Hydrogen-Like Atom in Spaces of Lower Dimensions

The problem of finding discrete energy values of a particle with negative charge equal in absolute value to the elementary charge in one-dimensional space (e → e ₁), located in the field of a nucleus with charge (Ze ₁) with potential corresponding to the space of this dimension (N = 1) and different from the potential of the nucleus in three-dimensional space (N = 3) is solved in the quasiclassical approximation. For the one-dimensional case, the corresponding Schrödinger equation is solved and exact energy values are obtained, coincident with the quasiclassical approximation in the limit of large quantum numbers, and the wave function, expressed in terms of the Airy function, is found. In this latter approach, the energy values depend on the zeros of the Airy function. Considerations are discussed, touching on the possibility of solution of the Schrödinger equation for a hydrogen-like atom in two-dimensional space (N = 2).