Efficient interpolating wavelet collocation scheme for quantum mechanical models in $$\mathbb {R}$$ R

This work deals with the development of an efficient interpolating wavelet collocation scheme for obtaining highly accurate eigenstates of Sturm–Liouville problem with a view to divulge some hidden bound state spectrum of quasi-exactly solvable models in non-relativistic quantum mechanics in $$\mathbb {R}$$ . The properties of scale functions in the interpolating wavelet basis generated by scale function and wavelets of symlets in Daubechies family have been judiciously used to provide an estimate of a posteriori error in the approximation of eigen functions. The efficiency of the scheme has been examined on a variety of exactly solvable models (e.g., harmonic oscillator problem having an infinite number of bound states, Schrodinger equation with Poschl–Teller potential having a finite number of bound states, the quantum version of Mathews–Lakshmanan oscillator with bound states decaying algebraically in the asymptotic region and some quasi-exactly solvable models) and found highly efficient. The scheme has been subsequently applied to reveal some hidden bound states of a number of quasi-exactly solvable models with polynomial and non-polynomial potentials. From the careful analysis of results of the examples exercised here, it is observed that the proposed scheme seems to be quite useful and efficient compared to other methods [e.g., double exponential sinc collocation method of Gaudreau et al. (Ann Phys 360:520–538, 2015)] available in the literature to expose highly accurate approximation of bound state energies and wave functions of quasi-exactly solvable or non-solvable non-relativistic quantum mechanical models in $$\mathbb {R}$$ .