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We obtain eigenvalues and eigenfunctions of the Schr\"{o}dinger equation with a hyperbolic double-well potential. We consider exact polynomial solutions for some particular values of the potential-strength parameter and also numerical energies for arbitrary values of this model parameter. We test the numerical method by means of a suitable exact asymptotic expression for the eigenvalues and also calculate critical values of the strength parameter that are related to the number of bound states supported by the potential.

We discuss a recent test of the performance of the kinetic energy partition method (KEP) through its application to two separable quantum-mechanical models. We argue that one of the benchmark models is exceedingly simple for testing any realistic approximate method and that almost any reasonable approach yields better results. In the second example our exact benchmark eigenvalues disagree considerably with those chosen by the authors for comparison, which casts doubts on the accuracy of their KEP approach.

We solve an eigenvalue equation that appears in several papers about a wide range of physical problems. The Frobenius method leads to a three-term recurrence relation for the coefficients of the power series that, under suitable truncation, yields exact analytical eigenvalues and eigenfunctions for particular values of a model parameter. From these solutions some researchers have derived a variety of predictions like allowed angular frequencies, allowed field intensities and the like. We also solve the eigenvalue equation numerically by means of the variational Rayleigh-Ritz method and compare the resulting eigenvalues with those provided by the truncation condition. In this way we prove that those physical predictions are merely artifacts of the truncation condition., Comment: arXiv admin note: text overlap with arXiv:2009.07039, arXiv:2008.03376

We analyze recent results for a harmonic oscillator in an environment with a pointlike defect. We show that the allowed oscillator frequencies predicted by the authors stem from a misinterpretation of the exact solutions of a conditionally solvable eigenvalue equation. Also the exact eigenvalues derived by those authors are meaningless because they belong to different quantum-mechanical models., Comment: arXiv admin note: text overlap with arXiv:2008.13604

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 analyze the calculation of bound states for a nonrelativistic spin-half neutral particle under the influence of a Coulomb-like potential induced by Lorentz symmetry breaking effects. We show that the truncation condition proposed by the authors only provides one energy eigenvalue for a particular model potential and misses all the other bound-state energies. The dependence of the cyclotron frequency on the quantum numbers is a mere artifact of the truncation condition that is by no means necessary for the existence of bound states.

We analyze the results obtained from a model consisting of the interaction etween the electric quadrupole moment of a moving particle and an electric field. We argue that the system does not support bound states because the motion along the $z$ axis is unbounded. It is shown that the author obtains a wrong bound-state spectrum for the motion in the $x-y$ plane and that the existence of allowed cyclotron frequencies is an artifact of the approach.

We analyze the results of a paper on "the arising of a Rashba-like coupling, a Zeeman-like term and a Darwin-like term induced by Lorentz symmetry breaking effects in the non-relativistic quantum dynamics of a spin-1/2 neutral particle interacting with external fields". 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., Comment: arXiv admin note: substantial text overlap with arXiv:2008.03376

We analyze a recent treatment of the interaction of a magnetic quadrupole moment with a radial electric field for a non-relativistic particle in a rotating frame and show that the derivation of the equations in the paper is anything but rigorous. The authors presented eigenvalues and eigenfunctions for two sets of quantum numbers as if they belonged to the same physical problem when they are solutions for two different models. In addition to it, the authors failed to comment on the possibility of multiple solutions for every set of quantum numbers.

We apply the Frobenius method to the Schr\"{o}dinger equation with a truncated Coulomb potential. By means of the tree-term recurrence relation for the expansion coefficients we truncate the series and obtain exact eigenfunctions and eigenvalues. From a judicious arrangement of the exact eigenvalues we derive useful information about the whole spectrum of the problem and can obtain other eigenvalues by simple and straightforward interpolation.

We analyze the distribution of the eigenvalues of the quantum-mechanical rotating harmonic oscillator by means of the Frobenius method. A suitable ansatz leads to a three-term recurrence relation for the expansion coefficients. Truncation of the series yields some particular eigenvalues and eigenfunctions in exact analytical form. The former can be organized in such a way that one obtains suitable information about the whole spectrum of the 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.

We analyze two conditionally solvable quantum-mechanical models: a one-dimensional sextic oscillator and a perturbed Coulomb problem. Both lead to a three-term recurrence relation for the expansion coefficients. We show diagrams of the distribution of their exact eigenvalues with the addition of accurate ones from variational calculations. We discuss the symmetry of such distributions. We also comment on the wrong interpretation of the exact eigenvalues and eigenfunctions by some researchers that has led to the prediction of allowed cyclotron frequencies and field intensities.

In a recent paper Castellanos-Jaramillo and Castellanos-Moreno proposed a simple quantum-mechanical model for an electron in the vicinity of an ionized nanostructure with a permanent electric dipole. They chose the interaction of the electron with the charge and the dipole in such a way that the resulting Schr\"{o}dinger equation is separable into radial and angular parts. In this comment we show that those authors did not solve the angular eigenvalue equation with proper periodic boundary conditions and that they also made a mistake in the elimination of the first derivative in the radial equation. Such errors invalidate their results of the Einstein coefficients for the $\left( GaAs\right)_{3}$ system considered.

The purpose of this comment is to present the perturbation approach proposed by Maiz J. Math. Phys. 60, 012103 (2019) in a clearer way. The results of our straightforward procedure agree with those obtained by that author except for one case in which we obtain the exact result while he obtained an approximate one. In addition to it, we show that for sufficiently deep double-well potentials the perturbation approach deteriorates considerably.

In this comment we discuss the application of homotopy perturbation method to a nonlinear differential mass equation that solves the Tolman-Oppenheimer-Volkoff equation for an isotropic and spherically symmetric system. We show that one obtains the same results, more easily and straightforwardly, by means of a textbook power-series method.

We discuss the numerous advantages of using dimensionless equations in non-relativistic quantum mechanics. Dimensionless equations are considerably simpler and reveal the number of relevant parameters in the models. They are less prone to round-off errors when applying numerical methods because all the quantities are of the other of unity. A dimensionless equation facilitates the application of perturbation theory and provides a glimpse of the sort of solution we are going to obtain beforehand.

In this paper we develop the Hellmann-Feynman theorem in statistical mechanics without resorting to the eigenvalues and eigenvectors of the Hamiltonian operator. Present approach does not require the quantum-mechanical version of the theorem at $T=0$ and bypasses any discussion about degenerate states.

We apply the algebraic method to the Bateman Hamiltonian and obtain its natural frequencies and ladder operators from the adjoint or regular matrix representation of that operator. Present analysis shows that the eigenfunctions compatible with the complex eigenvalues obtained earlier by other authors are not square integrable. In addition to this, the ladder operators annihilate an infinite number of such eigenfunctions.

In this comment we show that the eigenvalues of a quartic anharmonic oscillator obtained recently by means of the asymptotic iteration method may not be as accurate as the authors claim them to be.

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.

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 calculate the exceptional points of the eigenvalues of several parameter-dependent Hamiltonian operators of mathematical and physical interest. We show that the calculation is greatly facilitated by the application of the discriminant to the secular determinant. In this way the problem reduces to finding the roots of a polynomial function of just one variable, the parameter in the Hamiltonian operator. As illustrative examples we consider a particle in a one-dimensional box with a polynomial potential, the periodic Mathieu equation, the Stark effect in a polar rigid rotor and in a polar symmetric top.

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 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.

It is well known that the allowed wavefunctions for an $N$-electron system should be antisymmetric with respect to the permutation of any pair of electron labels. On the other hand, the Hamiltonian for such system is invariant under any permutation of electron labels and, consequently, its eigenfunctions are basis for the irreducible representations of the symmetric group $S_N$.Here, we investigate which symmetry species of the $S_N$ group are compatible with the antisymmetry principle. We illustrate the conclusions by means of simple $N$-particle one-dimensional models with harmonic interactions.

In this paper we analyze a recently proposed approach for the construction of antisymmetric functions for atomic and molecular systems. It is based on the assumption that the main problems with Hartree-Fock wavefunctions stem from their lack of proper permutation symmetry. This alternative building approach is based on products of a space times a spin function with opposite permutation symmetry. The main argument for devising such factors is that the eigenfunctions of the non-relativistic Hamiltonian are either symmetric or antisymmetric with respect to the transposition of the variables of a pair of electrons. However, since the eigenfunctions of the non-relativistic Hamiltonian are basis for the irreducible representations of the symmetric group they are not necessarily symmetric or antisymmetric, except in the trivial case of two electrons. We carry out a simple and straightforward general analysis of the symmetry of the eigenfunctions of the non-relativistic Hamiltonian and illustrate our conclusions by means of two exactly-solvable models of $N=2$ and $N=3$ identical interacting particles.

In this comment we show that the approach presented by Foj\'on et al~\cite{Fojon10} is not as accurate as they claim. A straightforward calculation using the models considered buy those authors clearly shows that the spectral method, which the authors criticize, proves to be considerably better., Comment: 12 pages, 11 figures

The well known hypervirial perturbation method (HPM)\ based on hypervirial relations and the Hellmann-Feynman theorem is suitable for the calculation of perturbation corrections of large order for the two-dimensional hydrogen-like atom in a uniform magnetic field. We show analytical results in terms of the quantum numbers and large order corrections for particular states. This approach appears to be simpler and more efficient than the recently proposed one based on Green functions.

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 show that it is possible to extract useful information from the straightforward perturbation theory in classical mechanics. Although the secular terms make the perturbation series useless for large time, these expansions yield the perturbation corrections for the period exactly and may even be useful for sufficiently short time. We think that present analysis may be suitable for an advanced undergraduate course on classical mechanics.

We present the first exact calculation of the energy of the bound state of a one dimensional Dirac massive particle in weak short-range arbitrary potentials, using perturbation theory to fourth order (the analogous result for two dimensional systems with confinement along one direction and arbitrary mass is also calculated to second order). We show that the non--perturbative extension obtained using Pad\'e approximants can provide remarkably good approximations even for deep wells, in certain range of physical parameters. As an example, we discuss the case of two gaussian wells, comparing numerical and analytical results, predicted by our formulas., Comment: 11 pages, 4 figures

We apply the Lindstedt-Poincar\'{e} method to the Lotka-Volterra model and discuss alternative implementations of the approach. By means of an efficient systematic algorithm we obtain an unprecedented number of perturbation corrections for the two dynamical variables and the frequency. They enable us to estimate the radius of convergence of the perturbation series for the frequency as a function of the only model parameter. The method is suitable for the treatment of systems with any number of dynamical variables.

We have applied the Lindstedt-Poincar\'e method to study the limit cycle of the van der Pol oscillator, obtaining the numerical coefficients of the series for the period and for the amplitude to order $859$. Hermite-Pad\'e approximants have been used to extract the location of the branch cut of the series with unprecendented accuracy ($100$ digits). Both series have then been resummed using an approach based on Pad\'e approximants, where the exact asymptotic behaviors of the period and the amplitude are taken into account. Our results improve drastically all previous results obtained on this subject., Comment: 14 pages, 8 figures, 1 table

We obtain the eigenvalues of the harmonic oscillator in a space with a screw dislocation. By means of a suitable nonorthogonal basis set with variational parameters we obtain sufficiently accurate eigenvalues for an arbitrary range of values of the space-deformation parameter. The energies exhibit a rich structure of avoided crossings in terms of such model parameter.

We propose to introduce the concept of exceptional points in intermediate courses on mathematics and classical mechanics by means of simple textbook examples. The first one is an ordinary second-order differential equation with constant coefficients. The second one is the well known damped harmonic oscillator. They enable one to connect the occurrence of linearly dependent exponential solutions with a defective matrix that cannot be diagonalized but can be transformed into a Jordan canonical form.

We show that the well known Kronecker product is a suitable tool for the construction of matrix representations of widely used spin Hamiltonians. In this way we avoid the explicit use of basis sets for the construction of the matrix elements. As illustrative examples we discuss two isotropic models and an anisotropic one.

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 revisit the problem posed by an anharmonic oscillator with a potential given by a polynomial function of the coordinate of degree six that depends on a parameter $\lambda $. The ground state can be obtained exactly and its energy $E_{0}=1$ is independent of $\lambda $. This solution is valid only for $\lambda >0$ because the eigenfunction is not square integrable otherwise. Here we show that the perturbation series for the expectation values are Pad\'{e} and Borel-Pad\'{e} summable for $\lambda >0$. When $% \lambda <0$ the spectrum exhibits an infinite number of avoided crossings at each of which the eigenfunctions undergo dramatic changes in their spatial distribution that we analyze by means of the expectation values $\langle x^{2}\rangle $.