Asymptotics for Klein — Gordon equation

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.


Introduction
In [5], we find the Klein-Gordon equation where ∆ is the Laplacian in dimension n, perturbed by a potential U = U (x) to We look for the solution of the equation Phi in the form Φ = exp(iωt)Ψ(x), (1.2) where ω is the frequency.If we replace PhiSi in Phi, then we obtain the equation When m = 0, we have the Schrödinger equation (−∆ + U )Ψ = EΨ, (1.4) that in the case when U describes a shallow potential well (i.e., dx ≤ 0 and the dimension n of the configuration space is 1 or 2. This was established for n = 1 and in the radially symmetric case for n = 2 in the famous book of Landau and Lifshitz [4] and it was demonstrated in the general case in dimension 2 by Simon [6].Close results to the limit behavior of the resolvent can be found in [1], [3].In [8], a different method was used for obtaining the asymptotics of the eigenfunctions.
It is based on a construction of eigenfunctions.It happens that this construction is elemental, when we pass to the momentum representation.Also, this method is efficient for the Schrödinger and Klein-Gordon equation.

Mathematical formulation
The mathematical formulation of the problem under consideration is as follows.We look for non trivial solutions Φ ∈ L 2 (R), of the problem where ε → 0 and V is such that R ∞ −∞ V (x)dx ≤ 0 and V has compact support, then V (x) = 0 for |x| > R with R sufficiently big.Given that the operator of multiplication by a function of compact support is compact in L 2 , the continuous spectrum of (2.1) coincides with the continuous spectrum of the non perturbed equation (ε = 0) and the last is the interval [m 2 , ∞).We prove the following theorem.
Then the problem (2.1) has an eigenvalue where is the solution of the secular equation for β (5.8).The contour Γ is defined by the equation (5.2).

Heuristic considerations
Denoting the Fourier transform by As in [8], the formulas that appear in the Theorem 2.1 are based on the following heuristic reasoning: For E = −β 2 + m 2 , the solution of (2.1) for |x| > R is given by Φ(x) ∼ e −β|x| .We obtain a function that is ˝almost con-stant˝, when β → 0. Since being ˝almost constant˝, its Fourier transform is a sequence of delta type when β → 0.

Reduction to an integral equation
Taking the Fourier transform in the equation L1, we obtain Φ(p) Here W (p, p 0 ) is given by W(p,p')= e V (p − p 0 ), where the tilde denotes the Fourier transform.

Demonstration of the Theorem 2.1
Proof.
Taking E = −β 2 + m 2 , β → 0+, we look for a solution of the equation ( 4) in the form (5.1) Denoting Ω the space of analytic functions on B 1 and continuous on B 1 with the standard norm of the supreme, kϕk = sup z∈B 1 ϕ(z) for all ϕ ∈ Ω.