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We analyse a recent application of the Cornell potential with spin-spin interaction to the mass spectra of quarkonia and show that the authors have in fact used the Kratzer-Fues potential. They inadvertently converted one potential into the other by means of an invalid transformation.

We discuss a systematic construction of dimensionless quantum-mechanical equations. The process reduces the number of independent model parameters to a minimum and, at the same time, provides the natural units of length, energy, etc. in a clear, straightforward way. We compare this systematic procedure with the widely adopted one that consists of setting $\hbar=1$. As illustrative examples, we choose some simple one-dimensional models proposed recently for the study of localized states in inhomogeneous media.

We study a general one-mode non-Hermitian quadratic Hamiltonian that does not exhibit $\mathcal{PT}$-symmetry. By means of an algebraic method we determine the conditions for the existence of real eigenvalues as well as the location of the exceptional points. We also put forward an algebraic alternative to the generalized Bogoliubov transformation that enables one to convert the quadratic operator into a simpler form in terms of the original creation and annihilation operators. We carry out a similar analysis of a two-mode oscillator that consists of two identical one-mode oscillators coupled by a quadratic term.

We discuss results obtained recently for a quantum-mechanical model given by a neutral particle with a magnetic quadrupole moment in a radial electric field and a scalar potential proportional to the radial variable in cylindrical coordinates that also includes the noninertial effects of a rotating reference frame. We show that the conjectured allowed values of the cyclotron frequency are a mere artifact of the truncation of the power series used to solve the radial eigenvalue equation. Our analysis proves that the analytical expression for the eigenvalues are far from correct.

We discuss some simple H\"uckel-like matrix representations of non-Hermitian operators with antiunitary symmetries that include generalized $\mathcal{PT}$ (parity transformation followed by time-reversal) symmetry. One of them exhibits extremely broken antiunitary symmetry (complex eigenvalues for all nontrivial values of the model parameter) because of the degeneracy of the operator in the Hermitian limit. These examples illustrate the effect of point-group symmetry on the spectrum of the non-Hermitian operators. We construct the necessary unitary matrices by means of simple graphical representations of the non-Hermitian operators.

We analyze a recent application of two Adomian-based approximate methods to the Chen-Lee-Liu equation. We prove that the outcome of these approaches is merely the Taylor expansion of the solution about $t=0$ and, consequently, they are valid only for sufficiently small values of the time variable $t$.

We study the behavior of a quantum particle trapped in a confining potential in one dimension under multiple sudden changes of velocity and/or acceleration. We develop the appropriate formalism to deal with such situation and we use it to calculate the probability of transition for simple problems such as the particle in an infinite box and the simple harmonic oscillator. For the infinite box of length $L$ under two and three sudden changes of velocity, where the initial and final velocity vanish, we find that the system undergoes quantum revivals for $\Delta t = \tau_0 \equiv \frac{4mL^2}{\pi\hbar}$, regardless of other parameters ($\Delta t$ is the time elapsed between the first and last change of velocity). For the simple harmonic oscillator we find that the states obtained by suddenly changing (one change) the velocity and/or the acceleration of the potential, for a particle initially in an eigenstate of the static potential, are {\sl coherent} states. For multiple changes of acceleration or velocity we find that the quantum expectation value of the Hamiltonian is remarkably close (possibly identical) to the corresponding classical expectation values. Finally, the probability of transition for a particle in an accelerating harmonic oscillator (no sudden changes) calculated with our formalism agrees with the formula derived long time ago by Ludwig and recently modified by Dodonov~\cite{Dodonov21}, but with a different expression for the dimensionless parameter $\gamma$. Our probability agrees with the one of ref.~\cite{Dodonov21} for $\gamma \ll 1$ but is not periodic in time (it decays monotonously), contrary to the result derived in ref.~\cite{Dodonov21}., Comment: 29 pages, 10 figures

We discuss a recently proposed approach termed Q-BOR-FDTD method and develop its main equations in a clearer and more rigorous way. We show that it is unsuitable for the calculation of the eigenfunctions in the case of degenerate states and propose an improvement to overcome such limitation.

We give a simple proof of the well known fact that the approximate eigenvalues provided by the Rayleigh-Ritz variational method are increasingly accurate upper bounds to the exact ones. To this end, we resort to the variational principle, mentioned in most textbooks on quantum mechanics and quantum chemistry, and to a simple set of projection operators. We think that present approach may be suitable for an advanced course on quantum mechanics or quantum chemistry.

We apply the Frobenius (power-series) method to some simple exactly-solvable and conditionally-solvable quantum-mechanical models with supposed physical interest. We show that the supposedly exact solutions to radial eigenvalue equations derived in recent papers are not correct because they do not satisfy some well-known theorems. We also discuss the origin of the mistake by means of the approach indicated above.

We show that the effective secular equation proposed several years ago is suitable for estimating the location of the exceptional points of eigenvalue equations. As an illustrative example we choose the well known Mathieu

We show that the straightforward application of the discriminant to some physical problems may yield a trivial useless result. If the symmetry of the model matrix does not change with variations of the model parameter the discriminant may vanish for all values of the parameter due to degeneracy. We illustrate this problem by means of a simple $6\times6$ matrix representation of an Hermitian Hamiltonian operator.

We obtain the eigenvalues and eigenfunctions of the singular harmonic oscillator $V(x)=\alpha/(2x^2)+x^2/2$ by means of the simple and straightforward Frobenius (power-series) method. From the behaviour of the eigenfunctions at origin we derive two branches for the eigenvalues for negative values of $\alpha$. We discuss the well known fact that there are square-integrable solutions only for some allowed discrete values of the energy.

We analyze recently published results about the Aharonov-Bohm effect for bound states from the interaction of the magnetic quadrupole moment of a neutral particle with axial fields. We show that the eigenvalues obtained by the authors from an arbitrary truncation of the Frobenius power series correspond to more than one model and, consequently, are unsuitable for drawing any sound physical conclusion. Besides, the prediction of allowed oscillator frequencies is a mere consequence of the truncation condition just mentioned., Comment: arXiv admin note: text overlap with arXiv:2010.04997

We show a parameter-dependent $3\times 3$ non-Hermitian matrix that exhibits both degeneracy and coalescence of eigenvalues at an exceptional point (Hermitian and non-Hermitian degeneracies). This simple non-Hermitian model is suitable for the discussion of those concepts in an undergraduate or graduate course on quantum-mechanics. We also study the symmetry group responsible for the degeneracy.

We show that the author of a recent paper [arXiv:2008.04929] put forward some false statements about the eigenstates of Hermitian and non-Hermitian systems. We conjecture that one of the non-Hermitian Hamiltonians for a one-dimensional lattice is similar to an Hermitian one and, consequently, exhibits real eigenvalues. Present theoretical analysis of the eigenvalue equation suggests that one of the sets of numerical results in the criticized paper may not be correct.

We analyze the results obtained from a model consisting of the interaction between a moving electric quadrupole moment and a magnetic field. We argue that there are no bound states contrary to what the author stated. It is shown that the proposed method is unsuitable for obtaining all the bound states and that the existence of allowed cyclotron frequencies is an artifact of the approach.

We analyze recent results on a Dirac oscillator. We show that the truncation of the Frobenius series does not yield all the eigenvalues and eigenfunctions of the radial equation. For this reason the eigenvalues reported by the authors are useless and the prediction of allowed oscillator frequencies meaningless.

We analyse the solution to the Thomas-Fermi equation discovered by Majorana. We show that the series for the slope at origin enables one to obtain results of accuracy far beyond those provided by available methods. We also estimate the radius of convergence of this series and conjecture that the singularity closest to origin is a square-root branch point.

We analyze the solution of the Schr��dinger equation arising in the treatment of a geometric model introduced to explain the origin of the observed shallow levels in semiconductors threaded by a dislocation density. We show (contrary to what the authors claimed) that the model does not support bound states for any chosen set of model parameters. Assuming a fictitious motion in the $x-y$ plane there are bound states provided that $k\neq 0$ and not only for $k>0$ as the authors believed. The truncation condition proposed by the authors yields only one particular energy for a given value of a chosen model parameter and misses all the others (conditionally solvable problem)

We analyze recent results concerning the hypothesis of a privileged direction in the space-time that is made by considering a background of the Lorentz symmetry violation determined by a fixed spacelike vector field and the analysis of quantum effects of this background on the interaction of a nonrelativistic electron with a nonuniform electric field produced by a uniform electric charge distribution. We show that the conclusions derived by the authors are an artifact of the truncation of the Frobenius series by means of the tree-term recurrence relation for the expansion coefficients. Thus, the existence of allowed angular frequencies stemming from this procedure is meaningless and unphysical., Comment: arXiv admin note: text overlap with arXiv:2010.04997

We analyze the eigenvalues and eigenfunctions stemming from a recent study of the interaction of a scalar particle with a Coulomb potential in the presence of a background of the violation of the Lorentz symmetry established by a tensor field. We show, beyond any doubt, that the physical conclusions drawn by the authors from a truncation of a power series, coming from the application of the Frobenius method, are meaningless and nonsensical.

We discuss the diagonalization of a general Hamiltonian operator for a set of coupled harmonic oscillators and determine the conditions for the existence of bound states. We consider the particular cases of two and three oscillators studied previously and show the conditions for bound states in the latter example that have been omitted in an earlier treatment of this model.

We analyze the results of a paper on ``Relativistic quantum dynamics of a charged particle in cosmic string spacetime in the presence of magnetic field and scalar potential''. We show that the authors did not obtain the spectrum of the eigenvalue equation but only one eigenvalue for a specific relationship between model parameters. In particular, the existence of allowed cyclotron frequencies conjectured by the authors is a mere artifact of the truncation condition used to obtain exact solutions to the radial eigenvalue equation.

In this paper we discuss the validity of the Hellmann-Feynman theorem (HFT) for degenerate states. We derive it in a general way and apply it to simple illustrative examples. We also analyze a recent paper that shows results that apparently suggest that the HFT does not apply to degenerate states.

We analyze the results obtained recently by Khazael and Sebastiani (J. Chem. Phys. \textbf{147}, 194303 (2017)) for two coupled methyl rotors. We show that one of the avoided crossings obtained by those authors is in fact a true crossing between states of different symmetry. Present analysis is supposed to be clearer because it is based on an independent calculation of eigenvalues for each irreducible representation. The application of group theory also enables a more accurate calculation by means of smaller matrix representations of the Hamiltonian operator.

We consider the Hamiltonian for a charged particle in a harmonic potential in the presence of a magnetic field. The most symmetric case depends on one parameter, the variation of which leads from a spectrum bounded from below to an unbounded spectrum. At the transition point the spectrum is bounded from below but each eigenvalue has infinite multiplicity. The algebraic method proves to be a remarkable tool for the analysis of this quadratic Hamiltonian.

We study the behavior of a quantum particle, trapped in localized potential, when the trapping potential starts suddenly to move with constant velocity. In one dimension we have reproduced the results obtained by Granot and Marchewka, Ref. \cite{Granot09}, for an attractive delta function, using an approach based on a spectral decomposition, rather than on the propagator. We have also considered the cases of P\"oschl-Teller and simple harmonic oscillator potentials (in one dimension) and to the hydrogen atom (in three dimensions). In this last case we have calculated explicitly the leading contribution to the ionization probability for the hydrogen atom due to a sudden movement., Comment: 13 pages; 4 figures

We analyze the simple model of a rigid rotor with $C_{3}$ symmetry and show that the use of parity simplifies considerably the calculation of its eigenvalues. We also consider a non-Hermitian space-time-symmetric counterpart that exhibits real eigenvalues and determine the exceptional point at which the antiunitary symmetry is broken.

We provide a pedagogical approach to the problem of avoided crossings between electronic molecular curves and to diabatic and adiabatic transitions when the nuclei of a diatomic molecule move according to classical mechanics. For simplicity we restrict the analysis to only two electronic states.

We investigate a simple model for the prediction of the splitting of the $3d$ orbitals of a metal ion in the environment of ligands. The electrons are considered to be independent and their interaction with the ligands is represented by the Dirac delta function in three dimensions. We discuss several cases where the model is successful and a few ones where it is not.

By means of a suitable rational approximation to the logarithmic derivative of the wavefunction we obtain tight upper and lower bounds to the eigenvalues and critical parameters of the quartic double-well potential.

We have studied the emergence of bound states in weakly deformed and/or heterogeneous waveguides, comparing the analytical predictions obtained using a recently developed perturbative method, with precise numerical results, for different configurations (a homogeneous asymmetric waveguide, a heterogenous asymmetric waveguide and a homogeneous broken-strip). In all the examples considered in this paper we have found excellent agreement between analytical and numerical results, thus providing a numerical verification of the analytical approach., Comment: 11 pages, 6 figures

We generalize a recently proposed approach for the construction of pseudo-Hermitian Hamiltonians with real spectra. Present technique is based on a simple and straightforward similarity transformation of the coordinate and momentum.

We show that the resonances of the Stark effect in hydrogen reported in Phys. Rev. A \textbf{88}, 022509 (2013) are considerably less accurate than the number of digits appear to suggest. In particular, the imaginary part of the lowest resonance is several orders of magnitude greater than it should be. We compare the results of that paper with those provided by the Riccati-Pad\'{e} method, perturbation theory and an asymptotic expansion for the resonance width. The inaccuracy of those results can be traced back to the lack of precision in the calculation of the matrix elements of the secular equation. We carry out a more accurate calculation with the same method and show that the agreement with earlier results of other authors is greatly improved.

We show that parity-time and partial parity-time symmetries are particular cases of antiunitary symmetry. This point is illustrated by means of a recently discussed system of non-Hermitian coupled harmonic oscillators that also exhibits other types of antiunitary symmetries. We also show that a combination of group and perturbation theory is a useful tool for predicting broken antiunitary symmetry.

The purpose of this paper is the discussion of a pair of coupled linear oscillators that has recently been proposed as a model of a system of two optical resonators. By means of an algebraic approach we show that the frequencies of the classical and quantum-mechanical interpretations of the optical phenomenon are exactly the same. Consequently, if the classical frequencies are real, then the quantum-mechanical eigenvalues are also real.

We analyse some PT-symmetric oscillators with $T_{d}$ symmetry that depend on a potential parameter $g$. We calculate the eigenvalues and eigenfunctions for each irreducible representation and for a range of values of $g$. Pairs of eigenvalues coalesce at exceptional points $g_c$; their magnitude roughly decreasing with the magnitude of the eigenvalues. It is difficult to estimate whether there is a phase transition at a nonzero value of $g$ as conjectured in earlier papers. Group theory and perturbation theory enable one to predict whether a given space-time symmetry leads to real eigenvalues for sufficiently small nonzero values of $g$.

We calculate the eigenvalues of some two-dimensional non-Hermitian Hamiltonians by means of a pseudospectral method and straightforward diagonalization of the Hamiltonian matrix in a suitable basis set. Both sets of results agree remarkably well but differ considerably from the eigenvalues obtained some time ago by other authors. In particular, we do not observe the multiple phase transitions claimed to occur in one of the anharmonic oscillators.

In this paper we show that the point-group (geometrical) symmetry is insufficient to account for the degeneracy of the energy levels of the particle in a cubic box. The discrepancy is due to hidden (dynamical symmetry). We obtain the operators that commute with the Hamiltonian one and connect eigenfunctions of different symmetries. We also show that the addition of a suitable potential inside the box breaks the dynamical symmetry but preserves the geometrical one.The resulting degeneracy is that predicted by point-group symmetry.

We discuss a PT-symmetric Hamiltonian with complex eigenvalues. It is based on the dimensionless Schr\"{o}dinger equation for a particle in a square box with the PT-symmetric potential $V(x,y)=iaxy$. Perturbation theory clearly shows that some of the eigenvalues are complex for sufficiently small values of $|a|$. Point-group symmetry proves useful to guess if some of the eigenvalues may already be complex for all values of the coupling constant. We confirm those conclusions by means of an accurate numerical calculation based on the diagonalization method. On the other hand, the Schr\"odinger equation with the potential $V(x,y)=iaxy^{2}$ exhibits real eigenvalues for sufficiently small values of $|a|$. Point group symmetry suggests that PT-symmetry may be broken in the former case and unbroken in the latter one.

We discuss the uncertainty relation for the azimuthal angle $\phi$ and the $z$-component of the angular momentum $L_z$. To this end we derive the uncertainty relation for an arbitrary pair of observables and discuss the conditions for its validity. By means of a simple parameter-dependent state we illustrate the well known fact that the standard uncertainty relation for the coordinate and its conjugate momentum does not apply the pair of observables $\phi-L_z$. This analysis is motivated by a discussion of the motion of an electron in a constant magnetic field appeared recently in this journal (Eur. J. Phys. \textbf{33} (2012) 1147) where the author assumed the validity of the standard uncertainty relation for the pair $\phi-L_z$.

We show that the authors of the commented paper draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In one of the studied examples the authors missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operator and focused their attention on the complex ones that do not. We also show that the authors misread Bender's argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-$x$ plane (Rep. Prog. Phys. {\bf 70} (2007) 947)., Comment: 7 pages, 1 table

We analyse a nonadiabatic self-consistent field method by means of an exactly-solvable model. The method is based on nuclear and electronic orbitals that are functions of the cartesian coordinates in the laboratory-fixed frame. The kinetic energy of the center of mass is subtracted from the molecular Hamiltonian operator in the variational process. The results for the simple model are remarkably accurate and show that the integration over the redundant cartesian coordinates leads to couplings among the internal ones.

We show that some sum rules for the hydrogen atom derived recently are incorrect because the authors did not take into account the continuous part of the spectrum in the sum over intermediate states.

We show that Wronskians between properly chosen linearly independent solutions of the Schr\"odinger equation greatly facilitate the study of quantum scattering in one dimension. They enable one to obtain the necessary relationships between the coefficients that determine the asymptotic behavior of the wavefunction. As illustrative examples we calculate the transmission probability for the penetration of a Gaussian barrier and the scattering resonances for a Gaussian well.

We analyze a simple textbook approach to nonlinear oscillators proposed recently, disclose its errors, limitations and misconceptions and complete the calculations that the authors failed to perform.

We discuss two recent applications of homotopy analysis method and homotopy perturbation method and conclude that the results are completely useless from both mathematical and physical points of view.

We analyze a model proposed recently for the calculation of the energy of an exciton in a quantum dot and show that the authors made a serious mistake in the solution to the Schr\"{o}dinger equation.

We apply the Born--Oppenheimer approximation to a harmonic diatomic molecule with one electron. We compare the exact and approximate results not only for the internal degrees of freedom but also for the motion of the center of mass. We address the problem of the permutation symmetry of identical nuclei and discuss other applications of the model and its limitations.