Okan Ozer

Supersymmetric approach to exactly solvable systems with position-dependent effective masses

We discuss the relationship between exact solvability of the Schrodinger equation with a position-dependent mass and the ordering ambiguity in the Hamiltonian operator within the framework of supersymmetric quantum mechanics. The one-dimensional Schrodinger equation, derived from the general form of the effective mass Hamiltonian, is solved exactly for a system with exponentially changing mass in the presence of a potential with similar behaviour, and the corresponding supersymmetric partner Hamiltonians are related to the effective-mass Hamiltonians proposed in the literature.

An exactly solvable Schrödinger equation with finite positive position-dependent effective mass

The solution of the one-dimensional Schrodinger equation is discussed in the case of position-dependent mass. The general formalism is specified for potentials that are solvable in terms of generalized Laguerre polynomials and mass functions that are positive and bounded on the whole real x axis. The resulting four-parameter potential is shown to belong to the class of “implicit” potentials. Closed expressions are obtained for the bound-state energies and the corresponding wave functions, including their normalization constants. The constant mass case is obtained by a specific choice of the parameters. It is shown that this potential contains both the harmonic oscillator and the Morse potentials as two distinct limiting cases and that the original potential carries several characteristics of these two potentials. Possible generalizations of the method are outlined.

NEW EXACT TREATMENT OF THE PERTURBED COULOMB INTERACTIONS

A novel method for the exact solvability of quantum systems is discussed and used to obtain closed analytical expressions in arbitrary dimensions for the exact solutions of the hydrogenic atom in the external potential ΔV(r) = br + cr2, which is based on the recently introduced supersymmetric perturbation theory.A novel method for the exact solvability of quantum systems is discussed and used to obtain closed analytical expressions in arbitrary dimensions for the exact solutions of the hydrogenic atom in the external potential ∆V (r) = br + cr2, which is based on the recently introduced supersymmetric perturbation theory. One of the challenging problems in nonrelativistic quantum mechanics is to find exact solutions to the Schrödinger equation for potentials that are used in different fields of physics. In particular, the perturbed Coulomb potentials represent simplified models of many situations found in atomic, molecular, condensed matter and particle physics. There has been much interest in obtaining analytical solutions of such potentials in arbitrary dimensions. These problems have been studied for years and a general solution has not yet been found. Such class of potentials, V (r) = −a/r + br + cr , (1) are possible candidates for the quarkonium potential as has been indicated by the quarkonium spectroscopy. In the special case of c = 0 and b > 0 such potentials reduce to the well known charmonium potential. Apart from its relevance in heavy quarkonium spectroscopy, this class of potentials with c = 0 has important applications in atomic physics. The Stark effect in a hydrogen atom in one dimension is given exactly by the charmonium like potential (b being the electric field parameter). The more general class of these potentials with c > 0 is also relevant in atomic physics. This could be interpreted as the potential seen by an electron of an atom exposed to a suitable admixture of electric and magnetic fields. In addition, nuclei in the presence of an electron background form a system which is important for condensed matter physics and for laboratory and stellar plasmas. The potential between two nuclei embedded in such a plasma is approximately Coulomb plus harmonic oscillator, which corresponds to b = 0 in (1). As the exact form of such interactions are being unknown to a great extent, it is thus desirable to study the general analytical properties of a large class of potentials in (1). In connection with this, the analyticity of the energy levels for these kind of potentials was investigated rigorously by many authors using different theories [1]-[14] in relation to their potential applications in spectroscopic problems. In this letter, we introduce an alternative, simple formalism for an algebraic solution of the Schrödinger equation with the perturbed Coulomb potential and find exact solutions in N dimesional space. The new formalism is based on the supersymmetric quantum mechanics and

Supersymmetry and the relationship between a class of singular potentials in arbitrary dimensions

The eigenvalues of the potentials V1(r) = A1/r + A2/r2 + A3/r3 + A4/r4 and V2(r) = B1r2 + B2/r2 + B3/r4 + B4/r6 and of the special cases of these potentials such as Kratzer and Goldman-Krivchenkov potentials, are obtained in N-dimensional space. The explicit dependence of these potentials in higher-dimensional space is discussed, which has not been previously covered.

The asymptotic iteration method applied to certain quasinormal modes and non Hermitian systems

We study the Schrödinger equation with potentials admitting quasinormal modes using the asymptotic iteration method (AIM). We also study non-Hermitian PT symmetric potentials using AIM. The spectra, in all cases, are found to be in excellent agreement with exact results.

Asymptotic iteration approach to supersymmetric bistable potentials

We examine quasi exactly solvable bistable potentials and their supersymmetric partners within the framework of the asymptotic iteration method (AIM). It is shown that the AIM produces excellent approximate spectra and that sometimes it is found to be more useful to use the partner potential for computation. We also discuss the direct application of the AIM to the Fokker–Planck equation.

Eigenvalue Problems of non-Hermitian Systems via Improved Asymptotic Iteration Method

We simply use the relation between the asymptotic iteration method and the Nikiforov–Uvarov method for the analytical solution of the second order linear ordinary differential equations. We apply this relation to study the Schrodinger equation with potentials admitting quasinormal modes. Non-Hermitian PT symmetric potentials have also been studied. Energy eigenvalues in all the cases by the relation are found to be consistent with exact results.

Unified treatment of screening Coulomb and anharmonic oscillator potentials in arbitrary dimensions

A mapping is obtained relating radial screened Coulomb systems with low screening parameters to radial anharmonic oscillators in N-dimensional space. Using the formalism of supersymmetric quantum mechanics, it is shown that exact solutions of these potentials exist when the parameters satisfy certain constraints.

Application of the Asymptotic Taylor Expansion Method to Bistable Potentials

A recent method called asymptotic Taylor expansion (ATEM) is applied to determine the analytical expression for eigenfunctions and numerical results for eigenvalues of the Schrodinger equation for the bistable potentials. Optimal truncation of the Taylor series gives a best possible analytical expression for eigenfunctions and numerical results for eigenvalues. It is shown that the results are obtained by a simple algorithm constructed for a computer system using symbolic or numerical calculation. It is observed that ATEM produces excellent results consistent with the existing literature.

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

1. IntroductionSince the wave function contains all the necessary information to describe aquantum system fully, it is of high importance of obtaining exact or approximatesolutions of Schr¨odinger equation in quantum mechanics. It is known that there arenot so many potentials that can be solved exactly. Therefore, many techniques aresuggested (and in use) to find the approximate solutions of the potentials that arenot exactly solvable. A recent technique, called the Asymptotic Iteration Method(AIM),