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The supersymmetric quantization condition is used to study the wave functions of SWKB equivalent q-deformed harmonic oscillator which are obtained by using only the knowledge of bound-state spectra of q-deformed harmonic oscillator. We have also studied the nonuniqueness of the obtained interactions by this spectral inverse method.

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The linear delta expansion technique has been developed for solving the differential equation of motion for symmetric and asymmetric anharmonic oscillators. We have also demonstrated the sophistication and simplicity of this new perturbation technique.We have presented the linear delta expansion (LDE) technique [1] for the resolution of oscillatory non-linear problems. This is a powerful technique which has been originally introduced to deal with problems of strong coupling in quantum field theory. This method has also been applied to a wide class of problems [2-9]. In the original formulation of LDE technique, the Lagrangian density L δ which is not exactly solvable, is interpolated with a solvable Lagrangian L 0 . The full Lagrangian was written as L δ = L 0 + δL. For δ = 0, one obtains solvable Lagrangian L 0 . Here δL is treated as a perturbation and δ is used to keep track of the perturbative order.In the theory of harmonics, there is an important phenomenon which should be pointed out because of its practical importance in demonstrating non-linear effects. The thermal expansion of a crystalline solid can be understood from the non-linear force acting between the atoms. The restoring force between a pair of atoms is asymmetric.For large amplitude of vibration, the restoring force often contains additional terms involving higher power of displacement x. In such case, the restoring force is expressed aswhere s, a 2 , a 3 , a 4 , a 5 etc. are constants. For this case, the oscillator becomes nonlinear and the vibration contains a fundamental frequency and higher harmonics. The oscillations of a non-linear oscillator are called anharmonic oscillations. The differential equation of motion for this type of oscillation is 117

An approximation method based on the iterative technique is developed within the framework of linear delta expansion (LDE) technique for the eigenvalues and eigenfunctions of the one-dimensional and three-dimensional realistic physical problems. This technique allows us to obtain the coefficient in the perturbation series for the eigenfunctions and the eigenvalues directly by knowing the eigenfunctions and the eigenvalues of the unperturbed problems in quantum mechanics. Examples are presented to support this. Hence, the LDE technique can be used for nonperturbative as well as perturbative systems to find approximate solutions of eigenvalue problems.Keywords. Formulation of iterative technique for one dimension and applications; formulation of iterative technique for three dimensions and applications.

A unified approach in the light of supersymmetric quantum mechanics (SSQM) has been suggested for generating multidimensional quasi-exactly solvable (QES) potentials. This method provides a convenient means to construct isospectral potentials of derived potentials.

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