scholarly journals Exact and approximate solutions to Schrödinger’s equation with decatic potentials

Open Physics ◽  
2013 ◽  
Vol 11 (3) ◽  
Author(s):  
David Brandon ◽  
Nasser Saad
Keyword(s):  
Sufficient Conditions ◽  
Approximate Solutions ◽  
Asymptotic Iteration Method ◽  
Polynomial Potential ◽  
Schrödinger’S Equation ◽  
Polynomial Solutions ◽  
Polynomial Coefficients ◽  
Linear Differential ◽  
Polynomial Potentials ◽  
Schrödinger's Equation

AbstractThe one-dimensional Schrödinger’s equation is analysed with regard to the existence of exact solutions for decatic polynomial potentials. Under certain conditions on the potential’s parameters, we show that the decatic polynomial potential V (x) = ax 10 + bx 8 + cx 6 + dx 4 + ex 2, a > 0 is exactly solvable. By examining the polynomial solutions of certain linear differential equations with polynomial coefficients, the necessary and sufficient conditions for corresponding energy-dependent polynomial solutions are given in detail. It is also shown that these polynomials satisfy a four-term recurrence relation, whose real roots are the exact energy eigenvalues. Further, it is shown that these polynomials generate the eigenfunction solutions of the corresponding Schrödinger equation. Further analysis for arbitrary values of the potential parameters using the asymptotic iteration method is also presented.

scholarly journals Exact and approximate solutions of Schrödinger’s equation for a class of trigonometric potentials

Open Physics ◽  
2013 ◽  
Vol 11 (1) ◽  
Author(s):  
Hakan Ciftci ◽  
Richard Hall ◽  
Nasser Saad
Keyword(s):  
Differential Equation ◽  
Coordinate Transformation ◽  
Iteration Method ◽  
Order Differential Equation ◽  
Approximate Solutions ◽  
Asymptotic Iteration Method ◽  
Perturbation Expansions ◽  
One Dimensional ◽  
Schrödinger’S Equation ◽  
Schrödinger's Equation

AbstractThe asymptotic iteration method is used to find exact and approximate solutions of Schrödinger’s equation for a number of one-dimensional trigonometric potentials (sine-squared, double-cosine, tangent-squared, and complex cotangent). Analytic and approximate solutions are obtained by first using a coordinate transformation to reduce the Schrödinger equation to a second-order differential equation with an appropriate form. The asymptotic iteration method is also employed indirectly to obtain the terms in perturbation expansions, both for the energies and for the corresponding eigenfunctions.

scholarly journals On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application

Algorithms ◽  
2020 ◽  
Vol 13 (11) ◽  
pp. 286
Author(s):  
Kyle R. Bryenton ◽  
Andrew R. Cameron ◽  
Keegan L. A. Kirk ◽  
Nasser Saad ◽  
Patrick Strongman ◽  
...  
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Differential Equations ◽  
Theoretical Physics ◽  
Necessary Condition ◽  
Polynomial Solutions ◽  
Polynomial Coefficients ◽  
Linear Differential ◽  
Necessary And Sufficient

The analysis of many physical phenomena is reduced to the study of linear differential equations with polynomial coefficients. The present work establishes the necessary and sufficient conditions for the existence of polynomial solutions to linear differential equations with polynomial coefficients of degree n, n−1, and n−2 respectively. We show that for n≥3 the necessary condition is not enough to ensure the existence of the polynomial solutions. Applying Scheffé’s criteria to this differential equation we have extracted n generic equations that are analytically solvable by two-term recurrence formulas. We give the closed-form solutions of these generic equations in terms of the generalized hypergeometric functions. For arbitrary n, three elementary theorems and one algorithm were developed to construct the polynomial solutions explicitly along with the necessary and sufficient conditions. We demonstrate the validity of the algorithm by constructing the polynomial solutions for the case of n=4. We also demonstrate the simplicity and applicability of our constructive approach through applications to several important equations in theoretical physics such as Heun and Dirac equations.

scholarly journals Asymptotic iteration method for solving Hahn difference equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Lucas MacQuarrie ◽  
Nasser Saad ◽  
Md. Shafiqul Islam
Keyword(s):  
Difference Equations ◽  
Iteration Method ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Asymptotic Iteration Method ◽  
Order Linear ◽  
Polynomial Solutions ◽  
Hahn Difference Equations ◽  
Iteration Methods ◽  
Necessary And Sufficient

AbstractHahn’s difference operator $D_{q;w}f(x) =({f(qx+w)-f(x)})/({(q-1)x+w})$ D q ; w f ( x ) = ( f ( q x + w ) − f ( x ) ) / ( ( q − 1 ) x + w ) , $q\in (0,1)$ q ∈ ( 0 , 1 ) , $w>0$ w > 0 , $x\neq w/(1-q)$ x ≠ w / ( 1 − q ) is used to unify the recently established difference and q-asymptotic iteration methods (DAIM, qAIM). The technique is applied to solve the second-order linear Hahn difference equations. The necessary and sufficient conditions for polynomial solutions are derived and examined for the $(q;w)$ ( q ; w ) -hypergeometric equation.

scholarly journals Bound State of Heavy Quarks Using a General Polynomial Potential

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Hesham Mansour ◽  
Ahmed Gamal
Keyword(s):  
Bound States ◽  
Bound State ◽  
Heavy Quarks ◽  
Polynomial Potential ◽  
Schrödinger’S Equation ◽  
General Polynomial ◽  
L Value ◽  
Schrödinger's Equation ◽  
Better Than ◽  
Good Agreement

In the present work, the mass spectra of the bound states of heavy quarks cc-,bb-, and Bc meson are studied within the framework of the nonrelativistic Schrödinger’s equation. First, we solve Schrödinger’s equation with a general polynomial potential by Nikiforov-Uvarov (NU) method. The energy eigenvalues for any L- value is presented for a special case of the potential. The results obtained are in good agreement with the experimental data and are better than previous theoretical studies.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

scholarly journals Correctness conditions for high-order differential equations with unbounded coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kordan N. Ospanov
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Order Linear Differential Equation ◽  
Unbounded Coefficients ◽  
Weighted Norms ◽  
Uniqueness Of The Solution ◽  
Linear Differential

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

A Recursion Formula for the Coefficients of Entire Functions Satisfying an ODE with Polynomial Coefficients

2004 ◽  
Vol 11 (3) ◽  
pp. 409-414
Author(s):  
C. Belingeri
Keyword(s):  
Differential Equations ◽  
Sum Rules ◽  
Entire Functions ◽  
Recursion Formula ◽  
Linear Differential Equations ◽  
Bell Polynomials ◽  
Polynomial Coefficients ◽  
Linear Differential

Abstract A recursion formula for the coefficients of entire functions which are solutions of linear differential equations with polynomial coefficients is derived. Some explicit examples are developed. The Newton sum rules for the powers of zeros of a class of entire functions are constructed in terms of Bell polynomials.

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