Solutions to the Painlev\'e V equation through supersymmetric quantum mechanics

In this paper we shall use the algebraic method known as supersymmetric quantum mechanics (SUSY QM) to obtain solutions to the Painlev\'e V (PV) equation, a second-order non-linear ordinary differential equation. For this purpose, we will apply first the SUSY QM treatment to the radial oscillator. In addition, we will revisit the polynomial Heisenberg algebras (PHAs) and we will study the general systems ruled by them: for first-order PHAs we obtain the radial oscillator, while for third-order PHAs the potential will be determined by solutions to the PV equation. This connection allows us to introduce a simple technique for generating solutions of the PV equation expressed in terms of confluent hypergeometric functions. Finally, we will classify them into several solution hierarchies.Comment: 39 pages, 18 figures, 4 tables, 70 reference

Harmonic Oscillator SUSY Partners and Evolution Loops

Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one. If applied to the harmonic oscillator, a family of Hamiltonians ruled by polynomial Heisenberg algebras is obtained. In this paper it will be shown that the SUSY partner Hamiltonians of the harmonic oscillator can produce evolution loops. The corresponding geometric phases will be as well studied

Gevrey expansions of hypergeometric integrals II

We study integral representations of the Gevrey series solutions of irregular hypergeometric systems under certain assumptions. We prove that, for such systems, any Gevrey series solution, along a coordinate hyperplane of its singular support, is the asymptotic expansion of a holomorphic solution given by a carefully chosen integral representation.Comment: 27 pages, 2 figure

Nonsensical models for quantum dots

We analyze a model proposed recently for the calculation of the energy of an exciton in a quantum dot and show that the authors made a serious mistake in the solution to the Schr\"{o}dinger equation

Wronskian differential formula for confluent supersymmetric quantum mechanics

A Wronskian differential formula, useful for applying the confluent second-order SUSY transformations to arbitrary potentials, will be obtained. This expression involves a parametric derivative with respect to the factorization energy which, in many cases, is simpler for calculations than the previously found integral equation. This alternative mechanism shall be applied to the free particle and the single-gap Lame potential.Comment: 11 pages, 4 figure

Complex oscillator and Painlev\'e IV equation

Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one. In this article the first- and second- order supersymmetric transformations will be used to obtain new exactly solvable potentials departing from the complex oscillator. The corresponding Hamiltonians turn out to be ruled by polynomial Heisenberg algebras. By applying a mechanism to reduce to second the order of these algebras, the connection with the Painlev\'{e} IV equation is achieved, thus giving place to new solutions for the Painlev\'{e} IV equation.Comment: 23 pages, 13 figure

Perturbation theory by the moment method and point-group symmetry

We analyze earlier applications of perturbation theory by the moment method (also called inner product method) to anharmonic oscillators. For concreteness we focus on two-dimensional models with symmetry $C_{4v}$ and $C_{2v}$ and reveal the reason why some of those earlier treatments proved unsuitable for the calculation of the perturbation corrections for some excited states. Point-group symmetry enables one to predict which states require special treatment

Electron impact excitation of N IV: calculations with the DARC code and a comparison with ICFT results

There have been discussions in the recent literature regarding the accuracy of the available electron impact excitation rates (equivalently effective collision strengths $\Upsilon$) for transitions in Be-like ions. In the present paper we demonstrate, once again, that earlier results for $\Upsilon$ are indeed overestimated (by up to four orders of magnitude), for over 40\% of transitions and over a wide range of temperatures. To do this we have performed two sets of calculations for N~IV, with two different model sizes consisting of 166 and 238 fine-structure energy levels. As in our previous work, for the determination of atomic structure the GRASP (General-purpose Relativistic Atomic Structure Package) is adopted and for the scattering calculations (the standard and parallelised versions of) the Dirac Atomic R-matrix Code ({\sc darc}) are employed. Calculations for collision strengths and effective collision strengths have been performed over a wide range of energy (up to 45~Ryd) and temperature (up to 2.0$\times$10$^6$~K), useful for applications in a variety of plasmas. Corresponding results for energy levels, lifetimes and A-values for all E1, E2, M1 and M2 transitions among 238 levels of N~IV are also reported.Comment: This paper with 5 Figs. and 8 Tables will appear in MNRAS (2016