Partner Potentials Research Articles

The rotating Morse potential energy eigenvalues solved by using the analytical transfer matrix method

We study the eigenvalues of the rotating Morse potential by using the quantization condition from the analytical transfer matrix (ATM) method. A hierarchy of supersymmetric partner potentials is obtained with Pekeris approximation, which can be used to calculate the energies of higher rotational states from the energies of lower states. The energies of rotational states of the hydrogen molecule are calculated by the ATM condition, and comparison of the results with those from the hypervirial perturbation method reveals that the accuracy of the approximate expression of Pekeris for the eigenvalues of the rotating Morse potential can be improved substantially in the framework of supersymmetric quantum mechanics.

Equivalent Hermitian Hamiltonians for some non-Hermitian Hamiltonians

Six years ago, by using operator techniques and path-integral methods, it was shown that the complex non-Hermitian $PT$-symmetric Hamiltonian ${p}^{2}\ensuremath{-}g{x}^{4}$ is equivalent to a conventional Hermitian Hamiltonian ${p}^{2}+4g{x}^{4}\ensuremath{-}2\ensuremath{\hbar}\sqrt{g}x$. Further it was revealed that the linear term in the Hermitian Hamiltonian is anomalous in the sense that it has no classical analog. In this paper we show that the complex non-Hermitian $\mathit{PT}$-symmetric Hamiltonian ${p}^{2}\ensuremath{-}g{x}^{4}+4i\ensuremath{\hbar}\sqrt{g}x$ and the conventional Hermitian Hamiltonian ${p}^{2}+4g{x}^{4}+6\ensuremath{\hbar}\sqrt{g}x$ have the same eigenspectra. In this case, the anomalous terms in both Hamiltonians are different from the previous one and vanish in the semiclassical limit. Further these equivalent Hamiltonians have zero-energy ground states. The exact ground-state wave functions and supersymmetric partner potentials are derived.

Two-dimensional magnetic Schrödinger equations, Darboux transformations and solutions of associated auxiliary equations

We extend the formalism of Darboux transformations to magnetic Schrödinger equations in two dimensions. The partner potentials associated with the Darboux transformation are obtained in explicit form through an integral formula. We apply our construction to a particular spectral problem of Dirichlet type, obtaining the Darboux partner, its spectrum and the corresponding solutions in explicit form.

Experimental examination of women's selection criteria for sperm donors versus life partners

To evaluate factors shaping mate‐selection decisions, 2 age groups of women were randomly assigned to hypothetical scenarios wherein they sought a sperm donor or life partner. Respondents were asked to (a) construct an ideal man and (b) rate and rank the importance of traits associated with good genes, parenting ability, partner potential, and economic resources. Women seeking donors valued good genes more and partner potential less than women seeking partners. Younger women sought greater degree of physical resemblance to donors than to partners. Older women showed greater desire to be similar to partners than donors on sociocultural characteristics. The findings suggest women adjust their selection criteria as a function of context and that mate preferences may change as women mature.

Intertwining relations and Darboux transformations for the wave equations

Using the intertwining operator technique we construct Darboux transformations for the wave equation with position-dependent effective mass and with linearly energy-dependent potentials. The formally adjoint generators of supersymmetry and two formally self-adjoint superpartner Hamiltonians are constructed and they close a quadratic pseudo-superalgebra. The Darboux transformations are constructed in differential and integral forms and an interrelation is established between them. The approach is applied to generation of isospectral potentials with additional or removal bound states or construction of new partner potentials without changing the spectrum, i.e. fully isospectral potentials. The method is illustrated by some concrete examples. The influence of distance between levels on the form of potentials is investigated. In particular, asymmetric double well potentials are generated.

Factorization method in scalar field onAdS3in spherical coordinates

In this paper, we reformulate free scalar field on AdS3 with spherical coordinates. The suitable coordinates lead us to obtain the action for the massless free particle. This action admits to apply the variable separation and obtains the corresponding second order equation. So, we factorize this equation with respect to n and m and obtain the wave function and energy spectrum. In order to have the shape invariance condition, we show that the corresponding Hamiltonian must be factorized just in terms of m. This leads us to study the partner potential and super potential. Finally, we discuss the commutation relation between supercharge and corresponding Hamiltonian, which shows that the first order operators will be a form of generators algebra.

Phase Shift between Supersymmetric Partner Potentials

The relationship between phase shifts produced by two supersymmetric (SUSY) partner potentials has been examined critically. Phase shifts produced by two SUSY partner potentials bear a simple relation irrespective of being shape invariant potential (SIP) or not (non-SIP). In general, the phase shift cannot be obtained algebraically, even for standard SIPs except the Coulomb potential.

Asymptotic iteration approach to supersymmetric bistable potentials

We examine quasi exactly solvable bistable potentials and their supersymmetric partners within the framework of the asymptotic iteration method (AIM). It is shown that the AIM produces excellent approximate spectra and that sometimes it is found to be more useful to use the partner potential for computation. We also discuss the direct application of the AIM to the Fokker—Planck equation.

ON THE ALGEBRAIC DERIVATION OF PHASE SHIFTS FOR SHAPE INVARIANT POTENTIALS

Phase shifts produced by two supersymmetric partner potentials satisfy a simple relation. One expects that the phase shift can be obtained algebraically, if the potential is shape invariant. We show that this is true only for a special class of shape invariant potentials. Among known shape invariant potentials, only the Coulomb potential satisfy this requirement. The Coulomb phase shift can be obtained analytically, which agrees with the well-known standard result.

A new two-parameter family of potentials with a tunable ground state

In a previous paper (Fellows and Smith 2009 J. Phys. A: Math. Theor. 42 335303) we solved a countably infinite family of one-dimensional Schrödinger equations by showing that they were supersymmetric partner potentials of the standard quantum harmonic oscillator. In this work we extend these results to find the complete set of real partner potentials of the harmonic oscillator, showing that these depend upon two continuous parameters. Their spectra are identical to that of the harmonic oscillator, except that the ground state energy becomes a tunable parameter. We finally use these potentials to analyse the physical problem of Bose–Einstein condensation in an atomic gas trapped in a dimple potential.

Semiclassical aspects and supersymmetry of bound Dirac states for central pseudo-scalar potentials

Relativistic bound states for a linear, radial pseudo-scalar potential model are discussed. The two radial Dirac components are known to have a close connection to partner states in super-symmetric quantum mechanical theory. The pseudo-scalar potential plays the role of the ‘super potential’. Hence, the Dirac components satisfy decoupled Schrödinger-type equations with isospectral, so-called, ‘partner potentials’ except possibly for a single state; the ground state corresponding to one of the partner potentials. The energy spectrum of a confining linear radial potential is discussed in some detail. Accurate amplitude-phase computations and a novel semiclassical (phase-integral) approach are presented.

Quantum isotonic nonlinear oscillator as a Hermitian counterpart of Swanson Hamiltonian and pseudo-supersymmetry

Within the ideas of pseudo-supersymmetry, we have studied a non-Hermitian Hamiltonian , where α ≠ β and ξ is a first-order differential operator, to obtain the partner potentials V+(x) and V−(x) which are new isotonic and isotonic nonlinear oscillators, respectively, as the Hermitian equivalents of the non-Hermitian partner Hamiltonians H±. We have provided an algebraic way to obtain the spectrum and wavefunctions of a nonlinear isotonic oscillator. The solutions of V−(x) which are Hermitian counterparts of Swanson Hamiltonian are obtained under some parameter restrictions that are found. Also, we have checked whether the intertwining operator satisfies η1H− = H+η1, where and is the first-order differential operator, which factorizes Hermitian equivalents of H±.

Planarizable Supersymmetric Quantum Toboggans

In supersymmetric quantum mechanics the emergence of a singularity may lead to the breakdown of isospectrality between partner potentials. One of the regularization recipes is based on a topologically nontrivial, multisheeted complex deformations of the line of coordinate $x$ giving the so called quantum toboggan models (QTM). The consistent theoretical background of this recipe is briefly reviewed. Then, certain supersymmetric QTM pairs are shown exceptional and reducible to doublets of non-singular ordinary differential equations a.k.a. Sturm-Schr\"odinger equations containing a weighted energy $E\to EW(x)$ and living in single complex plane.

Tensions and transitions: effecting change towards sustainability at a mainstream university through staff living and learning at an alternative, civil society college

This paper explores the impact of short immersive residentials at a radical institution for staff currently working in a mainstream one; in this instance, at Schumacher College for those at the nearby University of Plymouth (UK). Schumacher is an independent, alternative college offering residential courses in ‘transformative learning for sustainable living’. Using a qualitative research design, we explore the potential of a University staff Scholarship Scheme partnering with Schumacher to catalyse change towards sustainability in the university's curriculum and pedagogy. Our findings suggest that whilst many of the scholarships realised their anticipated potential to effect change, there was no simple correlation between the scholars' learning and experience at Schumacher and their subsequent personal and professional change. However, since the Plymouth–Schumacher route offers a promising pathway to bring innovation and fresh ideas into thinking and practice, the paper also explores how the mainstream and the radical could interact to better effect, drawing on wider thinking on the principles, challenges and tensions of pursuing sustainability, partnership, institutional learning and change.

Shape Invariant Potentials in Second-Order Supersymmetric Quantum Mechanics

Shape invariance condition in the framework of second-order supersymmetric quantum mechanics is studied. Two classes of solvable shape invariant potentials are consequently constructed, in which the parameters a0 and a1 of partner potentials are related to each other by translation a1 = a0 + ?. In each class, general properties of the obtained shape invariant potentials are systematically investigated. The energy eigenvalues are algebraically determined and the corresponding eigenfunctions are expressed in terms of generalized associated Laguerre polynomials. It is found that these shape invariant potentials are inherently singular, characterized by the 1/x2 singularity at the origin. MSC 2010: 81Q60, 81Q80.

Near-exact supersymmetric partner potentials: Construction and exploitation

Precise supersymmetric (SUSY) partner potentials can be generated only for exactly solvable problems of the stationary Schrodinger equation. This is a severe restriction, as most problems are not amenable to exact solutions. We employ here a linear variational strategy to explicitly construct approximate SUSY partners of a few common, not exactly solvable potentials and subsequently examine their properties to explore the advantages in practical implementation. The efficacy of our proposed scheme is commendable. We demonstrate that, for symmetric potentials, the constructed partners may be so good that the overall recipe has the nicety of generating the whole eigenspectrum by employing only half of the full Hilbert space functions. A similar strategy is shown to work for the odd states too, with proper boundary conditions. Pilot calculations involve a number of low-lying states of some mixed oscillator and double-well potentials. Analysis of the results reveals a few interesting features of the problem of construction of approximate SUSY partners and their practical use. Particularly, we identify places where the operator-level approximations are involved and how far they affect the bounding properties of energies that are obtained as eigenvalues of a matrix diagonalization problem associated with linear variations. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

A strategic partnering framework analysis methodology for public‐private partnerships

PurposeThe purpose of this paper is to view and analyse public‐private partnerships (PPPs) under a strategic partnering approach between the key parties involved, i.e. public sector, private sector and lenders, and their business environment.Design/methodology/approachA strategic partnering framework analysis methodology has been devised based on existing and well‐known business strategic analysis tools (the political‐economic‐social‐technological (PEST) and strengths‐weaknesses‐opportunities‐threats (SWOT) analysis). The methodology consists of modules and may be used to identify the potential of strategic partnering in a sector and/or country and/or for a particular project in a procurement process. By using appropriate modules of the methodology, public sector partnering requirements or the market potential for PPPs, in general, may be assessed.FindingsThe small‐scale application of a module of the methodology is demonstrated through an international consultation on the influence of the present credit crunch. Findings indicate market changes and trends that may be pursued.Research limitations/implicationsThe proposed strategic partnering methodology may be used in all business partnership applications. Its scope is to identify the key elements on which the particular partnership should be based.Originality/valuePPPs have been viewed through finance‐ and service‐based approaches. The present study introduces the view of a partnership‐based approach. In the effort, the study also presents improvements to the commonly used PEST and SWOT analysis by adding a dynamic dimension to the analysis.

Ladder operators' normal ordering problem for quantum-deformed systems and the (q, p)-generalization of the Stirling and Bell numbers

We resolve the ladder operators normal ordering problem for strings in the form for (q, p)-deformed supersymmetric and shape-invariant potential systems, where n is a positive integer. We provide exact and explicit expressions for their normal form , where in all are at the right-hand side, and find that the solution involves expansion-coefficients sequence which corresponds to the generalization, for any (q, p)-deformed shape-invariant potential system, of the classical Stirling and Bell numbers. We show that these numbers are given for expressions depending on parameters related to the forms of supersymmetric partner potentials and the system quantum deformation. We apply the general formalism to Pöschl–Teller, Morse and Scarf potential systems.

Family of fish-eye-related models and their supersymmetric partners

A large family of potentials related to the Maxwell fish-eye model is derived with the help of conformal mappings. It is shown that the whole family admits square-integrable $E=0$ solutions of the Schr\"odinger equation for discrete values of the coupling constant. A corresponding supersymmetric family of partner potentials to the preceding ones is derived as well. Some applications of the considered potentials are also discussed.

Supersymmetry, PT-symmetry and spectral bifurcation

We demonstrate that large class of PT-symmetric complex potentials, which can have isospectral real partner potentials, possess two different superpotentials. In the parameter domain, where the superpotential is unique, the spectrum is real and shape-invariant, leading to translational shift in a suitable parameter by real units. The case of two different superpotentials, leading to same potential, yields broken PT-symmetry, the energy spectra in the two phases being separated by a bifurcation. Interestingly, these two superpotentials generate the two disjoint sectors of the Hilbert space. In the broken case, shape invariance produces complex parametric shifts.