Higher order quantum waves in fractal dimensions from nonlocal complex derivative operator

In this study, we introduced a new nonlocal complex derivative operator in fractal dimension based concurrently on the concept of “nonlocal generalized complex backward-forward coordinates” and the “product-like fractal measure”. The quantization of the theory in fractal dimension leads to a higher order Schrödinger equation characterized by a higher order energy operator. As an illustration, we have discussed the cases of infinite quantum well and power-law potentials. Their associated zero-point energies were found to depend on the numerical value of the fractal dimension. For the infinite well, the decrease in zero-point energy with fractal dimension may result in the emission of large wavelengths photons observed experimentally in X-ray laser bursts emitted from the solid.