An efficient interpolating wavelet collocation scheme for quasi‐exactly solvable Sturm–Liouville problems in ℝ

• Published in: Mathematical methods in the applied sciences Vol. 45; no. 7; pp. 4002 - 4023
• Main Authors: Singh, Debabrata, Saha, M. K., Banik, Sayan, Mohan Panja, Madan
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 15-05-2022
• Subjects: Approximation; Collocation methods; Daubechies wavelets; Eigenvalues; Eigenvectors; interpolating wavelets; Mathematical analysis; Quantum mechanics; quasi‐exactly solvable Sturm–Liouville problem; radial Schrödinger equation; wavelet collocation method
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.8028

This investigation is an attempt to obtain a highly accurate approximation of the spectrum of Sturm–Liouville problems in
ℝ+ by representing the unknown solution of the model in the interpolating wavelet basis of
L2(ℝ). To accomplish the goal, the domain
ℝ+ has been stretched to
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
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
ℝ+ with high accuracy.