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, Schrödinger equation with Pöschl–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 \(\mathbb {R}\).

这项工作致力于开发一种有效的内插子波配置方案，以获得Sturm-Liouville问题的高精度本征态，以期在\（\ mathbb {R} \）。在Daubechies族中，由尺度函数和symlet小波产生的插值小波尺度函数的性质已被明智地用于估计本征函数的后验误差。该方案的效率已在各种可完全求解的模型上进行了检验（例如，具有无限数量的结合态的谐波振荡器问题，具有有限数量的结合态的Pöschl–Teller势的Schrödinger方程，Mathews的量子形式–具有约束态的Lakshmanan振荡器在渐近区域和一些拟精确可解模型中代数衰减，并且发现是高效的。该方案随后被应用来揭示许多具有多项式和非多项式势的拟精确可解模型的一些隐藏界态。通过仔细分析此处执行的示例的结果，可以发现，与其他方法相比，提出的方案似乎非常有用和高效[例如，Gaudreau等人的双指数Sinc配置方法。（Ann Phys 360：520–538，2015）]可提供文献中的准精确可解或不可解非相对论量子力学模型的束缚态能和波函数的高精度近似。Gaudreau等人的双指数Sinc配置方法。（Ann Phys 360：520–538，2015）]可提供文献中的准精确可解或不可解非相对论量子力学模型的束缚态能和波函数的高精度近似。Gaudreau等人的双指数Sinc配置方法。（Ann Phys 360：520–538，2015）]可提供文献中的准精确可解或不可解非相对论量子力学模型的束缚态能和波函数的高精度近似。\（\ mathbb {R} \）。