General comparison theorems for the Klein-Gordon equation in d dimensions

Abstract

We study bound-state solutions of the Klein-Gordon equationfor bounded vector potentials which in one spatial dimension have the form V (x) = v f (x), where f (x) ≤ 0 is the shape of a finite symmetric central potential that is monotone non-decreasing on [0, ∞) and vanishes as x → ∞. Two principal results are reported. First, it is shown that the eigenvalue problem in the coupling parameter v leads to spectral functions of the form v = G(E) which are concave, and at most uni-modal with a maximum near the lower limit E = −m of the eigenenergy E ∈ (−m, m). This formulation of the spectral problem immediately extends to central potentials in d > 1 spatial dimensions. Secondly, for each of the dimension cases, d = 1 and d ≥ 2, a comparison theorem is proven, to the effect that if two potential shapes are ordered f1(r) ≤ f2(r), then so are the corresponding pairs of spectral functions G1(E) ≤ G2(E) for each of the existing eigenvalues. These results remove the restriction to positive eigenvalues necessitated by earlier comparison theorems for the Klein-Gordon equation.Hall and Harb at a function v(E), whereas E(v) is a two-valued expression. In section 2, we establish the principal features of the spectral curves v(E) for the class of negative bounded potentials that vanish at infinity. In section 3 we solve the Klein-Gordon equation analytically for the square-well potential in d ≥ 1 dimensions. In section 4 we prove some comparison theorems: the principal results claim that for any discrete eigenvalue E ∈ (−m, m) and negative potential-shape functions f 1 (r) and f 2 (r) we haveIn section 5 we exhibit a complete recipe for spectral bounds for this class of potentials based on comparisons with the exactly soluble square-well problem.II. GENERAL FEATURES OF THE SPECTRAL CURVE G(E) = v(E). A. One-dimensional caseThe Klein-Gordon equation in one dimension is given by:

Refining the general comparison theorem for Klein-Gordon equation

“…We fix E = 0.96 and obtain v 1 = 0.4895 and v 2 = 0.4799. Since f 1 (r) > f (r) for r ∈ [0, ∞) as shown in Figure 7, then according to our simple general comparison theorem [21], we find that v 1 > v. On the other hand, f and f 2 cross over at r 0 ≈ 1.2 as shown in the right graph of Figure 8, with r0 0 f 2 (r) − f (r) r 2 dx = 0.0108 > 0. Hence, applying our refined version of the general comparison theorem, we obtain v > v 2 .…”

“…We use the square-well potential and the exponential potential as a lower bound and an upper bound respectively. We have discussed a similar idea in our previous paper [21] based on square-well spectral bounds, but the results were limited by the condition that the graphs cannot cross over. Since we have been able to refine our previous comparison theorem, we can find better bounds now.…”

“…Since we have been able to refine our previous comparison theorem, we can find better bounds now. We have chosen the square-well and the exponential potentials because we know the exact solutions of the Klein-Gordon equation with each of these potentials [32,33], [21]. These solutions allow us to find the exact values for the corresponding eigenvalues v = v(E).…”

“…It is clear from these examples that the graphs of E(v) are certainly not monotone. However, by the use of monotonic analysis several relativistic comparison theorems have been established [15][16][17][18][19][20], but the proofs of such results for the Klein-Gordon equation were all restricted to positive energies. In a recent paper [21] we were able to generalize these comparison theorems by studying the eigen-problem in the coupling parameter v for a given E ∈ (−m, m).…”

“…However, by the use of monotonic analysis several relativistic comparison theorems have been established [15][16][17][18][19][20], but the proofs of such results for the Klein-Gordon equation were all restricted to positive energies. In a recent paper [21] we were able to generalize these comparison theorems by studying the eigen-problem in the coupling parameter v for a given E ∈ (−m, m). The spectral relations expressed in form v = G(E) are single-valued functions, which facilitate the design of the unrestricted Klein-Gordon comparison theorems.…”