HIGH ENERGY AND SMOOTHNESS ASYMPTOTIC EXPANSION OF THE SCATTERING AMPLITUDE D.Yafaev Department of Mathematics, University Rennes-1, Campus Beaulieu, 35042, Rennes, France (e-mail : [email protected]) Abstract We find an explicit expression for the kernel of the scattering matrix for the Schr¨odinger operator containing at high energies all terms of power order. It turns out that the same expression gives a complete description of the diagonal singular- ities of the kernel in the angular variables. The formula obtained is in some sense universal since it applies both to short- and long-range electric as well as magnetic potentials. 1. INTRODUCTION d−1 d−1 1. High energy asymptotics of the scattering matrix S(λ): L2(S ) → L2(S ) for the d Schr¨odinger operator H = −∆+V in the space H = L2(R ), d ≥ 2, with a real short-range potential (bounded and satisfying the condition V (x) = O(|x|−ρ), ρ > 1, as |x| → ∞) is given by the Born approximation. To describe it, let us introduce the operator Γ0(λ), −1/2 (d−2)/2 ˆ 1/2 d−1 (Γ0(λ)f)(ω) = 2 k f(kω), k = λ ∈ R+ = (0, ∞), ω ∈ S , (1.1) of the restriction (up to the numerical factor) of the Fourier transform fˆ of a function f to −1 −1 the sphere of radius k. Set R0(z) = (−∆ − z) , R(z) = (H − z) . By the Sobolev trace −r d−1 theorem and the limiting absorption principle the operators Γ0(λ)hxi : H → L2(S ) and hxi−rR(λ + i0)hxi−r : H → H are correctly defined as bounded operators for any r > 1/2 and their norms are estimated by λ−1/4 and λ−1/2, respectively. Therefore it is easy to deduce (see, e.g., [16, 27]) from the usual stationary representation ∗ S(λ) = I − 2πiΓ0(λ)(V − VR(λ + i0)V )Γ0(λ) (1.2) for the scattering matrix (SM) and the resolvent identity that N X n n ∗ S(λ) = I − 2πi (−1) Γ0(λ)V (R0(λ + i0)V ) Γ0(λ) + σN (λ), (1.3) n=0 −(N+2)/2 where ||σN (λ)|| = O(λ ) as λ → ∞. Moreover, the operators σN belong to suitable Schatten - von Neumann classes Sα(N) and α(N) → 0 as N → ∞. Nevertheless the Born expansion (1.3) has at least three drawbacks. First, the struc- ture of the nth term is extremely complicated already for relatively small n. Second, (1.3) definitely fails for long-range potentials, and, finally, it fails as λ → ∞ for a perturbation of the operator −∆ by first order differential operators even with short-range coefficients (magnetic potentials). 2. In the particular case when A = 0 and V belongs to the Schwartz class a conve- nient form of the high-energy expansion of the kernel of SM (called often the scattering amplitude) was obtained in [4] (see also the earlier paper [9]). The method of [4] relies on a preliminary study of the scattering solutions of the Schr¨odinger equation defined, for example, by the formula 2 ˆ d ψ±(ξ) = u0(ξ) − R(|ξ| ∓ i0)V u0(ξ), u0(x, ξ) = exp(ihx, ξi), ξ = ξ|ξ| ∈ R . It is shown in [4] that (at least on all compact sets of x) the function ψ±(x, ξ) has the ihx,ξi asymptotic expansion ψ±(x, ξ) = e b±(x, ξ) where N (N) X −n (±) b±(x, ξ) = b± (x, ξ) = (2i|ξ|) bn (x, ξ), b0(x, ξ) = 1,N → ∞. (1.4) n=0 The function b±(x, ξ) is determined by the transport equation (see subs. 2.3 below), and (±) (±) ˆ the coefficients bn (x, ξ) = bn (x, ξ) are quite explicit. Therefore it is easy to deduce from (1.2) that, for any N, the kernel of SM admits the asymptotic expansion s(ω, ω0; λ) = δ(ω, ω0) − πi(2π)−dkd−2 N Z × X(2ik)−n eikhω0−ω,xiV (x)b(−)(x, ω0)dx + O(kd−3−N ), (1.5) d n n=0 R where δ(·) is of course the Dirac-function on the unit sphere. We emphasize that the (−) 0 0 functions bn (x, ω ) are growing as |x| → ∞ in the direction of ω and the rate of growth increases as n increases. Thus, expansion (1.5) loses the sense (for sufficiently large N) if V (x) decreases only as some power of |x|−1. The generalization of the results of [4] to the case of short-range potentials V satis- α −ρv−|α| fying the condition ∂ V (x) = O(|x| ) for some ρv > 1 was suggested in [24] where the asymptotics of the scattering amplitude was also deduced from that of the scatter- ing solutions. We note finally the paper [3] where the leading term of the high-energy asymptotics of the scattering amplitude was found for short-range magnetic potentials. 3. In the present paper we suggest a new method which allows us to find an explicit 0 0 function s0(ω, ω ; λ) which describes with arbitrary accuracy the kernel s(ω, ω ; λ) of the SM S(λ) at high energies (as λ → ∞) both for short- and long-range electric and mag- 0 netic potentials. It turns out that the same function s0(ω, ω ; λ) gives also all diagonal 0 0 d−1 singularities of the kernel s(ω, ω ; λ) in the angular variables ω, ω ∈ S . We emphasize that our approach allows us to avoid a study of solutions of the Schr¨odingerequation. We consider the Schr¨odingeroperator H = (i∇ + A(x))2 + V (x) (1.6) 2 in the space H with electric V (x) and magnetic A(x) = (A1(x),...,Ad(x)) potentials satisfying the assumptions α −ρv−|α| ) |∂ V (x)| ≤ Cα(1 + |x|) , ρv > 0, α −ρa−|α| (1.7) |∂ A(x)| ≤ Cα(1 + |x|) , ρa > 0, for all multi-indices α. We suppose that potentials are real, that is V (x) = V (x) and Aj(x) = Aj(x), j = 1, . , d. Set ρ = min{ρv, ρa}, and 2 V0(x) = V (x) + |A(x)| ,V1(x) = V0(x) + idiv A(x). Then H = −∆ + 2ihA(x), ∇i + V1(x). (1.8) We emphasize that the cases ρ > 1 (short-range potentials) and ρ ∈ (0, 1] (long-range potentials) are treated in almost the same way. Let us formulate our main result. The answer is given in terms of approximate solutions of the Schr¨odinger equation 2 −∆ψ(x, ξ) + 2ihA(x), ∇iψ(x, ξ) + V1(x)ψ(x, ξ) = |ξ| ψ(x, ξ). (1.9) (N) To be more precise, we denote by u±(x, ξ) = u± (x, ξ) explicit functions (see Section 2, for their construction) iΘ±(x,ξ) u±(x, ξ) = e b±(x, ξ) (1.10) such that 2 iΘ±(x,ξ) (−∆ + 2ihA(x), ∇i + V1(x) − |ξ| )u±(x, ξ) = e r±(x, ξ) =: q±(x, ξ) (1.11) (N) −p −q and r±(x, ξ) = r± (x, ξ) tends to zero faster than |x| as |x| → ∞ and |ξ| as |ξ| → ∞ where p = p(N) → ∞ and q = q(N) → ∞ as N → ∞ off any conical neighborhood of the ˆ directionx ˆ = ∓ξ. Note that the phase Θ±(x, ξ) = hx, ξi if A(x) = 0 and V (x) is a short- range function and Θ±(x, ξ) satisfies approximately the eikonal equation in the general case. The function b±(x, ξ) is obtained as an approximate solution of the corresponding transport equation. As is well known (see [1]), off the diagonal ω = ω0, the kernel s(ω, ω0; λ) is a C∞- 0 d−1 −1 function of ω, ω ∈ S where it tends to zero faster than any power of λ as λ → ∞. Thus, it suffices to describe the structure of s(ω, ω0; λ) in a neighborhood of the diagonal 0 d−1 ω = ω . Let ω0 ∈ S be an arbitrary point, Πω0 be the plane orthogonal to ω0 and d−1 Ω±(ω0, δ) ⊂ S be determined by the condition ±hω, ω0i > δ > 0. Set x = ω0z + y, y ∈ Πω0 , (1.12) and 0 (N) 0 d−2 −d s0(ω, ω ; λ) = s0 (ω, ω ; λ) = ∓πik (2π) Z 0 0 × u+(y, kω)(∂zu−)(y, kω ) − u−(y, kω )(∂zu+)(y, kω) dy Πω0 Z 0 −2i hA(y), ω0iu+(y, kω)u−(y, kω )dy (1.13) Πω0 3 0 for ω, ω ∈ Ω± = Ω±(ω0, δ). Then, for any p, q and sufficiently large N = N(p, q), the kernel (N) 0 0 (N) 0 s˜ (ω, ω ; λ) = s(ω, ω ; λ) − s0 (ω, ω ; λ) (1.14) p p −q belongs to the class C (Ω×Ω) where Ω = Ω+ ∪Ω−, and its C -norm is O(λ ) as λ → ∞. Thus, all singularities of s(ω, ω0; λ) both for high energies and in smoothness are described 0 by the explicit formula (1.13). Let S0(λ) be integral operator with kernel s0(ω, ω ; λ). In view of representation (1.10), formula (1.13) shows that we actually consider the singular part S0(λ) of the SM as a Fourier integral or, more precisely, a pseudo-differential operator (PDO) acting on the unit sphere and determined by its amplitude. By our construction of functions (1.10), u+(x, ξ) = u−(x, −ξ) if A(x) = 0. Therefore 0 in the case A = 0 the singular part s0(ω, ω ; λ) satisfies the same symmetry relation (the time reversal invariance) s(ω, ω0; λ) = s(−ω0, −ω; λ) as kernel of the SM itself. Kernel (1.13) is also gauge invariant.