Levinson theorem in two dimensions

Abstract

A two-dimensional analogue of Levinson's theorem for nonrelativistic quantum mechanics is established, which relates the phase shift at threshold(zero momentum) for the $m$th partial wave to the total number of bound states with angular momentum $m\hbar(m=0,1,2,...)$ in an attractive central field.Comment: LaTeX, no figur

Generalized Levinson theorem for singular potentials in two dimensions

“…(41), one can obtain the Levinson theorem in the form (10). However, the method of the Green functions in the form proposed by Lin [23] does not work for singular potentials. The reason is that the difference of Green functions (41) has a singularity at the origin, hence it is not integrable.…”

“…The 2D Levinson theorem was established for different models, too: for the Schrödinger equation [23,24], the Klein-Gordon equation [25], and the Dirac equation [26]. Moreover there exists an extension of the Levinson theorem for the Schrödinger equation in D dimensions.…”

“…To generalize the Levinson theorem, we use the method of the Green functions, as it was done for regular potentials by Lin [23]. We consider the noncritical case, when the Schrödinger equation has no half bound states.…”

“…Moreover there exists an extension of the Levinson theorem for the Schrödinger equation in D dimensions. There are several methods for studying the lower-dimensional Levinson theorem: the Jost function method [27], the Green function method [20,23,26,28], and the Sturm-Liouville theorem [17,18,19,24,25,29]. Levinson's relation for the partial wave phase has the usual form, as for the 3D case; but the halfbound state for the p-wave (l = 1) contributes exactly like the bound state and gives an additional π to Levinson's relation [28].…”

Levinson's theorem for the Klein-Gordon equation in one dimension

“…Most of works mainly studied the Levinson theorem in the three-dimensional space. With the wide interest in lower-dimensional field theories recently, the two-dimensional Levinson theorem has been studied numerically [18] as well as theoretically [19][20][21][22][23][24][25]. …”