-spectral nature of energy eigenvalues. All the quantum operators are one-dimensional in nature. The present computation on negative quartic operator yields excellent agreement with the previously computed results (Bender and Boettcher in Phys Rev Lett 80:5243, 1998) and equivalent result of Jones and Mateo (Phys Rev D 73:085002, 2006).

New technique suggested by Almeida and Martin, can be only used to calculate eigenvalues of single well oscillators. As such the method will fail drastically even to compute eigenvalues of single properly non-linear oscillator of the type H = p 2 + x 2 + x 2 > x 4 . Further new technique can hardly ever be used to calculate eigenvalues of any double-well oscillator. Without the calculation of wave characteristics mod square | Ψ | 2 or oblique way by way of expectation value A > the stated paper loses its intention as widely usual method in eigenvalue relation.

Classical and quantum mechanical analysis have been carried out on harmonic like oscillator with asymmetric position dependent mass. Phase space analysis are performed both classically and quantum mechanically for a plausible understanding of the subject., Comment: 8 pages, 5 figures, 1 table We have performed numerical calculation through MATLAB and added the numerical results to this new version in order to have better comparison the classical results with that of quantum. In this manuscript, we added one co-author (M.E. Samei) for his contribution on numerical calculation

We address an intriguing question on spectral invariance in quantum mechanics onexchange of co-ordinate and momentum x ⇀ ↽ p considering general oscillator as anexample.Key words:exchange of co-ordinate and momentum , spectral invariance , generaloscillator .

A new way of constructing SUSY partner potentials with PT symmetry is proposed. In this construction, PT symmetric superpotentials generate PT symmetric SUSY partners and supercharges that satisfy commutator-anti-commutator relations of close superalgebra SL(1/1). Conversely, every PT symmetric Hamiltonian having zero or non zero ground state energy can be generated using PT symmetric superpotentials. Further, superpotentials having separate P and T symmetries can generate strictly isospectral SUSY partners if the wedge of integration can accommodate two asymptotically opposite quantization contours.

We show that proper use of similarity transformation in PT-symmetry operator can justify real spectrum. Further we discuss its limitation. As an example of this we consider harmonic oscillator to demonstrate this. PACS no: 03.65.G; 11.30.Pb

We present a complete energy and wavefunction analysis of a Harmonic oscillator with simultaneous non-hermitian transformations of co-ordinate ( x → ( x + i λ p ) ( 1 + β λ ) ) $(x \rightarrow \frac{(x + i\lambda p)}{\sqrt{(1+\beta \lambda)}})$ and momentum ( p → ( p + i β x ) ( 1 + β λ ) ) $(p \rightarrow \frac {(p+i\beta x)}{\sqrt{(1+\beta \lambda)}})$ using perturbation theory under iso-spectral conditions. We observe that two different frequencies of oscillation (w 1, w 2)correspond to the same energy eigenvalue, - which can also be verified using a Lie algebraic approach.

A general non-linear equation describing damped oscillator, positive van der Pol oscillator, negative van der Pol oscillator, hybrid van der Pol oscillator and a new non-linear oscillator has been solved by optimal virtual work method. In all these cases the time period depends on nonlinear parameter and for the damped oscillator, exact result is obtained.

For the emission of charged particles from metastable nuclei, a general decay formula is developed based on the basic phenomenon of resonances occurring in quantum scattering process under Coulomb-nuclear potential. It relates the half-lives of monopole radioactive decays with the Q -values of the outgoing elements in different angular momentum states as well as the masses and charges of the nuclei involved in the decay. The relation is found to be a generalization of the Geiger–Nuttall law in α radioactivity and it explains well all known emission of charged particles including clusters, alpha and proton carrying angular momenta.

Stable operation of superconducting magnets de- pend critically on the balance of heat deposition rate versus heat extraction rate by the cryo-coolant. Thus, the mass flow rate of the coolant in case of force-flow cooled supercon- ducting magnet with Cable-In-Co-nduit-Conductor (CICC) construction becomes an important factor for optimum sta- bility of magnets. The Toroidal and Poloidal Field mag- nets of Steady-state Superconducting Tokamak-1 (SST-1) is made of superconducting CICC with a void fraction of 40 % ± 2 %. For adequate cooling of magnets, supercritical helium at 4 bar and 4.5 K is forced-flown through the voids. Effect of temperature on mass flow rate and pressure drop in