Untitled

Quasi-exact solution of the problem of relativistic bound states in the (1+1)-dimensional case

“…We note that this result has a rigorous justification for Eq. (1.1) outside the framework of any approximation [7]. The coefficients A n for the interior solution are found using relations (2.24) straightforwardly from the values of f (z) at z = ±iω n .…”

“…In both cases, because of these purely imaginary finite argument shifts, the dynamical equations in fact turn out to be functional equations, and they differ radically from their differential analogues (the nonrelativistic Schrödinger equation and the Dirac equation) in several fundamental aspects. In particular, the integral form of these equations involves a specific nonlocality of the order of the Compton wavelength of the system whose consistent inclusion requires an essentially nonperturbative approach, while the straightforward perturbation theory turns out to be effective only under some rather stringent constraints on the problem parameters [7]. As a consequence, the solution properties for these equations differ considerably from those of their differential analogues in several features and lead to some qualitatively new effects in the behavior of the observed physical characteristics of the system.…”

“…Here, as in [7], we proceed from the analysis in [10] in the framework of the problem of covariantly quantizing the field in Lorentz bases and consider only those φ(ξ) and χ(ξ) that have an analytic continuation to the strip |Im ξ| < . In this case, we have…”

“…The analyticity condition for φ(ξ) and χ(ξ) in the strip |Im ξ| < can be given a rather clear physical interpretation [7], [10]. For brevity, we confine ourself here to only the boson case.…”

“…(1.1) and (1.7) in powers of 2 , which leads to the representation of these equations in the form of the corresponding series beginning with the Schrödinger or Dirac equation ((1.3) or (1.6)) may not be meaningful for 1. Moreover, as already shown in [7], this perturbation theory is in fact inapplicable even for constructing approximate solutions of these finitedifference equations because truncating this power series in 2 at a finite term is equivalent to replacing the exponential (finite-shift operator) with a polynomial. But a polynomial does not uniformly approximate the exponential throughout the number line, and it is consequently quite possible that the solutions of the equation thus truncated have nothing in common with the true solutions.…”