Related papers: An ubiquitous three-term recurrence relation
We solve an eigenvalue equation that appears in several papers about a wide range of physical problems. The Frobenius method leads to a three-term recurrence relation for the coefficients of the power series that, under suitable truncation,…
We apply the Frobenius method to the Schr\"{o}dinger equation with a truncated Coulomb potential. By means of the tree-term recurrence relation for the expansion coefficients we truncate the series and obtain exact eigenfunctions and…
The history of linear differential equations is over 350 years. By using Frobenius method and putting the power series expansion into linear differential equations, the recursive relation of coefficients starts to appear. There can be…
We analyze the distribution of the eigenvalues of the quantum-mechanical rotating harmonic oscillator by means of the Frobenius method. A suitable ansatz leads to a three-term recurrence relation for the expansion coefficients. Truncation…
We apply the Frobenius (power-series) method to some simple exactly-solvable and conditionally-solvable quantum-mechanical models with supposed physical interest. We show that the supposedly exact solutions to radial eigenvalue equations…
We review our new method, which might be the most direct and efficient way for approaching the continuum physics from Hamiltonian lattice gauge theory. It consists of solving the eigenvalue equation with a truncation scheme preserving the…
Analytic solutions for the energy eigenvalues are obtained from a confined potentials of the form $br$ in 3 dimensions. The confinement is effected by linear term which is a very important part in Cornell potential. The analytic eigenvalues…
We derive a concise closed-form solution for a linear three-term recurrence relation. Such recurrence relations are very common in the quantitative sciences, and describe finite difference schemes, solutions to problems in Markov processes…
We show that a perturbed Coulomb problem discussed recently is conditionally solvable. We obtain the exact eigenvalues and eigenfunctions and compare the former with eigenvalues calculated by means of a numerical method. We discuss the…
This paper presents a new approach to determine the number of solutions of three variable Frobenius related problems and to find their solutions by using order reducing methods. Here, the order of a Frobenius related problem means the…
For the generalized eigenvalue problem, a quotient function is devised for estimating eigenvalues in terms of an approximate eigenvector. This gives rise to an infinite family of quotients, all entirely arguable to be used in estimation.…
We first show the existence and nature of convergence to a limiting set of roots for polynomials in a three-term recurrence of the form $p_{n+1}(z) = Q_k(z)p_{n}(z)+ \gamma p_{n-1}(z)$ as $n$ $\rightarrow$ $\infty$, where the coefficient…
The fractional Sturm-Liouville eigenvalue problem appears in many situations, e.g., while solving anomalous diffusion equations coming from physical and engineering applications. Therefore to obtain solutions or approximation of solutions…
We analyze the application of the "tridiagonal representation approach" (TRA) to the Schr\"{o}dinger equation for some simple, exactly-solvable, quantum-mechanical models. In the case of the Kratzer-Fues potential the mathematical reasoning…
Three fundamental variational principles used for solving elastodynamic eigenvalue problems are studied within the context of elastic wave propagation in periodic composites (phononics). We study the convergence of the eigenvalue problems…
In this paper, we rigorously investigate the truncation method for the Cauchy problem of Helmholtz equations which is widely used to model propagation phenomena in physical applications. The method is a well-known approach to the…
It is well known that a family of $n\times n$ commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the $n$ joint eigenvalues of the family. In…
We present some necessary and/or sufficient conditions for the positivity problem of three-term recurrence sequences. As applications we show the positivity of diagonal Taylor coefficients of some rational functions in a unified approach.…
We introduce a new strategy in solving the truncated complex moment problem. To this aim we investigate recursive doubly indexed sequences and their characteristic polynomials. A characterization of recursive doubly indexed \emph{moment}…
The method reducing the solution of the Schroedinger equation for several types of power potentials to the solution of the eigenvalue problem for the infinite system of algebraic equations is developed. The finite truncation of this system…