Supersymmetric Potentials Research Articles

Supersymmetric quantum potentials analogs of classical electrostatic fields

A relation between classical electrostatic fields and Shrödinger-like Hamiltonians is evidenced. Hence, supersymmetric quantum potentials analogous to classical electrostatic fields can be constructed. Proposing an ansatz for the electrostatic potential as the natural logarithm of a nodeless function, it is demonstrated that the electrostatic fields fulfil the Bernoulli equation associated to a second-order confluent supersymmetric transformation. By using the so-called confluent algorithm, it is possible, given a charge density, to find the corresponding electrostatic field as well as the supersymmetric potentials. Furthermore, the associated charge density and the electrostatic field profile of Schrödinger-like solvable potentials can be determined.

Quantum control and quantum speed limits in supersymmetric potentials

Supersymmetry allows one to build a hierarchy of Hamiltonians that share the same spectral properties and which are pairwise connected through common super-potentials. The iso-spectral properties of these Hamiltonians imply that the dynamics and therefore control of different eigenstates are connected through supersymmetric intertwining relations. In this work we explore how this enables one to study general dynamics, shortcuts to adiabaticity and quantum speed limits for distinct states of different supersymmetric partner potentials by using the infinite box as an example.

Sensitivity of transfer cross sections to the bound-state wave functions

We test the sensitivity of transfer reactions to the bound state wave functions within a distorted wave Born approximation formalism. Using supersymmetric transformations, we remove the Pauli-forbidden states from the two-body potentials and generate an equivalent supersymmetric partner. Wave functions from these potentials have the same asymptotics, but they differ in the nuclear interior. This allows us to study the influence of the nuclear interior on transfer cross sections. We apply the calculations to the O16(d,p)17O and C12(7Li, t)16O reactions, which are typical examples of nucleon and α transfer, respectively. The spectroscopic factors for 17O are decreased by about 30% when using supersymmetric potentials. For 16O, the differences are smaller. However, we show that ambiguities exist in the determination of the spectroscopic factors, due to the choice of the angular range where the fit is performed.

The stabilizing effect of gravity made simple

A new approach to vacuum decay in quantum field theory, based on a simple variational formulation in field space using a tunneling potential, is ideally suited to study the effects of gravity on such decays. The method allows to prove in new and simple ways many results, among others, that gravitational corrections tend to make Minkowski or Anti de Sitter false vacua more stable semiclassically or that higher barriers increase vacuum lifetime. The approach also offers a very clean picture of gravitational quenching of vacuum decay and its parametric dependence on the features of a potential and allows to study the BPS domain-walls between vacua in critical cases. Special attention is devoted to supersymmetric potentials and to the discussion of near-critical vacuum decays, for which it is shown how the new method can be usefully applied beyond the thin-wall approximation.

Spontaneous breaking of SO(3) to finite family symmetries with supersymmetry \u2014 an A4 model

We discuss the breaking of SO(3) down to finite family symmetries such as A4, S4 and A5 using supersymmetric potentials for the first time. We analyse in detail the case of supersymmetric A4 and its finite subgroups Z3 and Z2. We then propose a supersymmetric A4 model of leptons along these lines, originating from SO(3) × U(1), which leads to a phenomenologically acceptable pattern of lepton mixing and masses once subleading corrections are taken into account. We also discuss the phenomenological consequences of having a gauged SO(3), leading to massive gauge bosons, and show that all domain wall problems are resolved in this model.

Nonlinear Schrödinger equation with complex supersymmetric potentials

Using the concept of supersymmetry we obtain exact analytical solutions of nonlinear Schrodinger equation with a number of complex supersymmetric potentials and power law nonlinearity. Linear stability of these solutions for self-focusing as well as de-focusing nonlinearity has also been examined.

Generically large nongaussianity in small multifield inflation

If forthcoming measurements of cosmic photon polarization restrict the primordial tensor-to-scalar ratio to r < 0.01, small field inflation will be a principal candidate for the origin of the universe. Here we show that small multifield inflation, without the hybrid mechanism, typically results in large squeezed nongaussianity. Small multifield potentials contain multiple flat field directions, often identified with the gauge invariant field directions in supersymmetric potentials. We find that unless these field directions have equal slopes, large nongaussianity arises. After identifying relevant differences between large and small two-field potentials, we demonstrate that the latter naturally fulfill the Byrnes-Choi-Hall large nongaussianity conditions. Computations of the primordial power spectrum, spectral index, and squeezed bispectrum, reveal that small two-field models which otherwise match observed primordial perturbations, produce excludably large nongaussianity if the inflatons' field directions have unequal slopes.

Bent waveguides for matter-waves: supersymmetric potentials and reflectionless geometries.

Non-zero curvature in a waveguide leads to the appearance of an attractive quantum potential which crucially affects the dynamics in matter-wave circuits. Using methods of supersymmetric quantum mechanics, pairs of bent waveguides are found whose geometry-induced potentials share the same scattering properties. As a result, reflectionless waveguides, dual to the straight waveguide, are identified. Strictly isospectral waveguides are also found by modulating the depth of the trapping potential. Numerical simulations are used to demonstrate the efficiency of these approaches in tailoring and controlling curvature-induced quantum-mechanical effects.

Directed transport induced byα-stable Lévy noises in weakly asymmetric periodic potentials

We study the motion of a particle in spatially periodic potentials with broken mirror symmetry under the influence of white α-stable Lévy noises. We consider both time-independent and fluctuating potentials. We focus on cases in which the spatial asymmetry of the potential is due not to a difference between the distances from an absolute minimum to the absolute maximum on its left and to the absolute maximum on its right but only to the curvatures of the potential profiles. The analysis is performed using the fractional Fokker-Planck formalism. Consistent results from Langevin simulations are also presented. We analyze the influence of the symmetry properties of the potentials in combination with the fluctuating characteristics of the system in the determination of the current. We find different situations in which both the absolute value and the direction of the current depend not only on the properties of the potential but also on the parameters characterizing the α-stable Lévy noise. Among other features, we analyze the case of supersymmetric potentials. In particular, we show that a static supersymmetric potential produces no current, and we analyze the conditions for observing a nonvanishing current when the potential fluctuates between different supersymmetric profiles.

Spectral properties of supersymmetric shape invariant potentials

We present the spectral properties of supersymmetric shape invariant potentials (SIPs). Although the folded spectrum is completely random, unfolded spectrum shows that energy levels are highly correlated and absolutely rigid. All the SIPs exhibit harmonic oscillator-type spectral statistics in the unfolded spectrum. We conjecture that this is the reflection of shape invariant symmetry.

Supersymmetric improvement of the Pekeris approximation for the rotating Morse potential

It is demonstrated that the accuracy of the approximate expression of Pekeris for the eigenvalues of the rotating Morse potential can be improved substantially by supersymmetry. A hierarchy of approximate supersymmetric potentials is obtained which allows one to calculate the energies of higher rovibrational states from the energies of lower rovibrational states for which Pekeris’ expression is accurate. This new formulation is tested by calculating the energies of rovibrational states of the hydrogen molecule and comparing the results with those of accurate numerical calculations.

D=5 M-theory radion supermultiplet dynamics

We show how the bosonic sector of the radion supermultiplet plus d=4, N=1 supergravity emerge from a consistent braneworld Kaluza–Klein reduction of D=5 M-theory. The radion and its associated pseudoscalar form an SL(2, R)/U(1) nonlinear sigma model. This braneworld system admits its own brane solution in the form of a 2-supercharge supersymmetric string. Requiring this to be free of singularities suggests an SL(2, Z) identification of the sigma model target space. The radion mode then gives a minimum length between branes; we suggest that this could be used to avoid the occurrence of singularities in brane–brane collisions. We discuss possible supersymmetric potentials for the radion supermultiplet and their relation to cosmological models such as the cyclic universe or hybrid inflation.

KdV conservation laws for some supersymmetric potentials

It is shown that a limited number of supersymmetric potentialsobtained from factorization methods, which are reflectionless,lead to the description of KdV Hamiltonians and their related KdVconservation laws are derived. Also, single-soliton solutions ofthe KdV equations corresponding to these potentials are presented.

New exactly solvable supersymmetric periodic potentials

Using the formalism of supersymmetric quantum mechanics, we give an exact solution for a family of one-dimensional periodic potentials, which are the supersymmetric partners of the potential proportional to the trigonometric function cos(2x) such that the Schrodinger equation for this potential is named the Mathieu equation mathematically. We show that the new potentials are distinctly different from their original ones. However, both have the same energy band structure. All the potentials obtained in this paper are free of singularities.

Exercises in exact quantization

The formalism of exact 1D quantization is reviewed in detail and applied to the spectral study of three concrete Schr\"odinger Hamiltonians $[-\d^2/\d q^2 + V(q)]^\pm$ on the half-line $\{q>0\}$, with a Dirichlet (-) or Neumann (+) condition at q=0. Emphasis is put on the analytical investigation of the spectral determinants and spectral zeta functions with respect to singular perturbation parameters. We first discuss the homogeneous potential $V(q)=q^N$ as $N \to +\infty$vs its (solvable) $N=\infty$ limit (an infinite square well): useful distinctions are established between regular and singular behaviours of spectral quantities; various identities among the square-well spectral functions are unraveled as limits of finite-N properties. The second model is the quartic anharmonic oscillator: its zero-energy spectral determinants $\det(-\d^2/\d q^2 + q^4 + v q^2)^\pm$ are explicitly analyzed in detail, revealing many special values, algebraic identities between Taylor coefficients, and functional equations of a quartic type coupled to asymptotic $v \to +\infty$ properties of Airy type. The third study addresses the potentials $V(q)=q^N+v q^{N/2-1}$ of even degree: their zero-energy spectral determinants prove computable in closed form, and the generalized eigenvalue problems with v as spectral variable admit exact quantization formulae which are perfect extensions of the harmonic oscillator case (corresponding to N=2); these results probably reflect the presence of supersymmetric potentials in the family above.

Investigation of phase-equivalent potentials by a halo transfer reaction

Using the supersymmetric quantum mechanics we investigate the wave function-sensitive properties of the supersymmetric potentials which have received a lot of attention in the literature recently. We show that a superdeep potential and its phase-equivalent shallow-partner potential give very similar "rms" values for the weakly bound systems such as the deuteron and 11Be nuclei. Although the corresponding eigenstates differ in the node-number, our investigation on the 11Be(p,d)10Be single nucleon halo transfer reaction at 35 MeV show that also other physical quantities such as the cross section angular distributions calculated using these wave functions reflect the nodal structure rather weakly. This lends support to two nearly equivalent treatments of the Pauli principle.

Anharmonicity deformation and curvature in supersymmetric potentials

An algebraic description of the class of 1D supersymmetric shape invariant potentials is investigated in terms of the shape-invariant-potential (SIP) deformed algebra, the generators of which act both on the dynamical variable and on the parameters of the potentials. The phase space geometry associated with SIP's is studied by means of a coherent state (SIP-CS) path integral and the ray metric of the SIP-CS manifold. The anharmonicity of SIP's results in a inhomogeneous phase space manifold with one Killing vector and with a modified symplectic Kahler structure, and it induces a non constant curvature into the generalized phase space. Analogous results from the phase space geometry of someq-deformed quantum algebras are briefly contrasted with those for the supersymmetric potentials. A star product operating between phase space symbols of operators is also suggested.

Berry phase for supersymmetric shape-invariant potentials

Berry phases for shape invariant potentials are discussed. A recurrence relation linking the Berry connections for different states is derived.

Shape invariance with application to the momentum representation

The authors formulate the method of shape invariance of supersymmetric potentials in a manner that is suitable for application to spherically symmetric problems in the momentum representation. Three examples are discussed: the isotropic nonrelativistic oscillator, a relativistic oscillator, and the Coulomb problem. In these shape invariance is used to determine eigenvalues and normalized momentum-space eigenfunctions for bound states.

Surface excitations of a nonideal bose gas

Surface excitations of a nonideal Bose gas are investigated within the Gross-Pitaevskii equation linearized near the kink solution. The resulting matrix Schrödinger equation with supersymmetric potentials describes two Goldstone (gapless) modes which are localized by these potentials near the free surface. The first mode represents an ordinary capillary wave (ripplon) and the second one corresponds to a surface weakly trapped phonon. The last one might additionally contribute to the surface tension of helium II explaining the discrepancy between experiment and theory.