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In this paper, we analyze the results for the ground state of a Klein–Gordon oscillator under gravitational effects in a topologically charged Ellis–Bronnikov wormhole derived recently. We show that the authors failed to truncate a power-series and, consequently, could not obtain the desired solution to a differential equation. For this reason, their analytical expression for the ground-state eigenvalue is incorrect, a fact that affects all the conclusions drawn by the authors for the physical problem. [ABSTRACT FROM AUTHOR]

We discuss some simple Hückel-like matrix representations of non-Hermitian operators with antiunitary symmetries that include generalized P T (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. [ABSTRACT FROM AUTHOR]

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 is incorrect. [ABSTRACT FROM AUTHOR]

In this paper, we argue that the results shown in [Mod. Phys. Lett. A 34, 1950319 (2019)] are not correct. The truncation of the Frobenius method only yields some particular bound states from which one cannot derive any sound physical conclusion. The existence of allowed model parameters is an artifact of such procedure. Besides, the authors failed to obtain most of the solutions given by the truncation of the Frobenius series as well as their connection with the actual eigenvalues of the problem. [ABSTRACT FROM AUTHOR]

We apply the Frobenius method to the Schrödinger 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 such as 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. [ABSTRACT FROM AUTHOR]

We generalize a recently proposed algebraic method in order to treat non-Hermitian Hamiltonians. The approach is applied to several quadratic Hamiltonians studied earlier by other authors. Instead of solving the Schrödinger equation we simply obtain the eigenvalues of a suitable matrix representation of the operator. We take into account the existence of unitary and antiunitary symmetries in the quantum-mechanical problem. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We analyze some polynomial solutions to the Schrödinger equation with a central-field potential derived recently by means of the so-called extended Nikiforov–Uvarov method. We show that this approach does not produce the spectrum of the non-relativistic quantum-mechanical model but merely some eigenvalues for particular values of the model parameters. In addition, the approach just mentioned appears to overlook the restrictions that reveal the problem as conditionally solvable. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this comment we show that the claims by Cassidy et al (J Math Chem 56: 477, 2018) about the limitations of the Riccati Padé method (RPM) are unfounded. To this end we compare the performance of the RPM and their DESCM by means of the calculation of the eigenvalues of a class of one-dimensional potentials. We show that the RPM is simple and extremely accurate. In addition to the exponential convergence of its approximate eigenvalues the RPM does not require any variable transformation, adjustable parameters or scaling that are necessary for the improvement of the performance of the DESCM. An additional remarkable feature of the RPM is that it provides upper and lower bounds to the eigenvalues of the perturbed Coulomb potentials chosen for this test. [ABSTRACT FROM AUTHOR]

A new method is developed for studying real singularities of functions from their power series expansions. It is based on an appropriate calculation of the least-squares deviation of the Taylor coefficients from their asymptotic behavior. Under certain conditions the procedure is shown to lead to the widely used ratio method and it is applied to perturbation-theory problems and virial series. The convergence radii of the Rayleigh–Schrödinger perturbation series for the ground-state energy levels of the plane and linear polar rigid rotators in uniform electric fields are calculated. The occurrence of the singularities in the compressibility factor for hard disks and spheres predicted by the Percus–Yevick and scaled-particle theory is verified. The critical parameters characterizing the Kirkwood ‘‘phase transition’’ near close-packing density and the asymptotic form of the equation of state in the neighborhood of the singularities are verified. Present results elucidate a critical-exponent paradox. [ABSTRACT FROM AUTHOR]

A simple method for studying real singularities from power series expansions is developed. Singularity positions, critical exponents, and critical amplitudes are calculated for a number of quantum-mechanical eigenvalue problems. Special attention is devoted to the Z-1 perturbation series for two-electron atoms. Present results for the states (1s2) 1S, (2p2) 3P, and (1s2s) 3S reveal inaccuracies in previous calculations. In particular it is suggested that the singularity for the 3S state occurs at Z=1 and that the energy lies in the limit of the continuous spectrum of H-. [ABSTRACT FROM AUTHOR]

We study a general one-mode non-Hermitian quadratic Hamiltonian that does not exhibit 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. • Non-Hermitian quadratic Hamiltonian. • The Schrödinger equation can be solved exactly. • The algebraic method proves useful. • It can also be solved by means of a canonical transformation. • It is possible to define an Hermitian, positive-definite metric. [ABSTRACT FROM AUTHOR]

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 equation. Present results prove wrong recent unfounded criticisms of this approach. • Perturbation series may exhibit slow convergence. • Resummation methods improve the convergence properties. • Reconstruction of the effective secular equation is one of such methods. • It enables one to estimate the location of the exceptional points. [ABSTRACT FROM AUTHOR]

We show that the recently proposed kinetic energy partition method for the approximate solution of the Schrödinger equation is a particular case of the well known Rayleigh-Ritz variational method with an extremely restricted implementation. In addition to it, the quantum-mechanical example chosen by the authors to illustrate their approach is unsuitable for the proposed application. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We obtain eigenvalues and eigenfunctions of the Schrödinger 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. [ABSTRACT FROM AUTHOR]

We improve a recently developed implementation of perturbation theory for separable problems. On choosing a new ansatz for the perturbed wave function that takes into account the singularities of the eigenvalue equation explicitly, the resulting perturbation equations become simpler. As an illustrative example, we consider the Stark shift of the microwave spectra of a polarizable symmetric-top molecule. [ABSTRACT FROM AUTHOR]

The Hill determinant method is shown to be useful in obtaining the eigenvalues of central-field systems with potentials that can be expanded in the Taylor series. Some model potentials for diatomic molecules are considered as illustrative examples. [ABSTRACT FROM AUTHOR]

We argue that the analytical energies obtained recently in Mustafa (2022) from a radial eigenvalue equation are incorrect because the truncation method used by the author forces a relationship between two model parameters. This relationship (overlooked by the author) depends on the truncation number and as a result the exact analytical energies do not correspond to a single model but are solutions to different physical models. The reason is that the radial eigenvalue equation is not exactly solvable but conditionally solvable so that one can only obtain some particular exact solutions. We also show that the exact eigenvalues obtained for a particular simpler case are also incorrect because the author overlooked that the shift of the radial coordinate leads to a change in the left boundary condition. • Confined Klein–Gordon oscillators. • The radial eigenvalue equation can be solved by power-series. • One obtains a three-term recurrence relation. • Two truncation conditions yield particular solutions. • One obtains analtical eigenvalues and a relationship between the model parameters. [ABSTRACT FROM AUTHOR]

We prove that any symmetric Hamiltonian that is a quadratic function of the coordinates and momenta has a pseudo-Hermitian adjoint or regular matrix representation. The eigenvalues of the latter matrix are the natural frequencies of the Hamiltonian operator. When all the eigenvalues of the matrix are real, then the spectrum of the symmetric Hamiltonian is real and the operator is Hermitian. As illustrative examples we choose the quadratic Hamiltonians that model a pair of coupled resonators with balanced gain and loss, the electromagnetic self-force on an oscillating charged particle and an active LRC circuit. [ABSTRACT FROM AUTHOR]

We analyze a recent discussion of one-dimensional quantum-mechanical models with a perturbation produced by a delta-function potential. We show that the authors failed to derive a correct transcendental equation for the eigenvalues in the case of the Morse oscillator because they omitted the contribution of the continuum spectrum in the basis set. [ABSTRACT FROM AUTHOR]

We calculate accurate critical parameters for a class of non-hermitian Hamiltonians by means of the diagonalization method. We study three one-dimensional models and two perturbed rigid rotors with PT symmetry. One of the latter models illustrates the necessity of a more general condition for the appearance of real eigenvalues that we also discuss here. [ABSTRACT FROM AUTHOR]

We discuss space-time symmetric Hamiltonian operators of the form H = H 0 + i g H ′ , where H 0 is Hermitian and g real. H 0 is invariant under the unitary operations of a point group G while H ′ is invariant under transformation by elements of a subgroup G ′ of G . If G exhibits irreducible representations of dimension greater than unity, then it is possible that H has complex eigenvalues for sufficiently small nonzero values of g . In the particular case that H is parity-time symmetric then it appears to exhibit real eigenvalues for all 0 < g < g c , where g c is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether H may exhibit real or complex eigenvalues for g > 0 . We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries. [ABSTRACT FROM AUTHOR]

We discuss a parity-time (PT) symmetric Hamiltonian with complex eigenvalues. It is based on the dimensionless Schrodinger 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|. Pointgroup 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 Schrodinger equation with the potential V(x, y) = iaxy2 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. [ABSTRACT FROM AUTHOR]

Abstract: We study both analytically and numerically the spectrum of inhomogeneous strings with -symmetric density. We discuss an exactly solvable model of -symmetric string which is isospectral to the uniform string; for more general strings, we calculate exactly the sum rules , with and find explicit expressions which can be used to obtain bounds on the lowest eigenvalue. A detailed numerical calculation is carried out for two non-solvable models depending on a parameter, obtaining precise estimates of the critical values where pair of real eigenvalues become complex. [Copyright &y& Elsevier]

We consider a class of one-dimensional non-Hermitian oscillators and discuss the relationship between the real eigenvalues of PT-symmetric oscillators and the resonances obtained by different authors. We also show the relationship between the strong-coupling expansions for the eigenvalues of those oscillators. A comparison of the results of the complex rotation and the Riccati-Padé methods reveals that the optimal rotation angle converts the oscillator into either a PT-symmetric or a Hermitian one. In addition to the real positive eigenvalues, the PT-symmetric oscillators exhibit real positive resonances under different boundary conditions. These can be calculated by means of the straightforward diagonalization method. The Riccati-Padé method yields not only the resonances of the non-Hermitian oscillators but also the eigenvalues of the PT-symmetric ones. [ABSTRACT FROM AUTHOR]

Abstract: We show that the Riccati–Padé method is suitable for the calculation of the complex eigenvalues of the Schrödinger equation with a repulsive exponential potential. The accuracy of the results is remarkable for realistic potential parameters. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

We show that the connected-moments polynomial approach proposed recently is equivalent to the well known Rayleigh–Ritz variation method in the Krylov space. We compare the latter with one of the original connected-moments methods by means of a numerical test on an anharmonic oscillator. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009 [ABSTRACT FROM AUTHOR]

Abstract: We calculate accurate bound states and resonances of two interesting perturbed Coulomb models by means of the Riccati–Padé method. This approach is based on a rational approximation to a modified logarithmic derivative of the eigenfunction and produces sequences of roots of Hankel determinants that converge rapidly towards the eigenvalues of the equation. [Copyright &y& Elsevier]

We analyze recently derived eigenvalues of the Schrödinger equation for a harmonic oscillator in the presence of a spiral dislocation. We show that in the case of confinement those results do not take into account the limit of free oscillator properly. We show a simple alternative expression for the ground state that takes approximately into account the regimes of strong confinement and free oscillator. • Exact solutions to harmonic oscillator in the presence of a spiral dislocation. • Topological effects of a spiral dislocation on the spectrum of energy. • Absence of Aharonov–Bohm-type effect for bound states from the topology of the defect. [ABSTRACT FROM AUTHOR]

Serhan et al. [J. Math. Phys. 59, 082105 (2018)] derived a quantum-mechanical Hamiltonian operator from the classical equations of motion for the damped harmonic oscillator and obtained its eigenvalues and eigenfunctions by canonical quantization. They claim that the eigenvalues are real, but here we show that they are in fact complex. [ABSTRACT FROM AUTHOR]

We calculate accurate eigenvalues and eigenfunctions of the Schrödinger equation for a two-dimensional quantum dipole. This model proved useful for the study of elastic effects of a single edge dislocation. We show that the Rayleigh–Ritz variational method with a basis set of Slater-type functions is considerably more efficient than the same approach with the basis set of point-spectrum eigenfunctions of the two-dimensional hydrogen atom used in earlier calculations. [ABSTRACT FROM AUTHOR]

We apply second order finite differences to calculate the lowest eigenvalues of the Helmholtz equation, for complicated non-tensor domains in the plane, using different grids which sample exactly the border of the domain. We show that the results obtained applying Richardson and Padé–Richardson extrapolations to a set of finite difference eigenvalues corresponding to different grids allow us to obtain extremely precise values. When possible we have assessed the precision of our extrapolations comparing them with the highly precise results obtained using the method of particular solutions. Our empirical findings suggest an asymptotic nature of the FD series. In all the cases studied, we are able to report numerical results which are more precise than those available in the literature. [ABSTRACT FROM AUTHOR]