Exactly solvable models of supersymmetric quantum mechanics and connection to spectrum generating algebra

For nonrelativistic Hamiltonians which are shape invariant, analytic expressions for the eigenvalues and eigenvectors can be derived using the well known method of supersymmetric quantum mechanics. Most of these Hamiltonians also possess spectrum generating algebras and are hence solvable by an independent group theoretic method. In this paper, we demonstrate the equivalence of the two methods of solution by developing an algebraic framework for shape invariant Hamiltonians with a general change of parameters, which involves nonlinear extensions of Lie algebras.Comment: 12 pages, 2 figure

Algebraic Shape Invariant Models

Motivated by the shape invariance condition in supersymmetric quantum mechanics, we develop an algebraic framework for shape invariant Hamiltonians with a general change of parameters. This approach involves nonlinear generalizations of Lie algebras. Our work extends previous results showing the equivalence of shape invariant potentials involving translational change of parameters with standard $SO(2,1)$ potential algebra for Natanzon type potentials.Comment: 8 pages, 2 figure

Exceptional orthogonal polynomials, QHJ formalism and SWKB quantization condition

We study the quantum Hamilton-Jacobi (QHJ) equation of the recently obtained exactly solvable models, related to the newly discovered exceptional polynomials and show that the QHJ formalism reproduces the exact eigenvalues and the eigenfunctions. The fact that the eigenfunctions have zeros and poles in complex locations leads to an unconventional singularity structure of the quantum momentum function $p(x)$, the logarithmic derivative of the wave function, which forms the crux of the QHJ approach to quantization. A comparison of the singularity structure for these systems with the known exactly solvable and quasi-exactly solvable models reveals interesting differences. We find that the singularities of the momentum function for these new potentials lie between the above two distinct models, sharing similarities with both of them. This prompted us to examine the exactness of the supersymmetric WKB (SWKB) quantization condition. The interesting singularity structure of $p(x)$ and of the superpotential for these models has important consequences for the SWKB rule and in our proof of its exactness for these quantal systems.Comment: 10 pages with 1 table,i figure. Errors rectified, manuscript rewritten, new references adde

Mapping of non-central potentials under point canonical transformations

Motivated by the observation that all known exactly solvable shape invariant central potentials are inter-related via point canonical transformations, we develop an algebraic framework to show that a similar mapping procedure is also exist between a class of non-central potentials. As an illustrative example, we discuss the inter-relation between the generalized Coulomb and oscillator systems.Comment: 11 pages article in LaTEX (uses standard article.sty). Please check http://www1.gantep.edu.tr/~gonul for other studies of Nuclear Physics Group at University of Gaziante

Exceptional Orthogonal Polynomials, QHJ Formalism and SWKB Quantization Condition

We study the quantum Hamilton-Jacobi (QHJ) equation of the recently obtained exactly solvable models, related to the newly discovered exceptional polynomials and show that the QHJ formalism reproduces the exact eigenvalues and the eigenfunctions. The fact that the eigenfunctions have zeros and poles in complex locations leads to an unconventional singularity structure of the quantum momentum function p(x), the logarithmic derivative of the wave function, which forms the crux of the QHJ approach to quantization. A comparison of the singularity structure for these systems with the known exactly solvable and quasi-exactly solvable models reveals interesting differences. We find that the singularities of the momentum function for these new potentials lie between the above two distinct models, sharing similarities with both of them. This prompted us to examine the exactness of the supersymmetric WKB (SWKB) quantization condition. The interesting singularity structure of p(x) and of the superpotential for these models has important consequences for the SWKB rule and in our proof of its exactness for these quantal systems

Single-Stage Repair versus Traditional Repair of High Anorectal Malformations, Functional Results’ Correlation with Kelly’s Score and Postoperative Magnetic Resonance Imaging Findings

Introduction: Posterior sagittal anorectoplasty (PSARP) is the standard treatment for anorectal malformations. In the present study, the clinical evaluation of anal continence was carried out using Kelly’s scoring system and the results of primary PSARP or abdomino-PSARP were compared with the traditional three-stage procedure and the functional outcome was correlated with the findings of MRI, which was used as an objective method of evaluation.Patients and methods: A total of 40 patients with intermediate and high anorectal malformations were studied and were divided into two groups on the basis of a random number table. The patients in group A were treated with a single-stage operation, whereas the patients in group B were treated with a standard staged operation (either PSARP or abdominoperineal pull-through). After clinical evaluation using the Kelly score, patients were divided into three clinical groups irrespective of whether they were operated in one stage or in three stages. All patients were subjected to MRI at the age of 3 years and the findings were correlated with the clinical scoring system.Result: Patients were categorized according to their Kelly’s scores as follows: group 1: clinically good (score 5–6); group 2: clinically fair (score 3–4); and group 3: clinically poor (score 0–2). The proportions of good development of the muscles (puborectalis, external sphincter muscle, and levator muscle hammock) were 78.9% in group 1, 40% in group 2, and none in group 3. Development of muscles was found to be a significant factor for anal continence. Other significant factors for anal continence are rectal diameter and anorectal angle.Conclusion: Clinical assessment using the Kelly score was similar for the single-stage operation and the staged procedure, and this was supported by MRI findings. Therefore, we recommend the single-stage procedure to achieve a better outcome in intermediate and high anorectal malformation.Keywords: Anorectal Malformations, MRI, Posterior Sagittal Anorectoplast

Conservation Laws and Energy Transformations in a Class of Common Physics Problems

We analyze a category of problems that is of interest in many physical situations, including those encountered in introductory physics classes: systems with two well-delineated parts that exchange energy, eventually reaching a shared equilibrium with a loss of mechanical or electrical energy. Such systems can be constrained by a constant of the system (e.g., mass, charge, momentum, or angular momentum) that uniquely determines the mechanical or electrical energy of the equilibrium state, regardless of the dissipation mechanism. A representative example would be a perfectly inelastic collision between two objects in one dimension, for which momentum conservation requires that some of the initial kinetic energy is dissipated by conversion to thermal or other forms as the two objects reach a common final velocity. We discuss how this feature manifests in a suite of four well-known and disparate problems that all share a common mathematical formalism. These examples, in which the energy dissipated during the process can be difficult to solve directly from dissipation rates, can be approached by students in a first-year physics class by considering conservation laws and can therefore be useful for teaching about energy transformations and conserved quantities. We then illustrate how to extend this method by applying it to a final example

Weakly coupled states on branching graphs

We consider a Schr\"odinger particle on a graph consisting of $\,N\,$ links joined at a single point. Each link supports a real locally integrable potential $\,V_j\,$; the self--adjointness is ensured by the $\,\delta\,$ type boundary condition at the vertex. If all the links are semiinfinite and ideally coupled, the potential decays as $\,x^{-1-\epsilon}$ along each of them, is non--repulsive in the mean and weak enough, the corresponding Schr\"odinger operator has a single negative eigenvalue; we find its asymptotic behavior. We also derive a bound on the number of bound states and explain how the $\,\delta\,$ coupling constant may be interpreted in terms of a family of squeezed potentials.Comment: LaTeX file, 7 pages, no figure

Quantum Mechanics of Multi-Prong Potentials

We describe the bound state and scattering properties of a quantum mechanical particle in a scalar $N$-prong potential. Such a study is of special interest since these situations are intermediate between one and two dimensions. The energy levels for the special case of $N$ identical prongs exhibit an alternating pattern of non-degeneracy and $(N-1)$ fold degeneracy. It is shown that the techniques of supersymmetric quantum mechanics can be used to generate new solutions. Solutions for prongs of arbitrary lengths are developed. Discussions of tunneling in $N$-well potentials and of scattering for piecewise constant potentials are given. Since our treatment is for general values of $N$, the results can be studied in the large $N$ limit. A somewhat surprising result is that a free particle incident on an $N$-prong vertex undergoes continuously increased backscattering as the number of prongs is increased.Comment: 17 pages. LATEX. On request, TOP_DRAW files or hard copies available for 7 figure

Accuracy of Semiclassical Methods for Shape Invariant Potentials

We study the accuracy of several alternative semiclassical methods by computing analytically the energy levels for many large classes of exactly solvable shape invariant potentials. For these potentials, the ground state energies computed via the WKB method typically deviate from the exact results by about 10%, a recently suggested modification using nonintegral Maslov indices is substantially better, and the supersymmetric WKB quantization method gives exact answers for all energy levels.Comment: 7 pages, Latex, and two tables in postscrip