Solution of a Novel Quasi-Exactly Solvable Potential via Asymptotic Iteration Method

Abstract

Since the wave function contains all the necessary information to describe a quantum system fully, it is of high importance of obtaining exact or approximate solutions of Schrodinger equation in quantum mechanics. It is known that there are not so many potentials that can be solved exactly. Therefore, many techniques are suggested (and in use) to find the approximate solutions of the potentials that are not exactly solvable. A recent technique, called the Asymptotic Iteration Method (AIM),1),2) has been used to obtain eigenvalues of second order homogeneous differential equations. In the case of Schrodinger equation, it has been found that the method reproduces the energy spectrum for most exactly solvable potentials,2)–6) and it produces very good results for non-exactly solvable potentials.7)–10) It is well known in quantum mechanics that the potentials having bound states can be classified as solvable, non solvable and quasi exactly solvable (QES). The potentials in the last category are characterized by the fact that part of their spectrum can be found analytically while the remaining part is unknown. In a sense, they constitute the intermediate step between exactly solvable equations (associated with potentials such as the harmonic oscillator one, the Coulomb one, · · · ) whose all analytic solutions can be obtained, and the analytically unsolvable ones requiring a numerical treatment.11) Some QES potentials can be given in terms of a parameter J and then one can obtain exactly the first J -th energy values and wavefunctions, for positive integer (and half-integer) values of J. Thus, one can find the solution of the QES system in the framework of Lie-algebraic formalism.12)–16) In their study, Cho et al.17) have found a novel QES model with symmetric inverted potentials which are unbounded from below. They have obtained a discrete spectrum of bound states for these inverted potentials and have shown the solvable part of the spectrum is the QES states they have discovered. Using the AIM, we determine the energy eigenvalues and the eigenfunctions of