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A second microlocalization for the three-body calculus

Yilin Ma

TL;DR

This work develops a rigorous second microlocal framework for the quantum three-body problem by introducing the conormal three-cone algebra and its second microlocalization. It blends the conormal b-calculus, scattering calculus, and Vasy’s second microlocal calculus with a new fibered, three-cone structure to resolve diffraction and decoupled decay indices in the three-body Helmholtz operator $P=\Delta+V-\lambda^2$. The key contributions include constructing the three-cone algebra, connecting it to the three-body and scattering formalisms via iterated blow-ups, and establishing a robust symbol calculus with indicial operators, along with a further-resolved, variable-order second microlocalized calculus that yields Fredholm-type estimates. These developments provide a precise analytic framework for diffraction phenomena in three-body quantum systems and set the stage for sharpened Fredholm theory and spectral analysis in anisotropic Sobolev spaces. The innovations have broad impact for microlocal scattering theory, enabling rigorous treatment of decoupled decay directions and high-frequency limits in multi-particle quantum problems.

Abstract

This paper is the first in a series studying the quantum mechanical three-body problem within a modern microlocal framework. The project aims to demonstrate how the Helmholtz operator is a Fredholm map between suitable anisotropic Hilbert spaces. Notably, we will consider decoupled indices that measure decay at the spatial infinity. Despite the problem's rich history, new phenomena arise under this approach, particularly regarding diffraction. The treatment of this involves the second microlocalization framework introduced by Vasy for the two-body Helmholtz operator, which does not directly extend to the three-body case. This paper will clarify this structure. We will construct an algebra of operators, referred to as the three-cone algebra, which serves as a 'converse perspective' to the second microlocalization. This algebra exhibits a scattering structure at one boundary face and a specific fibered structure (corresponding to the scattering-fibered structure for the three-body algebra) at another, which are connected by a fibered cone. We show that introducing suitable blow-ups at the fiber infinity allows us to modify the three-cone algebra only at the symbolic level, thereby constructing the desired second microlocalized algebra. The presence of variable orders will also be considered. Finally, we prove basic properties for the second microlocalized algebra.

A second microlocalization for the three-body calculus

TL;DR

This work develops a rigorous second microlocal framework for the quantum three-body problem by introducing the conormal three-cone algebra and its second microlocalization. It blends the conormal b-calculus, scattering calculus, and Vasy’s second microlocal calculus with a new fibered, three-cone structure to resolve diffraction and decoupled decay indices in the three-body Helmholtz operator . The key contributions include constructing the three-cone algebra, connecting it to the three-body and scattering formalisms via iterated blow-ups, and establishing a robust symbol calculus with indicial operators, along with a further-resolved, variable-order second microlocalized calculus that yields Fredholm-type estimates. These developments provide a precise analytic framework for diffraction phenomena in three-body quantum systems and set the stage for sharpened Fredholm theory and spectral analysis in anisotropic Sobolev spaces. The innovations have broad impact for microlocal scattering theory, enabling rigorous treatment of decoupled decay directions and high-frequency limits in multi-particle quantum problems.

Abstract

This paper is the first in a series studying the quantum mechanical three-body problem within a modern microlocal framework. The project aims to demonstrate how the Helmholtz operator is a Fredholm map between suitable anisotropic Hilbert spaces. Notably, we will consider decoupled indices that measure decay at the spatial infinity. Despite the problem's rich history, new phenomena arise under this approach, particularly regarding diffraction. The treatment of this involves the second microlocalization framework introduced by Vasy for the two-body Helmholtz operator, which does not directly extend to the three-body case. This paper will clarify this structure. We will construct an algebra of operators, referred to as the three-cone algebra, which serves as a 'converse perspective' to the second microlocalization. This algebra exhibits a scattering structure at one boundary face and a specific fibered structure (corresponding to the scattering-fibered structure for the three-body algebra) at another, which are connected by a fibered cone. We show that introducing suitable blow-ups at the fiber infinity allows us to modify the three-cone algebra only at the symbolic level, thereby constructing the desired second microlocalized algebra. The presence of variable orders will also be considered. Finally, we prove basic properties for the second microlocalized algebra.

Paper Structure

This paper contains 39 sections, 29 theorems, 620 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The three-body problem
  3. Review of the two-body problem
  4. Motivating the second microlocalization
  5. Overview of second microlocalization
  6. Summary of main innovations
  7. Acknowledgements
  8. Microlocal preliminaries
  9. The conormal b-calculus
  10. The scattering calculus
  11. The three-body calculus
  12. Vasy's second microlocal calculus
  13. The conormal cone calculus
  14. Large-parameters families of operators
  15. The conormal three-cone operators
  16. ...and 24 more sections

Key Result

Theorem 1.1

For every $m,r,l,b \in \mathbb{R}$, we can construct a space of operators acting on $[ \overline{\mathbb{R}^{n}} ; \mathcal{C}_{\alpha} ]$, such that there exists a well-behaved principal symbol map which descends to an isomorphism Moreover, (main theorem algebra) is closed under composition and taking adjoint, i.e., In particular, we can realize as a filtered algebra. It is in fact a calculu

Figures (5)

  • Figure 1: A sketch of $\Sigma_{\mathrm{sc}}(\lambda)$ in the momentum space and how the bicharacteristic flow behaves there. The circle has radius $\lambda$. Forward bicharacterstics travel from one radial set $\mathcal{R}_{+}(\lambda)$ to another $\mathcal{R}_{-}(\lambda)$. The values of the variable orders $\mathsf{r}_{\pm}$ are restricted at these sets.
  • Figure 2: An image of $\overline{\mathbb{R}^{3}}$ in the case where $\mathcal{C}_{\alpha}$ has dimension 1, i.e., a great circle. Only the upper hemisphere of $\overline{\mathbb{R}^{3}}$ is shown here. The curve $\gamma$ intersects $\mathcal{C}_{\alpha}$ at the radial set $\mathcal{R}_{\mathrm{n},+}$.
  • Figure 3: An image of $[ \overline{\mathbb{R}^{3}} ; \mathcal{C}_{\alpha} ]$ in the case where $\mathcal{C}_{\alpha}$ has dimension 0, i.e., just a point. Microlocal regularities propagate along $\gamma_{\pm}$, and upon hitting $\mathcal{C}_{\alpha}$, propagate to the 'other side' of $\mathcal{C}_{\alpha}$ by traversing around $\partial \mathrm{ff}_{\alpha}$ via curves like $\gamma^{\alpha}$
  • Figure 4: A sketch of the upper half of $[ \overline{\mathbb{R}^{2}} ; \mathcal{C}_{\alpha} ]$ when $\mathcal{C}_{\alpha}$ is just a point.
  • Figure 5: A sketch of $[ \overline{\mathbb{R}^{2}} ; \mathcal{C}_{\alpha} ; \mathrm{mf} \cap \mathrm{ff}_{\alpha} ]$ near the lift of $\mathcal{C}_{\alpha}$ when $\mathcal{C}_{\alpha}$ is a point. The three-cone structure connects the microlocal structures at $\mathrm{dmf}$ and $\mathrm{dff}_{\alpha}$ through $\mathrm{cf}_{\alpha}$.

Theorems & Definitions (90)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.1
  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • ...and 80 more