  This paper presents a novel and efficient approach for the computation of
energy eigenvalues in quantum semiconductor heterostructures. Accurate
determination of the electronic states in these heterostructures is crucial for
understanding their optical and electronic properties, making it a key
challenge in semiconductor physics. The proposed method is based on utilizing
series expansions of zero-order Bessel functions to numerically solve the
Schr\"odinger equation using boundary integral method for bound electron states
in a computationally efficient manner. To validate the proposed technique, we
applied it to address previously explored issues by other research groups. The
results clearly demonstrate the computational efficiency and high precision of
our approach. Notably, the proposed technique significantly reduces the
computational time compared to the conventional method for searching the energy
eigenvalues in quantum structures.
