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quasi-exactly Solvable Time-Dependent Hamiltonians

Open Journal of Microphysics 2014年第3期4卷 26-34页

作者： Ancilla Nininahazwe Université du Burundi Institut de Pédagogie Appliquée Bujumbura Burundi

A generalized method which helps to find a time-dependent SchrÖdinger equation for any static potential is established. We illustrate this method with two examples. Indeed, we use this method to find the time-dependent Hamiltonian of quasi-exactly solvable Lamé equation and to construct the matrix 2 × 2 time-dependent polynomial Hamiltonian.

关键词： quasi-exactly Solvable Time-Dependent Hamiltonian

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quasi-exactly Solvable Differential Models: A Canonical Polynomials Approach

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American Journal of Computational Mathematics 2019年第2期9卷 48-60页

作者： Mohamad K. El-Daou Talal H. AlZanki Nadia S. Al-Mutawa Department of Applied Sciences The College of Technological Studies PAAET Kuwait Department of Electronic Engineering Technology The College of Technological Studies PAAET Kuwait

A quasi-exactly solvable model refers to any second order differential equation with polynomial coefficients of the form A(x)y’’(x)+B(x)y’(x)+C(x)y(x)=0 where a pair of exact polynomials {y(x), C(x)} with respective degrees {deg[y]=n, deg[C]=p} are to be found simultaneously in terms of the coefficients of two given polynomials {A(x), B(x)}. The existing methods for solving quasi-exactly solvable models require the solution of a system of nonlinear algebraic equations of which the dimensions depend on n, the degree of the exact polynomial solution y(x). In this paper, a new method employing a set of polynomials, called canonical polynomials, is proposed. This method requires solving a system of nonlinear algebraic equations of which the dimensions depend only on p, the degree of C(x), and do not vary with n. Several examples are implemented to testify the efficiency of the proposed method.

关键词： Polynomial Solutions Canonical Polynomials Tau Method quasi-exactly Solvable Models

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On the High-Order quasi exactly Solvable Differential Equations

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American Journal of Computational Mathematics 2019年第4期9卷 234-250页

作者： Talal H. Alzanki Mohamed S. Shaaban Mohamad K. El-Daou Department of Electronic Engineering College of Technological Studies PAAET Kuwait City Kuwait Department of Applied Sciences College of Technological Studies PAAET Kuwait City Kuwait

In this paper, we present a new method for solving a class of high-order quasi exactly solvable ordinary differential equations. With this method, the computed solution is expressed as a linear combination of the canonical polynomials associated with the given differential operator. An iterative algorithm summarizing the procedure is presented and its efficiency is demonstrated through considering two applied problems.

关键词： quasi-exactly Solvable High-Order Differential Equations Canonical Polynomials Tau Method

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Complex Classical Mechanics of a QES Potential

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Communications in Theoretical Physics 2015年第10期64卷 425-432页

作者： Bhabani Prasad Mandal Sushant S.Mahajan Department of Physics Banaras Hindu University Department of Physics Indian Institute of Technology Banaras Hindu University

We study a combined parity(P) and time reversal(T) invariant non-Hermitian quasi-exactly solvable(QES) potential, which exhibits PT phase transition, in the complex plane classically to demonstrate different quantum effects. The particle with real energy makes closed orbits around one of the periodic wells of the complex potential depending on the initial condition. However interestingly the particle escapes to an open orbits even with real energy if it is placed beyond a certain distance from the center of the well. On the other hand when the particle energy is complex the trajectory is open and the particle tunnels back and forth between two wells which are separated by a classically forbidden path. The tunneling time is calculated for different pair of wells and is shown to vary inversely with the imaginary component of energy. Our study reveals that spontaneous PT symmetry breaking does not affect the qualitative features of the particle trajectories in the analogous complex classical model.

关键词： PT symmetric complex systems quasi-exactly solvabl

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