An efficient interpolating wavelet collocation scheme for quasi-exactly solvable Sturm-Liouville problems in Double-struck capital R

This investigation is an attempt to obtain a highly accurate approximation of the spectrum of Sturm-Liouville problems in Double-struck capital R+ by representing the unknown solution of the model in the interpolating wavelet basis of L2(Double-struck capital R). To accomplish the goal, the domain Double-struck capital R+ has been stretched to Double-struck capital R to avoid the additional care of the elements in the basis containing boundary point 0. In addition, such transformation may judiciously be utilized to eliminate (up to quadratic) the singularity of the equation. The equation in the new variable has been subsequently transformed into a generalized matrix eigenvalue problem by approximating the new (unknown) function in an appropriate (truncated) basis comprising interpolating scale functions generated by scale functions in Daubechies family. The (interpolating wavelet-collocation) scheme developed here has been applied to some solvable and quasi-exactly solvable Sturm-Liouville problems in Double-struck capital R+ appearing in quantum mechanical modeling in flat and curved spaces. It is observed that the approximation of eigenfunctions in the (compact support) interpolating wavelet basis obtained by using the collocation method can be reliably used to reveal a hidden spectrum of quasi-exactly solvable Sturm-Liouville problems in Double-struck capital R+ with high accuracy.

Automated summary

This investigation aims to obtain an accurate approximation of the spectrum of Sturm-Liouville problems in Double-struck capital R+ by representing the unknown solution in the interpolating wavelet basis of L2(Double-struck capital R). The study uses domain stretching and equation transformation techniques to handle the problem, and approximates the new function using an appropriate basis generated by scale functions. The results show that the interpolating wavelet basis obtained through the collocation method can reliably reveal the spectrum of solvable and quasi-exactly solvable Sturm-Liouville problems in Double-struck capital R+ with high accuracy.