The Spectral Density of the Scattering Matrix of the Magnetic Schrödinger Operator for High Energies

Abstract

The scattering matrix of the Schrödinger operator with smooth short-range electric and magnetic potentials is considered. The asymptotic density of the eigenvalues of this scattering matrix in the high energy regime is determined. An explicit formula for this density is given. This formula involves only the magnetic vector-potential. 1. Main result and discussion 1.1. Main result. Consider the Schrödinger operator H = (i∇+ A) + V in L(R), d ≥ 2; here V : R → R is an electric potential and A = (A1, . . . , Ad) : R → R is a magnetic vector-potential. We assume that both V and A are infinitely differentiable and satisfy the estimates (1.1) |∂V (x)| ≤ Cα〈x〉, |∂A(x)| ≤ Cα〈x〉, ρ > 1, for all multi-indices α; here 〈x〉 = (1 + |x|). Let H0 = −∆; we denote by S(k) the scattering matrix associated with the pair H,H0 at the energy λ = k 2 > 0. We recall the precise definition of the scattering matrix in Section 2.1; here we only note that S(k) is a unitary operator in L2(Sd−1) and the operator S(k)− I is compact (see e.g. [19, 18]). Thus, the spectrum of S(k) consists of eigenvalues on the unit circle T; all eigenvalues (apart from possibly 1) have finite multiplicities and can accumulate only to 1. Our aim is to describe the asymptotic behaviour of these eigenvalues as k →∞. If A ≡ 0, one has (1.2) ‖S(k)− I‖ = O(k−1), k →∞ (A ≡ 0), and so the eigenvalues of S(k) are located on an arc near 1 of length O(k−1). The large energy asymptotics of S(k) in this case is given by the Born approximation, see e.g. [19, Chapter 8]; this makes the analysis of S(k) rather explicit. In [6], using the Born approximation, we have determined the large energy asymptotic density of the spectrum of S(k) for A ≡ 0; we will say more about this in the next subsection. When A 6≡ 0, the situation is radically different: as k →∞, the norm ‖S(k)− I‖ does not tend to zero and the Born approximation is no longer valid. The high energy asymptotic expansion of the scattering amplitude (= the integral kernel of S(k) − I) for the magnetic Schrödinger operator was obtained (through a very difficult proof) in [17]. Our main result below is merely a spectral consequence of [17]. 2000 Mathematics Subject Classification. Primary: 81U20; Secondary: 35P25.

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