Complex Parameters in Quantum Mechanics

Abstract

The Schrodinger equation for stationary states in a central potential is studied in an arbitrary number of spatial dimensions, say q. After transformation into an equivalent equation, where the coefficient of the first derivative vanishes, it is shown that in such equation the coefficient of the second inverse power of r is an even function of a parameter, say lambda, depending on a linear combination of q and of the angular momentum quantum number, say l. Thus, the case of complex values of lambda, which is useful in scattering theory, involves, in general, both a complex value of the parameter originally viewed as the spatial dimension and complex values of the angular momentum quantum number. The paper ends with a proof of the Levinson theorem in an arbitrary number of spatial dimensions, when the potential includes a non-local term which might be useful to understand the interaction between two nucleons.