The traditional Schrödinger equation is expressed in smooth Euclidean space, wherein the Laplacian operator embodies integer-dimensional geometry. Nonetheless, some natural and quantum systems have irregular, self-similar structures that are more accurately described by non-integer, or fractal, dimensions. This study presents a theoretical enhancement of the Schrödinger equation by the introduction of a fractal Laplacian operator defined over a domain of fractal dimension Formula: see text. The new operator takes into consideration scale invariance and non-integer spatial dimensions, which makes it possible to rewrite the kinetic term in terms of fractal scaling. Analytical derivation shows that the energy eigenvalues scale as Formula: see text, which is not the same as the normal Formula: see text dependence seen in integer dimensions. The requirement for normalization is restated by employing integration over a fractal measure. This paradigm offers a purely mathematical basis for the examination of quantum behavior on fractal geometries, connecting spectral theory with fractal analysis without relying on any physical or experimental premises.
Valarmathi et al. (Mon,) studied this question.