Antibound State for Klein-Gordon Equation

Transcription

1 International Journal of Mathematical Analysis Vol. 8, 2014, no. 59, HIKARI Ltd, Antibound State for Klein-Gordon Equation Ana-Magnolia Marin-Ramirez Ruben-Dario Ortiz-Ortiz Randy Zabaleta-Mesino Copyright c 2014 Ana-Magnolia Marin-Ramirez, Ruben-Dario Ortiz-Ortiz and Randy Zabaleta-Mesino. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we construct an asymptotic for resonance of a wave function associated with the Klein-Gordon equation in presence of a potential barrier. To achieve this, we reduce the main differential equation to an integral equation using Green s function, Fourier transform and Neumann series. Mathematics Subject Classification: 34A34, 34C25 Keywords: Klein-Gordon Equation, Resonance, Green s function

2 2946 A. M. Marin, R. D. Ortiz and R. Zabaleta 1 Introduction It was constructed an asymptotic for the Klein-Gordon equation [1]. Unbounded states were associated with the wave equation [5]. It was studied the resonance for scattered waves [4]. It was connected the Fredholm determinant approach of Froese to the Fourier transform approach of Zworski [3]. It was studied resonance for water waves [2]. Our goal is to construct an asymptotic of an unbounded solution of the Klein-Gordon equation perturbed by a potential barrier V (x) C0 (R). The antibound state (it is known as a resonance) is related with a potential that satisfies V (x)dx > 0. 2 Preliminary Notes We study the continuous Klein-Gordon equation Ψ tt Ψ + m 2 Ψ + ɛv (x)ψ = 0, m > 0, ɛ 0. Now, we are looking for the solution that satisfies the definition (2.1) in the form Ψ = e iwt ϕ(x), where w is the frequency, we obtain ϕ xx (x) + m 2 ϕ(x) + ɛv (x)ϕ(x) = λϕ(x), λ = w 2, (1) 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 ɛ = 0 and it is given by [m 2, ). Definition 2.1. A solution ϕ(x) of equation (1) is called a resonance if ϕ satisfies ϕ(x) e β x x (2) with β > 0 and λ = m 2 β 2. 3 Main Result The main result is as follows Theorem 3.1. Let V (x)dx > 0. Then for ɛ sufficiently small, the equation (1) has a resonance for λ = m 2 β 2, where β = ɛ 2Ṽ (0) + O(ɛ2 ). (3)

3 Antibound state for Klein-Gordon equation 2947 Proof. We consider the problem (1) and taking λ = m 2 β 2 we obtain ϕ xx (x) + β 2 ϕ(x) = ɛv (x)ϕ(x). We define L = d2 + β 2 and g(x) = ɛv (x)ϕ(x). Using Green s functions dx 2 L(G(x, ξ)) = δ(x ξ), where G(x ξ) = 1 e β x ξ [1]. Thus, the solution for Lϕ = g is given by ϕ = G g, where g has compact support. Now, we are looking for the solution of problem (1) applying Fourier transform ϕ(x) = [G ( ɛv ϕ)](x), where G(p) = 1 p 2 +β 2 ϕ(p) = ɛ Ṽ ϕ(p), (4) p 2 + β2 (p 2 + β 2 ) ϕ(p) =. (5) We know that outside the support of V (x), x > r, ϕ(x) = A 1 e βx +A 2 e βx. So the sought solution is of the form ϕ(x) = 1 p 2 + β dp + A 1e βx + A 2 2 e βx, (6) for x. To study the behavior of (6), we define the following contours around the simple poles D + = { x 1, y = 0} {x + iy : x 2 + y 2 = 1, y > 0} D = { x 1, y = 0} {x + iy : x 2 + y 2 = 1, y < 0}. Applying the Cauchy residue theorem to (6), we have ϕ(x) = 1 ) (Ã(iβ) 2π p 2 + β dp A 1 e βx + A 2 e βx, (7) D + for x > 0. Considering the right hand side of (7) and A 1 = Ã(iβ), we have ϕ(x) = A 2 e βx + 1 e ip Ã(p + i) 2π e x D + {i} (p + i) 2 + β dp. 2 So, ϕ(x) = A 2 e βx + O(e x ), when x.

4 2948 A. M. Marin, R. D. Ortiz and R. Zabaleta Analogously, ϕ(x) = A 1 e βx + O(e x ), when x and A 2 = Ã( iβ). Therefore, ϕ(x) = 1 p 2 + β 2 dp Ã(iβ) The Fourier transform of (8) has the form ϕ(p) = e βx Ã( iβ) e βx. (8) p 2 + β 2 + 2πA 1δ(p iβ) + 2πA 2 δ(p + iβ). (9) Substituting (9) into (4), we obtain = ɛ Ã(ξ) Ṽ (p ξ) ξ 2 + β dξ ɛa 1Ṽ (p iβ) ɛa 2Ṽ (p + iβ). (10) 2 Applying the Cauchy residue theorem to the equation (10), we obtain = ɛ ( ) Ã(ξ) A( iβ) Ṽ (p ξ) 2π D + ξ 2 + β dξ + ɛ Ṽ (p + iβ). (11) 2 We define Ω as the set of bounded analytic functions on B 1 = {z C, I z < 1}, with the norm ϕ = sup z B1 ϕ(z), and the operator T β : Ω Ω by Ã(ξ) [T β Ã(ξ)](p) = Ṽ (p ξ) dξ, p Ω. (12) D + ξ 2 + β2 We can rewrite the equation (11) [(1 + ɛt β )Ã(ξ)](p) = ɛ (Ã( iβ) ) Ṽ (p + iβ). (13) Since the operator T β is bounded, therefore ɛt β is small, it corresponds to a contraction operator then we can take its inverse, thus: ) (Ã( iβ) = ɛ [(1 + ɛt β ) ξ p ] 1 Ṽ (ξ + iβ), (14) where 1 is the identity operator. Rewriting (14) in terms of the Neumann series ) (Ã( iβ) = ɛ ( 1) n ɛ n [Tβ n Ṽ (ξ + iβ)](p). (15) n=0

5 Antibound state for Klein-Gordon equation 2949 Now, let us evaluate (15) at p = iβ, we obtain 1 = 1 ( 1) n ɛ n+1 [Tβ n Ṽ (ξ + iβ)]( iβ). (16) n=0 Taking the main term of (16) Multiplying by β, we obtain (3). 1 = ɛ Ṽ (ξ + iβ) ξ= iβ + O(ɛ 2 ). Acknowledgements. The authors express their deep gratitude to Universidad de Cartagena for partial financial support References [1] A. M. Marin, R. D. Ortiz and J. A. Rodriguez Ceballos, Asymptotics of the Klein-Gordon equation, Far East J. Appl. Math., 70(2) (2012), [2] M. I. Romero Rodriguez and P. Zhevandrov, Trapped modes and resonances for water waves over a slightly perturbed bottom, Russian J. Math. Phys., 17(3) (2010), [3] B. Simon, Resonances in one dimension and Fredhlom determinants, J. Funct. Anal., 178(2) (2000), [4] S. H. Tang and M. Zworski, Resonance expansions of scattered waves, Com. Pure App. Math., 53(10) (2000), [5] M. Zworski, Resonances in physics and geometry, Notices Amer. Math. Soc., 46(3) (1999), Received: December 7, 2014; Published: December 30, 2014