Supersymmetric partners of the trigonometric Poschl-Teller potentials

The first and second-order supersymmetry transformations are used to generate Hamiltonians with known spectra departing from the trigonometric Poschl-Teller potentials. The several possibilities of manipulating the initial spectrum are fully explored, and it is shown how to modify one or two levels, or even to leave the spectrum unaffected. The behavior of the new potentials at the boundaries of the domain is studied.Comment: 20 pages, 4 figure

The SUSY partners of the QES sextic potential revisited

In this paper, the SUSY partner Hamiltonians of the quasi-exactly solvable (QES) sextic potential $V^{\rm qes}(x) = \nu\, x^{6} + 2\, \nu\, \mu\,x^{4} + \left[\mu^2-(4N+3)\nu \right]\, x^{2}$, $N \in \mathbb{Z}^+$, are revisited from a Lie algebraic perspective. It is demonstrated that, in the variable $\tau=x^2$, the underlying $\mathfrak{sl}_2(\mathbb{R})$ hidden algebra of $V^{\rm qes}(x)$ is inherited by its SUSY partner potential $V_1(x)$ only for $N=0$. At fixed $N>0$, the algebraic polynomial operator $h(x,\,\partial_x;\,N)$ 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 $V^{\rm qes}$ with a different non-integer (cohomology) parameter $N_1=N-\frac{3}{2}$. 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 ($\sim 20$ s. d.) of the energy $E_n=E_n(N)$ in the interval $N \in [-1,3]$ 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 $V^{\rm qes}$ 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.Comment: 25 pages, 20 figure

A case of pericardial effusion: Aid of bedside ultrasound in clinical decision making

Background: In life-threatening conditions, such as cardiac tamponade, we need to recognize signs of impending decompensation in a timely manner. Point of care ultrasound is non-invasive and is readily available, particularly in a resource limited setting. In this case, we present a patient with worsening pericardial effusion and the importance of bedside cardiac ultrasound. Case presentation: A 76-year-old male with stage IV adenocarcinoma of the lung was admitted for worsening dysphagia for 3 days and decreased oral intake. He was normotensive but was tachycardic with muffled heart sounds. CT of the chest demonstrated a significant pericardial effusion. Bedside ultrasound to evaluate the extent of pericardial effusion revealed significant circumferential effusion, right systolic atrial collapse, and non-variable IVC, signs of pre-tamponade physiology. Given these findings, cardiology was emergently consulted who recommended transfer to a higher level of care facility. Thus, point of care ultrasound helped in making an immediate decision prior to clinical decompensation of the patient. Conclusion: Point of care ultrasound can help in making critical decisions in resource-limited settings. In this patient, a large effusion with pre-tamponade physiology was identified on bedside ultrasound. Early recognition during bedside ultrasound can help reach a diagnosis before clinical deterioration. Our case highlights the importance of point of care ultrasound as a tool to reinforce critical thinking and help expedite rational decision-making processes

Analysis of the relevance of a 3D virtual programme to detect drug use behaviors and bullyng

p. 20The psychometric data of a new computer science tool of three-dimensional simulation (3D) designed for the detection of risk of drug taking behaviours and bullying in adolescents is presented. In partic- ular, Mii School (the name of this new program), recreates 17 conflicting situations of drug use and bullying, where the participant must respond when they are immersed in different situations of risk. The computer program has shown good reliability properties (α=.802) and validity (showing a facto- rial structure composed of 8 factors that explain 57.26% of the total variance). The study is con- ducted on the basis of differently realised studies: One discusses the capacity and utility of the pro- gram as a tool of detection of behaviours of drug consumption and bullying in young people, as well as the potential applicability of this program in educative contextsS