Sextic Potential Research Articles

A THEORETICAL INVESTIGATION OF THE SCHRÖDINGER EQUATION FOR A MODIFIED TWO DIMENSIONAL PURELY SEXTIC DOUBLE-WELL POTENTIAL

By Abdelmadjid Maireche. A theoretical analytical investigation for the exact solvability of non-relativistic quantum spectrum systems, at low energy for modified two-dimensional purely sextic double-well potential (2D-modified sextic DWAO potential) is discussed.

Chains of boson stars

We study axially symmetric multi-soliton solutions of a complex scalar field theory with a sextic potential, minimally coupled to Einstein's gravity. These solutions carry no angular momentum and can be classified by the number of nodes of the scalar field, $k_z$, along the symmetry axis; they are interpreted as chains with $k_z+1$ boson stars, bound by gravity, but kept apart by repulsive scalar interactions. Chains with an odd number of constituents show a spiraling behavior for their ADM mass (and Noether charge) in terms of their angular frequency, similarly to a single fundamental boson star, as long as the gravitational coupling is small; for larger coupling, however, the inner part of the spiral is replaced by a merging with the fundamental branch of radially excited spherical boson stars. Chains with an even number of constituents exhibit a truncated spiral pattern, with only two or three branches, ending at a limiting solution with finite values of ADM mass and Noether charge.

Warm bounce in loop quantum cosmology and the prediction for the duration of inflation

We study and estimate probabilistic predictions for the duration of the preinflationary and slow-roll phases after the bounce in loop quantum cosmology, determining how the presence of radiation in the prebounce phase affects these results. We present our analysis for different classes of inflationary potentials that include the monomial power-law chaotic type of potentials, namely, for the quadratic, quartic and sextic potentials and also for a Higgs-like symmetry breaking potential, considering different values for the vacuum expectation value in the latter case. We obtain the probability density function for the number of inflationary e-folds and for other relevant quantities for each model and produce probabilistic results drawn from these distributions. This study allows us to discuss under which conditions each model could eventually lead to observable signatures on the spectrum of the cosmic microwave background, or, else, be also excluded for not predicting a suffcient amount of accelerated expansion. The effect of radiation on the predictions for each model is explicitly quantified. The obtained results indicate that the number of inflationary e-folds in loop quantum cosmology is not a priori an arbitrary number, but can in principle be a predictable quantity, even though the results are dependent on the model and on the amount of radiation in the Universe prior to the start of the inflationary regime.

Some limitations on: Analytic approximate eigenvalues by a new technique. Application to sextic anharmonic potentials, D.D. Almeida and P. Martin, Results. in. Phys. 8, 140–145 (2018)

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=p2+x2+<x2>x4. 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.

Critical potentials and fluctuations phenomena with quartic, sextic, and octic anharmonic oscillator potentials

This is an investigation of the energy levels of three classes of two-dimensional anharmonic oscillator potentials characterized by free-, one-, and two- parameters with quartic, sextic, and octic anharmonicity. The wave functions and energy eigenvalues are found through the discretization using the nine-point finite-difference method. It is proved that the free-parameter potentials provide a “bridge” between the two-dimensional harmonic oscillator potential and two-dimensional infinite square well potential. However, the one- and two- parameter potentials show a transition from tunneling splitting mode to the spectra of free-parameter potentials crossing to the spectrum of infinite square well potential and in the same time present new features. One can distinguish cases of fluctuations phenomena in the ground state and excited states. We obtained, for the first time, a ground state with a four-peak structure. The quantum tunneling between the four minima of one- and two- parameter potentials depends on the characteristics of the separating barrier. These phase transitions are not of the ordinary thermodynamic category, but truly they are phase transitions in the structure of the low-lying energy spectra. The concept of critical potentials is associated with the potentials at the phase-transition points. The coherent quadrupole-octupole motion model, using these potentials, is applied to describe yrast and non-yrast energy sequences with alternating parity in several even-even nuclei from different regions, namely, in 100Mo, 146Ba, 226Ra, and 226Th.

Semi-exact solutions of sextic potential plus a centrifugal term

We revisit the one-dimensional quantum system of a sextic potential added with a centrifugal term, $$V(x)=a\, x^{-2}+b\, x^2+c\,x^4+d\,x^6$$ , where the parameters a, b, c and d are arbitrary. We find that its solutions can be expressed as a Biconfluent Heun function $$H_{B}(\alpha , \beta , \gamma ,\delta ; z)$$ , while the associated energy spectrum is determined by the parameter $$\delta$$ . The semi-exact solutions of wave functions fully consist with the properties of the potential.

γ-Unstable Bohr Hamiltonian with sextic potential for odd-A nuclei

In this paper, odd-A nuclei in the vibrational to γ-soft transitional region are studied in the collective model, in which the odd-A system is described by the Bohr Hamiltonian with the quasi exactly solvable sextic β-part potential for the even-even core coupled with a single nucleon in a j=3/2 orbit via the β-independent five-dimensional spin-orbit interaction and the total angular momentum degeneracy breaking term. To test the validity of the coupling scheme, we use the model to reproduce experimentally available level energies and B(E2) values of 187,189,191,193,195Ir. It is clearly shown from both the level energies and the known B(E2) values for 191,193Ir fitted that the model with the β-independent five-dimensional spin-orbit interaction and the total angular momentum degeneracy breaking term seems adequate to describe the low-lying level pattern and the structure of these odd-A nuclei.

Quasi-exact solutions for the Bohr Hamiltonian with sextic oscillator potential

A discussion on the quasi-exact solution of the Bohr Hamiltonian with sextic oscillator potential is made by attracting the attention on some recent results of its application to the phase transition from spherical vibrator to a γ-unstable system. More precisely, it is underlined the importance of the solvability order on the structure of the states, especially in the critical point, respectively, in the deformed region of the phase transition.

Quasi-exact description of the γ-unstable shape phase transition

The equation of the [Formula: see text]-unstable Bohr Hamiltonian, with particular forms of the sextic potential in the [Formula: see text] shape variable, is exactly solved for a finite number of states. The shape of the quasi-exactly solvable potential is then defined by the number of exactly determined states. The effect of exact solvability order on the spectral characteristics of the model is closely investigated, especially, concerning the critical point of the phase transition from spherical to deformed shapes. The energy spectra and the [Formula: see text] transition probabilities, up to a scaling factor, depend only on a single-free parameter, while for the critical point, parameter-free results are available. Several numerical applications are done for nuclei undergoing a [Formula: see text]-unstable shape phase transition in order to identify critical nuclei based on the most suitable exact solvability order.

Analytical study of the $$ \gamma $$-unstable Bohr Hamiltonian with quasi-exactly solvable decatic potential

In this work, the $$\gamma $$-unstable Bohr Hamiltonian with quasi-exactly solvable decatic potential with a centrifugal barrier is considered to describe nuclei near the critical point of the vibrational to $$\gamma $$-unstable (shape) phase transition. Analytical expression of the wavefunctions and the corresponding eigen-energies are derived under the two quasi-exactly solvable constraints on parameters of the potential. The theoretical results are compared with those of the E(5) model and those of the same Hamiltonian with the quasi-exactly solvable sextic potential. Low-lying level energy ratios and B(E2) ratios of some nuclei near the critical point are fitted and compared with the experimental data and the fitting quality of previous models. The fitting results show that the E(5) model is indeed the best in description of nuclei near or at this critical point, while the decatic model seems a little better than the sextic one near the critical point. Furthermore, though further check on the fitting quality to the B(E2) values is needed, the fitting results for even-even $$^{118{-}128}$$Xe show that the decatic model seems the best in fitting the energy ratios, while the fitting qualities of both the decatic model and the E(5) model to the B(E2) ratios are quite the same, which are always better than the sextic model.

Study of the sextic and decatic anharmonic oscillators using an interpolating scale function

Anharmonic oscillators with the sextic and decatic potentials are studied employing the refinable interpolating scale functions. This method yields highly accurate values of both energy eigenvalues and eigenfunctions for the sextic and decatic oscillator without constraining the potential parameters. Convergence of the solutions in the present method is noticed to be very fast.

Three-body closed chain of interactive (an)harmonic oscillators and the algebra

In this work we study 2- and 3-body oscillators with quadratic and sextic pairwise potentials which depend on relative distances, , between particles. The two-body harmonic oscillator is two-parametric and can be reduced to a one-dimensional radial Jacobi oscillator with hidden algebra , while in the 3-body case such a reduction is not possible in general. Our study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only (S-states). We pay special attention to the cases where all masses of the particles and spring constants are unequal as well as to the atomic, where one mass is infinite, and molecular, where two masses are infinite, limits; it is a non-integrable system. In general, a three-body harmonic oscillator is 7-parametric depending on three masses and three spring constants, and frequency; it is an exactly-solvable problem with spectra linear in three quantum numbers and with hidden algebra . In particular, the first and second order integrals of the 3-body oscillator for unequal masses are searched: it is shown that for certain relations involving masses and spring constants the system becomes maximally (minimally) superintegrable in the case of two (one) relations.

QUASI-EXACT SOLVABILITY OF THE D-DIMENSIONAL SEXTIC POTENTIAL IN TERMS OF TRUNCATED BI-CONFLUENT HEUN FUNCTIONS

The D-dimensional Schr¨odinger equation for an isotropic sextic potential is brought to a form compatible with the canonical bi-confluent Heun differential equation. The quasi-exactly solvable properties of the model are recovered by considering polynomial solutions for the bi-confluent Heun equation which constrains the potential parameters in terms of rotation quantum number, space dimension and order of the exact solvability. It is shown that the state independence of the potential can be maintained by using a see-saw adjustment between the rotation quantum number and the exact solvability order. An analysis on the exactly solvable instances of the sextic potential is presented in correlation with the extended set of exactly solvable states.

Application of the Bohr Hamiltonian with a double-well sextic potential to collective states in Mo isotopes

A diagonalization procedure for an SO(5) invariant Bohr Hamiltonian with a sextic potential for the β shape variable, having simultaneous spherical and deformed minima, is presented. The double-well potential allows for the study of shape coexistence and mixing phenomena from a geometrical perspective. The spectral and dynamical features of the model are investigated for different shapes of the potential amended with the centrifugal contribution from the kinetic term and subjected to the condition of degenerate minima. A generalization of the model is applied for the description of the low-lying energy spectra and shape mixing evolution in Mo nuclei. For each isotope, a shape transition from small to large deformation is found to occur between distinct states with various degrees of intensity and shape mixing.

Comment on “Elimination of degeneracy in the γ -unstable Bohr Hamiltonian in the presence of an extended sextic potential”

Comment on “Elimination of degeneracy in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>γ</mml:mi></mml:math> -unstable Bohr Hamiltonian in the presence of an extended sextic potential”

A family of analytically solvable Schrödinger equations related by Levi-Civita transformation

Some two-dimensional problems in non-relativistic quantum mechanics can connect to each other by certain spatial transformations such as Levi-Civita transformation. This property allows forming a series of two-dimensional problems into an interrelated family. Starting from two related problems namely Coulomb plus harmonic oscillator and sextic double-well anharmonic oscillator potentials, such family is constructed via repeatedly applying Levi-Civita transformations. Obviously, this family contains various of exactly analytically solvable problems. The quasi-exact solution for each unknown member of this family is also obtained and systematically investigated.

Spectral form factor in non-Gaussian random matrix theories

We consider Random Matrix Theories with non-Gaussian potentials that have a rich phase structure in the large $N$ limit. We calculate the Spectral Form Factor (SFF) in such models and present them as interesting examples of dynamical models that display multi-criticality at short time-scales and universality at large time scales. The models with quartic and sextic potentials are explicitly worked out. The disconnected part of the Spectral Form Factor (SFF) shows a change in its decay behavior exactly at the critical points of each model. The dip-time of the SFF is estimated in each of these models. The late time behavior of all polynomial potential matrix models is shown to display a certain universality. This is related to the universality in the short distance correlations of the mean-level densities. We speculate on the implications of such universality for chaotic quantum systems including the SYK model.

Four-body problem in d-dimensional space: Ground state, (quasi)-exact-solvability. IV

Due to its great importance for applications, we generalize and extend the approach of our previous papers to study aspects of the quantum and classical dynamics of a 4-body system with equal masses in d-dimensional space with interaction depending only on mutual (relative) distances. The study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only. The ground state (and some other states) in the quantum case and some trajectories in the classical case are of this type. We construct the quantum Hamiltonian for which these states are eigenstates. For d ≥ 3, this describes a 6-dimensional quantum particle moving in a curved space with special d-independent metric in a certain d-dependent singular potential, while for d = 1 it corresponds to a 3-dimensional particle and coincides with the A3 (4-body) rational Calogero model; the case d = 2 is exceptional and is discussed separately. The kinetic energy of the system has a hidden sl(7, R) Lie (Poisson) algebra structure, but for the special case d = 1, it becomes degenerate with hidden algebra sl(4, R). We find an exactly solvable 4-body S4-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasiexactly solvable 4-body sextic polynomial type potential with singular terms. The tetrahedron whose vertices correspond to the positions of the particles provides pure geometrical variables, volume variables, which lead to exactly solvable models. Their generalization to the n-body system as well as the case of nonequal masses is briefly discussed.

The Picard–Fuchs equation in classical and quantum physics: application to higher-order WKB method

The Picard–Fuchs equation is a powerful mathematical tool which has numerous applications in physics, for it allows one to evaluate integrals without resorting to direct integration techniques. We use this equation to calculate both the classical action and the higher-order WKB corrections to it, for the sextic double-well potential and the Lamé potential. Our development rests on the fact that the Picard–Fuchs method links an integral to solutions of a differential equation with the energy as a parameter. Employing the same argument we show that each higher-order correction in the WKB series for the quantum action is a combination of the classical action and its derivatives. From this, we obtain a computationally simple method of calculating higher-order quantum-mechanical corrections to the classical action, and demonstrate this by calculating the second-order correction for the sextic and the Lamé potential. This paper also serves as a self-consistent guide to the use of the Picard–Fuchs equation.

The fractional Allen–Cahn equation with the sextic potential

We extend the classical Allen–Cahn (AC) equation to the fractional Allen–Cahn equation (FAC) with triple-well potential. By replacing the spatial Laplacian and double-well potential with fractional Laplacian and triple-well potential, we observe different dynamics. This study leads us to understand different properties of the FAC equation. We seek the existence, boundedness, and unique solvability of numerical solutions for the FAC equation with triple-well potential. In addition, the inclusion principle for the Allen–Cahn equation is considered. These different properties make us enhance the applicability of the phase-field method to the mathematical modeling in materials science. In computation, the spectral decomposition for the fractional operator allows us to develop a numerical method for the fractional Laplacian problem. For the periodic and discrete Laplacian matrix and vector multiplication, circulant submatices are formed in more than two-dimensional case. Even if the fast Fourier transform (FFT) can be utilized in this modeling, we construct the inverse of the doubly-block-circulant matrix for the solution of the fractional Allen–Cahn equation. In doing so, it helps to straightforwardly understand the numerical treatment, and exploit the properties of the discrete Fourier transforms.