Levinson’s theorem for the Klein-Gordon equation in two dimensions

Abstract

In terms of the modified Sturm-Liouville theorem, the two-dimensional Levinson theorem for the Klein-Gordon equation with a cylindrically symmetric potential V(r) is established for an angular momentum m as a relation between the numbers n m Ϯ of the particle and antiparticle bound states and the phase shifts m (ϮM ): m ͑M͒Ϫ m ͑ϪM͒ϭ ͭ ͑n m ϩ Ϫn m Ϫ ϩ1͒ when a half-bound state occurs at EϭM for mϭ1 ͑ n m ϩ Ϫn m Ϫ Ϫ1 ͒ when a half-bound state occurs at EϭϪM for mϭ1 ͑ n m ϩ Ϫn m Ϫ ͒ the remaining cases. A solution of the Klein-Gordon equation with the energy M or ϪM is called a half-bound state if it is finite but does not decay fast enough at infinity to be square integrable.

Fermionic bound states in distinct kinklike backgrounds

“…Similar investigations have been carried out in Refs. [16][17][18][19][20][21]. It is worth mentioning that both potentials (2.7) and (2.8) reproduce the φ 4 model in the case p = n = 1.…”

“…We go on and investigate the threshold or half-bound states, which are states where the fermion field goes to a constant when x → ±∞. Although the wave function is finite when x → ±∞, these states do not decay fast enough to be square-integrable [17][18][19]. Anyway, to find threshold energies we solve the system of equations at x → ±∞.…”

Generalized Levinson theorem for singular potentials in two dimensions

“…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.…”

“…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]. …”