Quasi-exactly Solvable Research Articles

Davydov–Chaban Hamiltonian with deformation-dependent mass term for the sextic potential

This paper presents analytical expressions for spectra and wave functions derived from a Davydov–Chaban Hamiltonian, which models the collective motion of nonaxial nuclei within a sextic potential and incorporates the Deformation-Dependent Mass (DDM) formalism. The resulting solution, termed the Sextic and DDM Approach ([Formula: see text]-SDDMA), is attained through the combined application of Quasi-Exact Solvability (QES) and a Quantum Perturbation Method (QPM). Spectra and [Formula: see text] transition rates are provided for a selection of 15 nuclei: [Formula: see text]Xe, [Formula: see text]Pt and [Formula: see text]Pd, demonstrating a notable agreement with experimental data. Notably, the DDM parameter plays a crucial role in representing, for the first time, the [Formula: see text] band levels [Formula: see text] and [Formula: see text] as well as the higher levels within the natural ordering, resolving a previously identified anomaly present in all models related to the Davydov–Chaban Hamiltonian. Additionally, we explore the impact of DDM on the staggering effect of the [Formula: see text] band, moments of inertia, and shape phase transition, focusing on the prominent isotopic chains of Xe, Pt and Nd where some of the best candidates are obtained. Moreover, the DDM effect is found to facilitate the emergence of the backbending phenomenon in select nuclei in contrast with other models.

Generating QES potentials supporting zero energy normalizable states for an extended class of truncated Calogero Sutherland model

Motivated by recent interest in the search for generating potentials for which the underlying Schrödinger equation is solvable, we report in the recent work several situations when a zero-energy state becomes bound depending on certain restrictions on the coupling constants that define the potential. In this regard, we present evidence of the existence of regular zero-energy normalizable solutions for a system of quasi-exactly solvable (QES) potentials that correspond to the rationally extended many-body truncated Calogero–Sutherland (TCS) model. Our procedure is based upon the use of the standard potential group approach with an underlying so(2,1) structure that utilizes a point canonical transformation with three distinct types of potentials emerging having the same eigenvalues while their common properties are subjected to the evaluation of the relevant wave functions. These cases are treated individually by suitably restricting the coupling parameters.

Prepotential approach: a unified approach to exactly, quasi-exactly, and rationally extended solvable quantal systems

We give a brief overview of a simple and unified way, called the prepotential approach, to treat both exact and quasi-exact solvabilities of the one-dimensional Schrödinger equation. It is based on the prepotential together with Bethe ansatz equations. Unlike the the supersymmetric method for the exactly-solvable systems and the Lie-algebraic approach for the quasi-exactly solvable problems, this approach does not require any knowledge of the underlying symmetry of the system. It treats both quasi-exact and exact solvabilities on the same footing. In this approach the system is completely defined by the choice of two polynomials and a set of Bethe ansatz equations. The potential, the change of variables as well as the eigenfunctions and eigenvalues are determined in the same process. We illustrate the approach by several paradigmatic examples of Hermitian and non-Hermitian Hamiltonians with real energies. Hermitian systems with complex energies, called the quasinormal modes, are also presented. Extension of the approach to the newly discovered rationally extended models is briefly discussed.

Schrödinger equation as a confluent Heun equation

This article deals with two classes of quasi-exactly solvable (QES) trigonometric potentials for which the one-dimensional Schrödinger equation reduces to a confluent Heun equation (CHE) where the independent variable takes only finite values. Power series for the CHE are used to get polynomial and nonpolynomial eigenfunctions. Polynomials occur only for special sets of parameters and characterize the quasi-exact solvability. Nonpolynomial solutions occur for all admissible values of the parameters (even for values which give polynomials), and are bounded and convergent in the entire range of the independent variable. Moreover, throughout the article we examine other QES trigonometric and hyperbolic potentials. In all cases, for a polynomial solution there is a convergent nonpolynomial solution.

Bethe ansatz solutions and hidden [formula omitted] algebraic structure for a class of quasi-exactly solvable systems

The construction of analytic solutions for quasi-exactly solvable systems is an interesting problem. We revisit a class of models for which the odd solutions were largely missed previously in the literature: the anharmonic oscillator, the singular anharmonic oscillator, the generalized quantum isotonic oscillator, non-polynomially deformed oscillator, and the Schrödinger system from the kink stability analysis of ϕ6-type field theory. We present a systematic and unified treatment for the odd and even sectors of these models. We find generic closed-form expressions for constraints to the allowed model parameters for quasi-exact solvability, the corresponding energies and wavefunctions. We also make progress in the analysis of solutions to the Bethe ansatz equations in the spaces of model parameters and provide insight into the curves/surfaces of the allowed parameters in the parameter spaces. Most previous analyses in this aspect were on a case-by-case basis and restricted to the first excited states. We present analysis of the solutions (i.e. roots) of the Bethe ansatz equations for higher excited states (up to levels n=30 or 50). The shapes of the root distributions change drastically across different regions of model parameters, illustrating phenomena analogous to phase transition in context of integrable models. Furthermore, we also obtain the sl(2) algebraization for the class of models in their respective even and odd sectors in a unified way.

The SUSY partners of the QES sextic potential revisited

In this paper, the SUSY partner Hamiltonians of the quasi-exactly solvable (QES) sextic potential Vqes(x)=νx6+2νμx4+μ2−(4N+3)νx2 , N∈Z+ , are revisited from a Lie algebraic perspective. It is demonstrated that, in the variable z = x 2, the underlying sl2(R) hidden algebra of V qes(x) is inherited by its SUSY partner potential V 1(x) only for N = 0. At fixed N > 0, the algebraic polynomial operator h(x, ∂ x ; N) that governs the N exact eigenpolynomial solutions of V 1 is derived explicitly. These odd-parity solutions appear in the form of zero modes. The potential V 1 can be represented as the sum of a polynomial and rational parts. In particular, it is shown that the polynomial component is given by V qes with a different non-integer (cohomology) parameter N1=N−32 . A confluent second-order SUSY transformation is also implemented for a modified QES sextic potential possessing the energy reflection symmetry. By taking N as a continuous real constant and using the Lagrange-mesh method, highly accurate values (∼20 s. d.) of the energy E n = E n (N) in the interval N ∈ [ − 1, 3] are calculated for the three lowest states n = 0, 1, 2 of the system. The critical value N c above which tunneling effects (instanton-like terms) can occur is obtained as well. At N = 0, the non-algebraic sector of the spectrum of V qes is described by means of compact physically relevant trial functions. These solutions allow us to determine the effects in accuracy when the first-order SUSY approach is applied on the level of approximate eigenfunctions.

Nuclear Shape-Phase Transitions and the Sextic Oscillator

This review delves into the utilization of a sextic oscillator within the β degree of freedom of the Bohr Hamiltonian to elucidate critical-point solutions in nuclei, with a specific emphasis on the critical point associated with the β shape variable, governing transitions from spherical to deformed nuclei. To commence, an overview is presented for critical-point solutions E(5), X(5), X(3), Z(5), and Z(4). These symmetries, encapsulated in simple models, all model the β degree of freedom using an infinite square-well (ISW) potential. They are particularly useful for dissecting phase transitions from spherical to deformed nuclear shapes. The distinguishing factor among these models lies in their treatment of the γ degree of freedom. These models are rooted in a geometrical context, employing the Bohr Hamiltonian. The review then continues with the analysis of the same critical solutions but with the adoption of a sextic potential in place of the ISW potential within the β degree of freedom. The sextic oscillator, being quasi-exactly solvable (QES), allows for the derivation of exact solutions for the lower part of the energy spectrum. The outcomes of this analysis are examined in detail. Additionally, various versions of the sextic potential, while not exactly solvable, can still be tackled numerically, offering a means to establish benchmarks for criticality in the transitional path from spherical to deformed shapes. This review extends its scope to encompass related papers published in the field in the past 20 years, contributing to a comprehensive understanding of critical-point symmetries in nuclear physics. To facilitate this understanding, a map depicting the different regions of the nuclide chart where these models have been applied is provided, serving as a concise summary of their applications and implications in the realm of nuclear structure.

An algebraic approach for the Dunkl–Killingbeck problem from the bi-confluent Heun equation

In this paper, we study the Dunkl–Killingbeck problem in two dimensions. We apply the Lie algebraic approach within the framework of quasi-exact solvability to the radial part of the Dunkl–Killingbeck problem to find the general exact expressions for the energies and corresponding wave functions. The allowed values of the potential parameters are the representation space of sl(2) Lie algebra. In addition, we discuss that the effective potential of the Dunkl–Killingbeck is the same as the obtained from the bi-confluent Heun equation by a suitable variable transformation. Following earlier results, we follow the explicit solutions of this differential equation expressed as a series expansion of Hermite functions and obtain the expansion coefficients from a three-term recurrence relation. In the sequel, we present that this construction leads to the known quasi-exactly solvable (QES) form of the Dunkl–Killingbeck problem. Therefore, we find that the expressions for the energy eigenvalues and wave functions of the corresponding potential term are in agreement with those from the QES formalism. Then, we derive the ladder operators for the Dunkl–Killingbeck problem within the algebraic approach. It seems that this method is the Dunkl–Killingbeck rotation problem solved by operators of the su[Formula: see text] Lie algebra in a specific way.

Mapping atomic trapping in an optical superlattice onto the libration of a planar rotor in electric fields

We show that two seemingly unrelated problems—the trapping of an atom in an optical superlattice (OSL) and the libration of a planar rigid rotor in combined electric and optical fields (PR) – have isomorphic Hamiltonians. Formed by the interference of optical lattices whose spatial periods differ by a factor of two, OSL gives rise to a periodic potential that acts on atomic translation via the AC Stark effect. The PR system, also known as the generalized planar pendulum, is realized by subjecting a planar rigid rotor to combined orienting and aligning interactions due to the coupling of the rotor’s permanent and induced electric dipole moments to the combined fields. The PR system belongs to the class of conditionally quasi-exactly solvable problems and exhibits intriguing spectral properties that have been established in our previous work Becker et al (2017 Eur. Phys. J. D 71 149). Herein, we make use of both the PR OSL mapping and the quasi-exact solvability of the PR system to treat ultracold atoms in an OSL as a semifinite-gap system. The band structure of this system follows from the eigenenergies and their genuine and avoided crossings obtained previously for the PR system as solutions of the Whittaker-Hill equation. These solutions characterize both the squeezing and the tunneling of atoms trapped in an OSL and pave the way to unraveling their dynamics in analytic form. Furthermore, the PR OSL mapping makes it possible to establish correspondence between concepts developed for the two eigenproblems individually, such as localization on the one hand and orientation/alignment on the other.

Bohr Hamiltonian with sextic potential for γ -rigid prolate nuclei with deformation-dependent mass term

The main aim of the present paper is to solve the eigenvalues problem with the Bohr collective Hamiltonian for $\ensuremath{\gamma}$-rigid nuclei within a model we have elaborated by combining two model approaches: the quantum mechanical formalism, namely, deformation-dependent mass formalism (DDM), and the anharmonic sextic oscillator potential for the variable $\ensuremath{\beta}$ and $\ensuremath{\gamma}=0$. The model developed in this way is conventionally called the sextic and DDM approach. Analytical expressions for energy spectra are conjointly derived by means of quasiexact solvability and a quantum perturbation method. Due to the scaling property of the problem, the energy and $B(E2)$ transition ratios depend on two free parameters apart from an integer number which limits the number of allowed states. Numerical results are given for 35 nuclei---$^{98--108}\mathrm{Ru}, ^{100--102}\mathrm{Mo}, ^{116--130}\mathrm{Xe}, ^{180--196}\mathrm{Pt}, ^{172}\mathrm{Os}, ^{146--150}\mathrm{Nd}, ^{132--134}\mathrm{Ce}, ^{154}\mathrm{Gd}, ^{156}\mathrm{Dy}$, and $^{150--152}\mathrm{Sm}$---revealing a good agreement with experiment. Moreover, as proved for the first time by Bonatsos et al. [D. Bonatsos, P. Georgoudis, D. Lenis, N. Minkov, and C. Quesne, Phys. Lett. B 683, 264 (2010)], the dependence of the mass on the deformation with the sextic potential moderates the increase of the moment of inertia with the deformation, removing an important drawback that has been revealed in the constant mass case [Buganu and Budaca, J. Phys. G: Nucl. Part. Phys. 42, 105106 (2015)]. Additionally, the correlation between the DDM and the minimal length formalism persists for the sextic potential. Finally, the DDM effects on the shape phase transition for the most numerous isotopic chains, namely, Ru, Xe, Nd, and Pt, have been duly investigated.

Relativistic solutions of generalized-Dunkl harmonic and anharmonic oscillators

Dunkl derivative enriches solutions by discussing parity due to its reflection operator. Very recently, one of the authors of this manuscript presented one of the most general forms of Dunkl derivative that depends on three Wigner parameters to have a better tuning. In this manuscript, we employ the latter generalized Dunkl derivative in a relativistic equation to examine two dimensional harmonic and anharmonic oscillators solutions. We obtain the solutions by Nikiforov-Uvarov and quasi-exact solvability (QES) methods, respectively. We show that degenerate states can occur according to the Wigner parameter values.

Displaced Harmonic Oscillator V ∼ min [(x + d)2, (x − d)2] as a Benchmark Double-Well Quantum Model

For the displaced harmonic double-well oscillator, the existence of exact polynomial bound states at certain displacements d is revealed. The N-plets of these quasi-exactly solvable (QES) states are constructed in closed form. For non-QES states, the Schrödinger equation can still be considered “non-polynomially exactly solvable” (NES) because the exact left and right parts of the wave function (proportional to confluent hypergeometric function) just have to be matched in the origin.

Rational extension of many particle systems

In this talk, we briefly review the rational extension of many particle systems, and is based on a couple of our recent works. In the first model, the rational extension of the truncated Calogero-Sutherland (TCS) model is discussed analytically. The spectrum is isospectral to the original system and the eigenfunctions are completely expressed in terms of exceptional orthogonal polynomials (EOPs). In the second model, we discuss the rational extension of a quasi exactly solvable (QES) N-particle Calogero model with harmonic confining interaction. New long-range interaction to the rational Calogero model is included to construct this QES many particle system using the technique of supersymmetric quantum mechanics (SUSYQM). Under a specific condition, infinite number of bound states are obtained for this system, and corresponding bound state wave functions are written in terms of EOPs.

Investigation of two-electron quantum dot in a magnetic field by using of quasi-exact-solvable method

We study the non-relativistic particles of a two dimensional two-electron quantum dot. We consider the system in the presence of harmonic plus linear terms as well as a term related to spin interaction. The solution of the system is obtained by using of Quasi-Exactly-Solvable (QES) method. Next, we calculate the energy eigenstates and wave functions of the system in the presence of an external magnetic field.

Extended study on the application of the sextic potential in the frame of X(3)-sextic

The main aim of the present paper is to extensively study the γ-rigid Bohr Hamiltonian with anharmonic sextic oscillator potential for the variable β and γ = 0. For the corresponding spectral problem, a finite number of eigenvalues are explicitly found, by algebraic means, the so-called quasi-exact solvability (QES). The evolution of the spectral and electromagnetic properties by considering higher exact solvability orders is investigated, especially the approximate degeneracy of the ground and first two β bands at the critical point of the shape phase transition from a harmonic to an anharmonic prolate β-soft, as well as the shape evolution within an isotopic chain. The numerical results are given for 39 nuclei, namely, 98–108Ru, 100–102Mo, 116–130Xe, 180–196Pt, 172Os, 146–150Nd, 132–134Ce, 152–154Gd, 154–156Dy, 150–152Sm, 190Hg and 222Ra. Across this study, it seems that the higher QES order improves our results by decreasing the root mean square, mostly for deformed nuclei. The nuclei 100,104Ru, 118,120,126,128Xe, 148Nd and 172Os fall exactly at the critical point.

Quasi-exactly solvable hyperbolic potential and its anti-isospectral counterpart

We solve the eigenvalue spectra for two quasi exactly solvable (QES) Schrödinger problems defined by the potentials V(x;γ,η)=4γ2cosh4(x)+V1(γ,η)cosh2(x)+ηη−1tanh2(x) and U(x;γ,η)=−4γ2cos4(x)−V1(γ,η)cos2(x)+ηη−1tan2(x), found by the anti-isospectral transformation of the former. We use three methods: a direct polynomial expansion, which shows the relation between the expansion order and the shape of the potential function; direct comparison to the confluent Heun equation (CHE), which has been shown to provide only part of the spectrum in different quantum mechanics problems, and the use of Lie algebras, which has been proven to reveal hidden algebraic structures of this kind of spectral problems.

Examples of PT Phase Transition : QM to QFT

Parity Time Reversal (PT) phase transition is a typical characteristic of most of the PT symmetric non-Hermitian (NH) systems. Depending on the theory, a particular system and spacetime dimensionality PT phase transition has various interesting features. In this article we review some of our works on PT phase transitions in quantum mechanics (QM) as well as in Quantum Field theory (QFT). We demonstrate typical characteristics of PT phase transition with the help of several analytically solved examples. In one dimensional QM, we consider examples with exactly as well as quasi exactly solvable (QES) models to capture essential features of PT phase transition. The discrete symmetries have rich structures in higher dimensions which are used to explore the PT phase transition in higher dimensional systems. We consider anisotropic SHOs in two and three dimensions to realize some connection between the symmetry of original hermitian Hamiltonian and the unbroken phase of the NH system. We consider the 2+1 dimensional massless Dirac particle in the external magnetic field with PT symmetric non-Hermitian spin-orbit interaction in the background of the Dirac oscillator potential to show the PT phase transition in a relativistic system. A small mass gap, consistent with the other approaches and experimental observations is generated only in the unbroken phase of the system. Finally we develop the NH formulation in an SU(N) gauge field theoretic model by using the natural but unconventional Hermiticity properties of the ghost fields. Deconfinement to confinement phase transition has been realized as PT phase transition in such a non-hermitian model.

Collective states of even–even nuclei in γ-rigid quadrupole Hamiltonian with minimal length under the sextic potential

In the present paper, we study the collective states of even–even nuclei in γ-rigid mode within the sextic potential and the minimal length (ML) formalism in Bohr–Mottelson model. The eigenvalues problem for the latter is solved by combined means of quasi-exact solvability and a quantum perturbation method. Numerical calculations are performed for 35 nuclei: 98−108Ru, 100−102Mo, 116−130Xe, 180−196Pt, 172Os, 146−150Nd, 132−134Ce, 154Gd, 156Dy and 150−152Sm. Through this study, it appears that our elaborated model leads to an improved agreement of the theoretical results with the corresponding experimental data by reducing the rms with a rate going up to 63% for some nuclei. This comes from the fact that we have combined the sextic potential, which is a very useful phenomenological potential, with the formalism of the ML, which is based on the generalized uncertainty principle and which is in turn a quantum concept widely used in quantum physics. In addition, we investigate the effect of ML on energy ratios, transition rates, moments of inertia and a shape phase transition for the most numerous isotopic chains, namely Ru, Xe, Nd and Pt.

Quasi-exact and asymptotic iterative solutions of Dirac equation in the presence of some scalar potentials

In this paper, the Dirac equation in the presence of some scalar potentials based on sl(2) Lie algebra is solved by quasi-exact solvability theory. The configuration of the classes III and VI potentials in the Turbiner’s classification is constructed. Then, the Bethe ansatz equations are calculated so that the energy eigenvalues and eigenfunctions are obtained. Also, we study the problem by using asymptotic iteration method. Finally, we compare the results obtained by these two methods.

Quasi-Exactly Solvable Jacobi Elliptic Potential

A new example of 2×2 -matrix quasi-exactly solvable (QES) Hamiltonian which is associated to a Jacobi elliptic potential is constructed. We compute algebraically three necessary and sufficient conditions with the QES analytic method for the Jacobi Hamiltonian to have a finite dimensional invariant vector space. The matrix Jacobi Hamiltonian is called quasi-exactly solvable.