Recently developed approximation methods for quantum mechanical problems which treat the spatial dimension D as an expansion parameter offer approximations to energy levels at arbitrary D. Rather than simply being a detour to the D=3 case, there is physical interest in nonphysical values of D due to degeneracies between states in different dimensions. For example, such degeneracies make it possible to calculate some excited states of two-electron atoms in three dimensions from the ground state energy at nonphysical values of D. Such relationships can be exploited in a simple derivation of the hydrogen atom spectrum in arbitrary D, using only the solution at D=1 and a combination of inter- and intradimensional symmetry arguments. Applications to the Yukawa potential and an anharmonic oscillator are also presented. A large class of interdimensional degeneracies is found for two-electron atoms. Approximate degeneracies are also identified for these atoms which allow highly excited D=3 states to be treated as perturbed low-lying states in another dimension. The approximate degeneracies also serve to generalize the treatment of the hydrogen atom spectrum in a way appropriate to two-electron atoms.