A new definition of the function of a differential operator that leads to local operators of infinite order is proposed. It allows one to obtain an expression for the square root of the differential operator (the Hamiltonian of a free spinless particle) and to determine the relativistic Schr¨odinger equation as a close analog of the non-relativistic Schr¨odinger equation. It is shown that this equation does not lead to difficulties of the Klein-Gordon equation. Boundary conditions that lead to self-adjoint boundary problems similar to Sturm-Liouville problems, periodic boundary value problems, and singular boundary value problems are determined. Some problems of relativistic quantum mechanics are solved.
|Journal||IOP Conf. Series: Journal of Physics: Conf. Series 1124 (2018) 031002|
|Publication status||Published - 2019|
Головин, А. В., & Lagodinskiy, V. M. (2019). On the possibility of constructing relativistic quantum mechanics on the basis of the definition of the functions of differential operators. IOP Conf. Series: Journal of Physics: Conf. Series 1124 (2018) 031002, 1205(012019), . https://doi.org/10.1088/1742-6596/1205/1/012019