Nonsingular Potentials Research Articles

Compact stars admitting Finch-Skea symmetry in the presence of various matter fields* *PKS acknowledges National Board for Higher Mathematics (NBHM) under Department of Atomic Energy (DAE), Govt. of India for financial support to carry out the Research project No.: 02011/3/2022 NBHM (R.P.)/R#D II/2152 Dt.14.02.2022. Sokoliuk O. performed the work in frame of the "Mathematical modeling in interdisciplinary research of processes

In the present study, we investigate the anisotropic stellar solutions admitting Finch-Skea symmetry (viable and non-singular metric potentials) in the presence of some exotic matter fields, such as Bose-Einstein Condensate (BEC) dark matter, the Kalb-Ramond fully anisotropic rank-2 tensor field from the low-energy string theory effective action, and the gauge field imposing symmetry. Interior spacetime is matched with both Schwarzchild and Reissner-Nördstrom vacuum spacetimes for BEC, KB, and gauge fields. In addition, we study the energy conditions, Equation of State (EoS), radial derivatives of energy density and anisotropic pressures, Tolman-Oppenheimer-Volkoff equilibrium condition, relativistic adiabatic index, sound speed, and surface redshift. Most of the aforementioned conditions are satisfied. Therefore, the solutions derived in the current study lie in the physically acceptable regime.

Nonstationary Deformed Singular Oscillator: Quantum Invariants and the Factorization Method

New families of time-dependent potentials related with the stationary singular oscillator are introduced. This is achieved after noticing that a nonstationary quantum invariant can be constructed for the singular oscillator. Such a quantum invariant depends on coefficients related to solutions of the Ermakov equation, where the latter guarantees the regularity of the solutions at each time. In this form, after applying the factorization method to the quantum invariant rather than to the Hamiltonian, one manages to introduce the time parameter into the transformation, leading to factorized operators that become the constants of motion for the new time-dependent Hamiltonians. At the appropriate limit, the initial quantum invariant reproduces the stationary singular oscillator Hamiltonian. Some families of stationary potentials already reported by other authors are also recovered as particular cases. A striking feature of the method is that the singular barrier of the potential can be managed to vanish, which leads to non-singular time-dependent potentials.

Evolution by Schrödinger equation of Aharonov–Berry superoscillations in centrifugal potential

In recent years, we have investigated the evolution of superoscillations under Schrödinger equation with non-singular potentials. In all those cases, we have shown that superoscillations persist in time. In this paper, we investigate the centrifugal potential, which is a singular potential, and we show that the techniques developed to study the evolution of superoscillations in the case of the Schrödinger equation with a quadratic Hamiltonian apply to this setting. We also specify, in the case of the centrifugal potential, the notion of super-shift of the limit solution, a fact explained in the last section of this paper. It then becomes apparent that superoscillations are just a particular case of super-shift.

The Gaussian integration method of the Schrödinger equation and quantum 1-D theory of low gain free electron laser

We study the time-dependent solutions of Schrödinger equations ruled by different non-singular potentials. We employ a recently proposed integration procedure, assuming a time-dependent Gaussian shape for the wave function. The method is independent of the specific form of the potential and allows a straightforward separation of the time and spatial variables. Here, we reconsider the integration method by the use of the formalism of two-variable Hermite polynomials providing a very simple derivation of the relevant physical quantities. This method is eventually exploited to study different problems including anharmonic oscillators and pendulum-like potentials. Regarding the case of periodic potentials, we touch on the application of the method to the quantum free-electron laser dynamics. Finally, we comment on future developments of this line of research regarding the relevant comparison with other exact and approximate integration schemes.

Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators

The paper gives a review of recent progress in the classification of monodromy-free Schr\"odinger operators with rational potentials. We concentrate on a class of potentials constituted by generalized Hermite polynomials. These polynomials defined as Wronskians of classic Hermite polynomials appear in a number of mathematical physics problems as well as in the theory of random matrices and 1D SUSY quantum mechanics. Being quadratic at infinity, those potentials demonstrate localized oscillatory behavior near the origin. We derive an explicit condition of non-singularity of the corresponding potentials and estimate a localization range with respect to indices of polynomials and distribution of their zeros in the complex plane. It turns out that 1D SUSY quantum non-singular potentials come as a dressing of the harmonic oscillator by polynomial Heisenberg algebra ladder operators. To this end, all generalized Hermite polynomials are produced by appropriate periodic closure of this algebra which leads to rational solutions of the Painlev\'e IV equation. We discuss the structure of the discrete spectrum of Schr\"odinger operators and its link to the monodromy-free condition.

Higher order supersymmetric truncated oscillators

We study the supersymmetric partners of the harmonic oscillator with an infinite potential barrier at the origin and obtain the conditions under which it is possible to add levels to the energy spectrum of these systems. It is found that instead of the usual rule for non-singular potentials, where the order of the transformation corresponds to the maximum number of levels which can be added, now it is the integer part of half the order of the transformation which gives the maximum number of levels to be created.

Non-deformed singular and non-singular exponential-type potentials

The canonical transformation method applied to the Schrödinger equation to transform it into a second-order differential equation of hypergeometric-type is presented. Starting from there, those exactly solvable multiparameter exponential-type (ME-T) potentials with hypergeometric wavefunctions that belong to the families of radial (singular) and one-dimensional (non-singular) potentials, are obtained. Furthermore, we show how the choice of the involved parameters leads, as particular cases, to different deformed or non-deformed potential models already used in the study of electronic properties of diatomic molecules. Also, the analysis of parameters lets us identify the couple of potential partners (singular/non-singular) that correspond to each choice of the parameters appearing in the ME-T potential. As a useful application of the proposal, the most important non-deformed exponential potential models are considered for which it can be viewed as a unified treatment with the following advantages: (1) It is not necessary to use a special method to solve the Schrödinger equation for a specific potential model because solution is obtained as particular case by the simple choice of the involved parameters; (2) The families of singular and non-singular potentials are straightforward identified; (3) The corresponding associated partners, between radial and one-dimensional non-deformed potentials, are found; (4) New potentials, as interesting alternatives for quantum applications, are obtained. In addition, from the conditions that parameters must meet to have physically acceptable solutions, we establish the requirements for the existence or not of singular/non-singular potential partners.

From the generalized Morse potential to a unified treatment of the D-dimensional singular harmonic oscillator and singular Coulomb potentials

Bound-state solutions of the singular harmonic oscillator and singular Coulomb potentials in arbitrary dimensions are generated in a simple way from the solutions of the one-dimensional generalized Morse potential. The nonsingular harmonic oscillator and nonsingular Coulomb potentials in arbitrary dimensions with their additional accidental degeneracies are obtained as particular cases. Added bonuses from these mappings are the straightforward determination of the critical attractive singular potential, the proper boundary condition on the radial eigenfunction at the origin and the inexistence of bound states in a pure inversely quadratic potential.

Static spherically symmetric black holes with scalar field

Static spherically symmetric black holes and particle like solutions with self interacting minimally coupled scalar field $$\varphi $$ are analyzed. They are asymptotically flat or anti-de Sitter (AdS). We express them in terms of a single function $$\rho $$ which undergoes simple conditions. If $$\varphi $$ is nontrivial the ADM mass $$M$$ has to be positive. No-hair theorems are generalized to the AdS asymptotic. For both asymptotics the Killing horizon is nondegenerate and its radius cannot be bigger than $$2M$$ . Derivatives of $$\rho $$ at singularity determine properties of admissible potentials $$V(\varphi )$$ as regularity, boundedness and behavior for maximal values of $$\varphi $$ . Several classes of solutions with singular or nonsingular potentials are obtained. Their examples are presented in a form of plots.

Kinetic theory of spatially homogeneous systems with long-range interactions: II. Historic and basic equations

We provide a short historic of the early development of kinetic theory in plasma physics and synthesize the basic kinetic equations describing the evolution of systems with long-range interactions derived in Paper I. We describe the evolution of the system as a whole and the relaxation of a test particle in a bath of field particles at equilibrium or out of equilibrium. We write these equations for an arbitrary long-range potential of interaction in a space of dimension d . We discuss the scaling of the relaxation time with the number of particles for non-singular potentials. For always spatially homogeneous distributions, the relaxation time of the system as a whole scales like N in d > 1 and like N2 (presumably) or like eN (possibly) in d = 1 . For always spatially inhomogeneous distributions, the relaxation time of the system as a whole scales like N in any dimension of space. For 1D systems undergoing a dynamical phase transition from a homogeneous to an inhomogeneous phase, we expect a relaxation time of the form N δ with 1 < δ < 2 intermediate between the two previous cases. The relaxation time of a test particle in a bath always scales like N . We also discuss the kinetic theory of systems with long-range interactions submitted to an external stochastic potential. This paper gathers basic equations that are applied to specific systems in Paper III.

Factorization method and new potentials from the inverted oscillator

In this article we will apply the first- and second-order supersymmetric quantum mechanics to obtain new exactly-solvable real potentials departing from the inverted oscillator potential. This system has some special properties; in particular, only very specific second-order transformations produce non-singular real potentials. It will be shown that these transformations turn out to be the so-called complex ones. Moreover, we will study the factorization method applied to the inverted oscillator and the algebraic structure of the new Hamiltonians.

The time-dependent Schrödinger equation of dimension k + 1: explicit and rational solutions via GBDT and multinodes

A version of the binary Darboux transformation is constructed for a non-stationary Schrödinger equation of dimension k + 1, where k ⩾ 1 is the number of space variables. This is an iterated generalized Bäcklund–Darboux transformation version. New families of non-singular and rational potentials and solutions are obtained. Some results are also new for the case that k = 1. A certain generalization of a colligation introduced by M S Livšic and a generalization of the S-node introduced by L A Sakhnovich have been successfully used in our construction.

Nonsingular potentials from excited state factorization of a quantum system with position-dependent mass

The modified factorization technique of a quantum system characterized by a position-dependent mass Hamiltonian is presented. It has been shown that the singular superpotential defined in terms of a mass function and an excited state wavefunction of a given position-dependent mass Hamiltonian can be used to construct non-singular isospectral Hamiltonians. The method has been illustrated with the help of a few examples.

Fictitious-Time Wave-Packet Dynamics in Atomic Systems

Gaussian wave packets (GWPs) are well suited as basis functions to describe the time evolution of arbitrary wave functions in systems with non-singular smooth potentials. They are rare in atomic systems on account of the singular behaviour of the Coulomb potential.We present a time-dependent variational method that makes the use of GWPs possible in the description of propagation of quantum states also in these systems. We use a regularization of the Coulomb potential and introduce a fictitious-time coordinate in which the evolution of an initial state can be calculated exactly and analytically for a pure Coulomb potential. Therefore, in perturbed atomic systems variational approximations only arise from those parts of the potentials which deviate from the Coulomb potential. The method is applied to the hydrogen atom in external magnetic and electric fields. It can be adapted to systems with definite symmetries and thus allows for a wide range of applications.

Isospectral potentials from modified factorization

Factorization of quantum-mechanical potentials has a long history extending back to the earliest days of the subject. In the present article, the nonuniqueness of the factorization is exploited to derive new isospectral nonsingular potentials. Many one-parameter families of potentials can be generated from known potentials using a factorization that involves superpotentials defined in terms of excited states of a potential. For these cases an operator representation is available. If ladder operators are known for the original potential, then a straightforward procedure exists for defining such operators for its isospectral partners. The generality of the method is illustrated with a number of examples which may have many possible applications in atomic and molecular physics.

Second-order supersymmetric operators and excited states

Factorization of quantum mechanical Hamiltonians has been a useful technique for some time. This procedure has been given an elegant description by supersymmetric quantum mechanics, and the subject has become well developed. We demonstrate that the existence of raising and lowering operators for the harmonic oscillator (and many other potentials) can be extended to their supersymmetric partners. The double supersymmetry (or a factorization chain) is used to obtain non-singular isospectral potentials, and the explicit expressions for the ladder operators, wavefunctions and probability densities are provided. This application avoids the technical complexities of the most general approaches, and requires relatively modest methods from supersymmetric quantum mechanics.

On nonsingular potentials of Cox–Thompson inversion scheme

We establish a condition for obtaining nonsingular potentials using the Cox–Thompson inverse scattering method with one phase shift. The anomalous singularities of the potentials are avoided by maintaining unique solutions of the underlying Regge–Newton integral equation for the transformation kernel. As a by-product, new inequality sequences of zeros of Bessel functions are discovered.

Optical potentials using resonance states in supersymmetric quantum mechanics

Complex potentials are constructed as Darboux-deformations of short range, radial nonsingular potentials. They behave as optical devices which both refracts and absorbs light waves. The deformation preserves the initial spectrum of energies and it is implemented by means of a Gamow-Siegert function (resonance state). As straightforward example, the method is applied to the radial square well. Analytical derivations of the involved resonances show that they are ‘quantized’ while the corresponding wave-functions are shown to behave as bounded states under the broken of parity symmetry of the related one-dimensional problem.

Improving the convergence of an iterative algorithm proposed by Waxman

In the iterative algorithm recently proposed by Waxman for solving eigenvalue problems, we point out that the convergence rate may be improved. For many non-singular symmetric potentials which vanish asymptotically, a simple analytical relationship between the coupling constant of the potential and the ground state eigenvalue is obtained which can be used to make the algorithm more efficient.

Semiclassical approach to Regge poles trajectories calculations for nonsingular potentials: Thomas–Fermi type

A simple semiclassical approach, based on the investigation of anti-Stokes line topology, is presented for calculating Regge poles for nonsingular (Thomas–Fermi type) potentials, namely potentials with singularities at the origin weaker than order −2. The anti-Stokes lines for Thomas–Fermi potentials have a more complicated structure than those of singular potentials and require careful application of complex analysis. The explicit solution of the Bohr–Sommerfeld quantization condition is used to obtain approximate Regge poles. We introduce and employ three hypotheses to obtain several terms of the Regge pole approximation.Please note that the pdf of this article was replaced with a corrected version on 29 June 2004. Minor changes have been made to pages 6950–6953.