The series solution of the radial part of the Schrödinger equation for simultaneous coulomb and harmonic potential involves three-term recursion relation and is thus difficult to solve for bound states. We have suggested a simple method to solve for low lying states. Finite polynomial solutions exist only if the coulomb and oscillator potentials are nontrivially related.
We consider a large class of $q$-series that have the structure of Nahm sums, or equivalently motivic generating series for quivers. First, we initiate a systematic analysis and classification of classical and quantum A-polynomials associated to such $q$-series. These quantum quiver A-polynomials encode recursion relations satisfied by the above series, while classical A-polynomials encode asymptotic expansion of those series. Second, we postulate that those series, as well as their quantum quiver A-polynomials, can be reconstructed by means of the topological recursion. There is a large class…
We study the structure of series expansions of general spinning conformal blocks. We find that the terms in these expansions are naturally expressed by means of special functions related to matrix elements of Spin(d) representations in Gelfand-Tsetlin basis, of which the Gegenbauer polynomials are a special case. We study the properties of these functions and explain how they can be computed in practice. We show how the Casimir equation in Dolan-Osborn coordinates leads to a simple one-step recursion relation for the coefficients of the series expansion of general spinning conformal block. The…
We introduce a nine-parameter Heun-type differential equation and obtain three classes of its solutions as series of square integrable functions written in terms of the Jacobi polynomial. The expansion coefficients of the series satisfy three-term recursion relations, which are solved in terms of orthogonal polynomials with continuous and/or discrete spectra. Some of these are well-known polynomials while others are either new or modified versions of known ones.
Two Recursions for the Seiberg-Witten Prepotential: Zamolodchikov's Recursion, the AGT Correspondence and Topological Recursion on the Seiberg-Witten Curve
We introduce two ordinary second-order linear differential equations of the Laguerre- and Jacobi-type. Solutions are written as infinite series of square integrable functions in terms of the Laguerre and Jacobi polynomials, respectively. The expansion coefficients of the series satisfy three-term recursion relations, which are solved in terms of orthogonal polynomials with continuous and/or discrete spectra. Most of these are well-known polynomials whereas few are not. We present physical applications of these differential equations in quantum mechanics.
We use the Tridiagonal Representation Approach (TRA) to obtain exact bound states solution (energy spectrum and wavefunction) of the Schrödinger equation for a three-parameter short-range potential with 1/r, 1/r^2 and 1/r^3 singularities at the origin. The solution is a finite series of square integrable functions with weighted coefficients that satisfy a three-term recursion relation. The solution of the recursion is the discrete version of a non-conventional orthogonal polynomial. We are currently preparing to use the results of this work to study the binding of an electron to a molecule wit…
The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. Our problems have a powerful relatively recent tool in common, the so-called topological recursion (TR) introduced by Chekhov, Eynard and Orantin. We call a map fully simple if it has non self-intersecting disjoint boundaries, and ordinary if such a restriction is not imposed. We study the combinatorial relation between fully simple and ordinary maps with the topology of a disk or a cylinder, which reproduces relations between moments and free cumulants established in the contex…
We introduce a ten-parameter ordinary linear differential equation of the second order with four singular points. Three of these are finite and regular whereas the fourth is irregular at infinity. We use the tridiagonal representation approach to obtain a solution of the equation as bounded infinite series of square integrable functions that are written in terms of the Jacobi polynomials. The expansion coefficients of the series satisfy three-term recursion relation, which is solved in terms of a modified version of the continuous Hahn orthogonal polynomial.
Formal multidimensional integrals, stuffed maps, and topological recursion
我们证明了多迹1形式厄米矩阵模型中的大N展开是由[Eynard and Orantin，2007]的拓扑递归和初始条件决定的.在一维气体的本征值，这个模型包括-在范德蒙的平方-本征值之间的任何顺序的多线性相互作用。在这个问题中，拓扑递归的初始数据（W10，W20）的特征在于：对于W10，由一个非线性的，非局部的Riemann-Hilbert问题在不连续轨迹上确定;对于W20，由一个相关的，但线性的，非局部的Riemann-Hilbert问题在不连续轨迹上确定。在组合学中，这个模型列举了离散的表面（地图），其基本2-细胞可以有任何拓扑- W10是生成的一系列磁盘和W20的圆柱。特别是，通过替换，人们可以考虑其基本单元本身就是地图的地图，我们建议将其命名为“填充地图”。从某种意义上说，我们的结果完成了90年代开始的计算单厄米矩阵模型形式1/N的“矩量法”的程序。
This thesis is divided into two parts, where in the first part we investigate the computation of Virasoro 1-point blocks on the torus in the framework of Zamolodchikov's recursion relation. It is widely accepted that this recursion relation contains unphysical poles in the central charge $c$. At each order we conjecture how the pole free expressions depend on the internal and external conformal dimensions and central charge, and propose how to compute it numerically. In this thesis, we have calculated the pole free expression up to order 4. In the second part we introduce a conformal field the…
