Scattering for Schrödinger operators with conical decay

TL;DR

We address scattering for Schrödinger operators $H=H_0+V$ with potentials decaying inside a collection of cones, extending classical short-range scattering to geometries with anisotropic decay. Using a phase-space POVM $P_\delta$ and Enss-type arguments, we prove the existence of the wave operator $\Omega$ on a dense domain and identify $\operatorname{Ran}(\Omega)=\overline{\mathcal{H}_{\text{scat}}}$ with the interacting complement $\operatorname{Ran}(\Omega)^{\perp}=\mathcal{H}_{\text{int}}$, together with microlocal descriptions and, for wide cones, purely spatial characterizations. The framework relies on a detailed phase-space analysis of outgoing/incoming regions $\mathcal{W}_{n;\mathrm{out}}$ and $\mathcal{W}_{n,m;\mathrm{out}}$, and a generalized Enss condition $\|\chi_{\mathcal{A}_r}V\|_{op}\in L^1([0,\infty),dr)$. These results connect the geometry of the potential with the asymptotic dynamics, enabling spatial interpretations in broad cone geometries and providing a foundation for geometrically induced scattering phenomena and bound states in non-isotropic settings.

Abstract

We study the scattering properties of Schrödinger operators with potentials that have short-range decay along a collection of rays in $\bbR^d$. This generalizes the classical setting of short-range scattering in which the potential is assumed to decay along \emph{all} rays. For these operators, we show that any state decomposes into an asymptotically free piece and a piece which may interact with the potential for long times. We give a microlocal characterization of the scattering states in terms of the dynamics and a corresponding description of their complement. We also show that in certain cases these characterizations can be purely spatial.

Figures (6)

• Figure 1: Illustration of $\mathcal{C}_{x,\vec{v},\gamma}$ and $A_{r}(\mathcal{C}_{x,\vec{v},\gamma})$ for $d=2$: in orange we have the complement of $\mathcal{C}_{x,\vec{v},\gamma}$, which is where the potential is concentrated. In black we have the set $A_{r}(\mathcal{C}_{x,\vec{v},\gamma})$, in red we indicate $\vec{v}$ and $\gamma$.
• Figure 2: The geometry of the broken subspace: in orange we have the vectors $v_1,v_2$, in red we have the vectors $\vec{v}_*$ and $-\vec{v}_*$, in blue we have the outline of $T_r$- which contains $\mathop{\mathrm{supp}}\nolimits V$.
• Figure 3: Illustration of the momentum of $\hat{\psi}$, in red, with respect to $A_n(\mathcal{C})$, in orange. The dashed blue line corresponds to a classic trajectory from $x$ with momentum at the edge of the red cone.
• Figure 4: Illustration of the inclusion (\ref{['ConeInclsion']})
• Figure 5: Illustration of the phase space sets $W_{n;\textrm{out}}(\mathcal{C}_i)$ and $W_{n,m;\textrm{out}}(\mathcal{C}_i)$: space coordinates are inside the black cone while momentum coordinates point inside the red/blue cone, respectively.

Theorems & Definitions (44)

• Example 2.1: Single cone
• Example 2.2: Short-range scattering
• Example 2.3: Subspace potentials
• Example 2.4: Broken subspace
• Remark 3.1
• Theorem 3.2
• Theorem 3.3
• Proposition 4.1
• proof
• Lemma 4.2