[Paper Review] Remarks on scattering matrices for Schrödinger operators with critically long-range perturbations

Shu Nakamura|arXiv (Cornell University)|Apr 16, 2018

Spectral Theory in Mathematical Physics15 references4 citations

TL;DR

This paper establishes that the scattering matrix for Schrödinger operators with critically long-range potentials (V(x) = O(|x|^{-1})) is a pseudodifferential operator on the sphere S^{d-1}, computes its principal symbol explicitly, and analyzes its spectral properties. The key contribution is proving the scattering matrix has dense pure point spectrum under rotation symmetry and absolutely continuous spectrum in specific 2D examples, resolving a gap in prior work on μ=1 long-range scattering.

ABSTRACT

We consider scattering matrix for Schrödinger-type operators on $\mathbb{R}^d$ with perturbation $V(x)=O(\langle x angle^{-1})$ as $|x| o\infty$. We show that the scattering matrix (with time-independent modifiers) is a pseudodifferential operator, and analyze its spectrum. We present examples of which the spectrum of the scattering matrices have dense point spectrum, and absolutely continuous spectrum, respectively.

Motivation & Objective

• To establish the scattering matrix as a pseudodifferential operator for Schrödinger operators with critically long-range perturbations (μ=1).
• To compute the principal symbol of the scattering matrix in terms of the potential's integral along classical trajectories.
• To analyze the spectral properties of the scattering matrix, particularly the presence of dense pure point and absolutely continuous spectrum.
• To fill a gap in the literature by providing a refined analysis for the critical case μ=1, where previous results were incomplete.

Proposed method

• Use of time-independent modifiers to define the scattering matrix in the context of long-range perturbations.
• Application of pseudodifferential operator calculus to show S(λ) is a pseudodifferential operator on S^{d-1} with a specific symbol expansion.
• Derivation of the principal symbol via the integral of the potential along straight-line trajectories parametrized by energy λ and direction ξ.
• Employment of functional calculus for unitary pseudodifferential operators to analyze spectral properties.
• Use of asymptotic expansions and commutator estimates to construct a conjugate operator that approximates the scattering evolution.
• Application of the Kuroda-Birman theorem for unitary operators to relate trace class perturbations to spectral invariance of the absolutely continuous spectrum.

Experimental results

Research questions

• RQ1 Is the scattering matrix for Schrödinger operators with μ=1 long-range potentials a pseudodifferential operator on the sphere S^{d-1}?
• RQ2 What is the explicit form of the principal symbol of the scattering matrix in this critical case?
• RQ3 Under what conditions does the scattering matrix exhibit dense pure point spectrum?
• RQ4 Can the absolutely continuous spectrum of the scattering matrix be characterized in two dimensions for specific potentials?
• RQ5 How do spectral properties of the scattering matrix behave under small lower-order perturbations?

Key findings

• The scattering matrix S(λ) is a pseudodifferential operator on S^{d-1} with principal symbol s₀(λ,x,ξ) = exp(−i(2λ)^{-1/2} ∫_{-∞}^{∞} (V(x+tξ)−V(tξ)) dt).
• The principal symbol s₀(λ,x,ξ) belongs to the symbol class S^δ_{1,0}(T^*S^{d-1}) for any δ>0, and the difference S(λ)−s₀ is in S^{-1+δ}_{1,0}.
• Under rotation symmetry and the condition |x·∂ₓV(x)| ≥ c|x|^{-1} for large |x|, the scattering matrix has dense pure point spectrum on the entire unit circle.
• For d=2 and V(x) = a x₁ / ⟨x⟩² with a≠0, the essential spectrum of S(λ) is the arc {e^{iθ} : |θ| ≤ |a|π(2λ)^{-1/2}}, with absolutely continuous spectrum on the complement, except possibly at endpoints.
• The absolutely continuous spectrum is stable under small lower-order perturbations, and eigenvalues can accumulate only at the endpoints e^{±iπa(2λ)^{-1/2}}.
• The spectral invariance of the absolutely continuous spectrum under trace class perturbations is established via the unitary version of the Kuroda-Birman theorem.