Preface: New trends in representation theoretical methodologies

  • Metin Demiralp


The decomposition of the mathematical entities like functions, arrays, operators and some others is an important issue in many branches of science and engineering since it is the base for the application of so-called divide-and-conquer strategy. This gains a lot of importance when the target entity becomes multivariate (like multivariable functions and multiway arrays) because the decomposition techniques may take us to facing with many unexpected pitfalls due to the multidimensionality of the model under consideration. The type and the solvability of the problems arising from multidimensionality depends on the decomposition method structuring and the target entity properties.

In this special issue a coordinate bending paperĀ  authored by Semra Bayat is included. It is about the spatially univariate quantum systems having only discrete spectrum. Most famous and elementary quantum systems in
this direction are harmonic oscillators not only in univariance but also in multivariance. Their Schr\"odinger equations are all solvable analytically. When the models become having anharmonicity the same equations become quite
complicated. For all these cases the potential function forms an infinite well with a finite bottom. Their spectra are totally composed of discrete spectrum. The energy values should be greater than the minimum of the potential function. Harmonic and anharmonic oscillators are in this category. Even though all the anharmonic oscillators (mostly polynomially anharmonic ones) have quite complicated solutions; especially the exponentiality brings somehow awkward complications which may need quite rigorous asymptotic analyses. In this paperĀ  the target ODE's potential is constructed in such a way that a harmonic potential segment serves as spectrum controlling agent and a perturbation like segment also arises to facilitate the analysis for making a rather good device to handle the asymptotic behaviors at large values of the bent coordinate.