Analyticity of the scattering operator for the nonlinear Klein-Gordon equation with cubic nonlinearity

Kumlin, Peter

doi:10.1007/bf02101092

Cited by 3 publications

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“…± on the dense subspace F ⊂ H. Since W ± is an analytic diffeomorphism and V 0 (g) is continuous and linear, it suffices to show that V λ (g) is continuous to conclude the result for all v ∈ H. Writing V λ (g)v explicitly in terms of v in terms of an integral equation, continuity follows from the result of Kumlin [24] that the map from v to the solution φ is continuous from H to L 3 (IR, L 6 (IR 3 )).…”

Section: The Massive φ 4 Theorymentioning

confidence: 97%

“…Strauss [50,51] later proved the existence of wave operators on all of H, and inverted them at low energy, i.e., in a neighborhood of the origin of H. In 1985, Brenner [9] constructed inverses for the wave operators throughout H. Baez and Zhou proved that the wave operators are homeomorphisms, and analytic diffeomorphisms at low energy [7,8]. In a paper to be published, Kumlin [24] has proved that the wave operators are analytic diffeomorphisms throughout H. All this work relies primarily on hard analysis, and particularly on sharp decay estimates for solutions of the free theory. The technique for obtaining such estimates was found by Strichartz [53], and developed by Marshall, Strauss and Wainger [26,27].…”

Section: The Massive φ 4 Theorymentioning

confidence: 99%

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Scattering and complete integrability in four dimensions

Baez¹

1992

Contemporary Mathematics

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The method of 'linearization via the wave operator' establishes a close connection between scattering theory and complete integrability for nonlinear wave equations in four-dimensional Minkowski space. We review the proofs of complete integrability for the massive φ 4 theory on the space of solutions of finite energy, and for the massless φ 4 theory on the space of solutions all of whose images under conformal transformations have finite energy. We show that the complete integrability is associated to the infinite-dimensional symmetry group C(S 3 , U (1)) in the massive case, and a subgroup of C 1 (S 3 , U (1)) in the massless case. We review the construction of gauge-invariant conserved quantities for suitably regular solutions of the Yang-Mills equations in terms of asymptotic 'in' or 'out' fields, and discuss the prospects for complete integrability in this case. We wish to emphasize that by 'completely integrable' we do not mean 'exactly solvable,' which is a rather vague notion in any event. Instead, we mean the existence of a complete set of conserved quantities with vanishing Poisson brackets, arising from the action of a commutative Lie group of symmetries. Strictly speaking, this is a property not of an equation, but of a one-parameter group of canonical transformations of phase space. There are various ways of formulating this property precisely, especially when the phase space is infinite-dimensional, so we begin with some definitions. In all that follows we assume that the phase space M is a (smooth) Banach manifold. We say that a function from M to another Banach manifold is 'smooth' if it is infinitely Frechét-differentiable. We say that (M, ω) is a 'symplectic manifold' if ω is a smooth closed 2-form on M that is weakly nondegenerate, that is, for any nonzero u ∈ T x M there exists v ∈ T x M with ω(u, v) = 0. Given a symplectic manifold (M, ω), we say that a subspace V ⊂ T x M is 'isotropic' if ω|V = 0, and 'Lagrangian' if it is a maximal isotropic subspace. We say that a diffeomorphism f : M → M of a symplectic manifold is a 'symplectomorphism' (in the language of physics, a canonical transformation) if f * ω = ω. Let (M, ω) be a symplectic manifold. We say a function f ∈ C ∞ (M) is 'nice' if it generates a smooth vector field ζ f on M in the following sense: df = ω(ζ f , •). Since the map v → ω(v, •) need not define an isomorphism of T x M and T * x M , not every smooth function need be nice, but ζ f is unique if it exists. Given nice functions f 1 , f 2 generating vector fields ζ 1 , ζ 2 , the function {f 1 , f 2 } = −ω(ζ 1 , ζ 2) generates the vector field [v 1 , v 2 ]. In other words, the nice functions form a Lie algebra under the Poisson bracket {•, •}, and the map f → ζ f is a Lie algebra homomorphism. Let G be a Banach Lie group and ρ: G × M → M a smooth action of G on M as symplectomorphisms. The action ρ defines a homomorphism dρ from the Lie algebra g of G to the Lie algebra of smooth vector fields on M. We say that the action ρ is 'Hamiltonian' if for each v ∈ g there is a nice functio...

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Section: The Massive φ 4 Theorymentioning

confidence: 97%

Section: The Massive φ 4 Theorymentioning

confidence: 99%

Scattering and complete integrability in four dimensions

Baez¹

1992

Contemporary Mathematics

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“…[2,3,10]). However, we have to overcome some difficulties arising from loss of the good properties (1) and (2) for the Schrödinger equation.…”

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confidence: 99%

“…Based on these decay estimates and the arguments in Kumlin [10], we can prove Theorem 1.1 by applying the approximation theorem for analytic operator sequences (cf. [8]).…”

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confidence: 99%

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On the real analyticity of the scattering operator for the Hartree equation

Miao

Zhang

2009

Ann. Polon. Math.

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Abstract. We study the real analyticity of the scattering operator for the Hartree equation i∂tu = −∆u + u(V * |u| 2 ). To this end, we exploit interior and exterior cut-off in time and space, together with a compactness argument to overcome difficulties which arise from absence of good properties as for the Klein-Gordon equation, such as the finite speed of propagation and ideal time decay estimate. Additionally, the method in this paper allows us to simplify the proof of analyticity of the scattering operator for the nonlinear Klein-Gordon equation with cubic nonlinearity.

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Analyticity of the Scattering Operator for Semilinear Dispersive Equations

Carles

Gallagher

2008

Commun. Math. Phys.

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Abstract. We present a general algorithm to show that a scattering operator associated to a semilinear dispersive equation is real analytic, and to compute the coefficients of its Taylor series at any point. We illustrate this method in the case of the Schrödinger equation with powerlike nonlinearity or with Hartree type nonlinearity, and in the case of the wave and Klein-Gordon equations with power nonlinearity. Finally, we discuss the link of this approach with inverse scattering, and with complete integrability.

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Analyticity of the scattering operator for the nonlinear Klein-Gordon equation with cubic nonlinearity

Abstract: The wave and scattering operators for the equation with m > 0 and λ > 0 on four-dimensional Minkowski space are analytic on the space of finite-energy Cauchy data, i.e. L 2 (R 3 )®L 2 (R 3 ).

Cited by 3 publications

References 10 publications

Scattering and complete integrability in four dimensions

Scattering and complete integrability in four dimensions

On the real analyticity of the scattering operator for the Hartree equation

Analyticity of the Scattering Operator for Semilinear Dispersive Equations

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