Time-dependent scattering theory for Schrödinger operators on scattering manifolds

Abstract

We construct a time‐dependent scattering theory for Schrödinger operators on a manifold M with asymptotically conic structure. We use the two‐space scattering theory formalism, and a reference operator on a space of the form ℝ×∂M, where ∂M is the boundary of M at infinity. We prove the existence and the completeness of the wave operators, and show that our scattering matrix is equivalent to the absolute scattering matrix, which is defined in terms of the asymptotic expansion of generalized eigenfunctions. Our method is functional analytic, and we use no microlocal analysis in this paper.

Untitled

“…Then we havẽ A = W * − A 0 W − modulo S(h ∞ r −∞ ω −∞ , g 1 )-terms. Since there are no positive eigenvalues [Ito and Skibsted 2011;Melrose and Zworski 1996], we also have W ± f (P f )W * ± = f (P) by virtue of the intertwining property and asymptotic completeness [Ito and Nakamura 2010]. These imply…”

“…We recall the construction of the model introduced in [Ito and Nakamura 2010]. Let {ϕ α : U α → ‫ޒ‬ n−1 }, U α ⊂ ∂ M, be a local coordinate system of ∂ M. We take {φ α = I ⊗ ϕ α :Ũ α = ‫ޒ‬ + × U α → ‫ޒ‬ × ‫ޒ‬ n−1 } as the local coordinate system for M ∞ ∼ = ‫ޒ‬ + × ∂ M, and we use (r, θ ) ∈ ‫ޒ‬ × ‫ޒ‬ n−1 to represent a point in M ∞ .…”

“…It is known that P is essentially self-adjoint, that σ ess (P) = [0, ∞), and that P is absolutely continuous except on a countable discrete spectrum, the only possible accumulation point being 0 (see [Ito and Nakamura 2010] and references therein). We construct a time-dependent scattering theory for H as follows: We set…”

“…It is shown in [Ito and Nakamura 2010] that these operators exist and are complete in the following sense. Let Ᏺ be the Fourier transform in r , i.e.,…”

Geometric Scattering for Schrödinger Operators with Asymptotically Homogeneous Potentials of Order Zero

“…There are several formulation of geometric scattering and we employ the formulation of [7]. A key idea of the formulation in [7] is that they compare Schrödinger operators on the asymptotically conic manifolds with the simpler Schrödinger operators on the asymptotically tubic manifolds.…”