Asymptotics for Klein-Gordon equation

Abstract

We propose a simple method for constructing an asymptotic of an eigenvalue for the Klein-Gordon equation in the presence of a shallow potential well, reducing the initial problem to an integral equation and then by applying the method of Neumann series to solve it.

Antibound state for Klein-Gordon equation

“…where V (x) = 0 for x > r and x < −r with r sufficiently large. The continuous spectrum of equation (1) coincides with the continuous spectrum of the unperturbed equation when = 0 and it is given by [m 2 , ∞).…”

Approximation to the dissipative Klein-Gordon equation

“…It was found an antibound state for the Klein-Gordon equation [2]. In [3] was used a method with Green functions for constructing asymptotics of eigenvalues for the linear Klein-Gordon equation. In [4] was studied the Klein-Gordon equation.…”

“…In [4] was studied the Klein-Gordon equation. In [5] was given an explicit formula for the eigenvalue below the essential spectrum of discrete Klein-Gordon operator. We show there exists a solution for a small parameter and construct a traveling wave equation and a system without periodic orbits.…”

Antibound state for the discrete Schrodinger equation

“…It was constructed the asymptotic for natural frequencies of the Schrödinger equation using the (WKB) method [4]. It was encountered a formula explicitly to the eigenvalue that appears below the essential spectrum of the discrete equation Klein-Gordon [3]. It was found a resonance for the discrete shallow water equation in the case of an underwater trench [1] .…”