Fractal Laplacian approach to quantum wave localization

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.