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Low energy resolvent asymptotics of the multipole Aharonov--Bohm Hamiltonian

T. J. Christiansen, K. Datchev, M. Yang

TL;DR

This work derives low-energy resolvent expansions for the Aharonov–Bohm Hamiltonian with multiple poles, revealing that the total flux $\beta=\sum_k \alpha_k$ dictates distinct analytic structures. When $\beta$ is an integer, the operator is conjugate to a compact perturbation of the Laplacian, yielding a three-term resolvent expansion with logarithmic terms; for non-integer $\beta$, a model resolvent $P_{\beta}$ governs the near-zero behavior, and the resolvent exhibits a mixed expansion with powers determined by $|\beta|$ and $1-|\beta|$. The resolvent has meromorphic continuation to the logarithmic cover $\Lambda$ (and to $\Lambda_q$ for rational $\beta$), and these expansions underpin the long-time wave asymptotics, showing an interpolation between even- and odd-dimensional scattering depending on $\beta$. The results extend the low-energy scattering theory to multi-pole Aharonov–Bohm systems and provide explicit leading-order terms and structural insights into the zero-energy regime. This has implications for understanding wave propagation and decay in magnetic systems with multiple topological singularities.

Abstract

We compute low energy asymptotics for the resolvent of the Aharonov--Bohm Hamiltonian with multiple poles for both integer and non-integer total fluxes. For integral total flux we reduce to prior results in black-box scattering while for non-integral total flux we build on the corresponding techniques using an appropriately chosen model resolvent. The resolvent expansion can be used to obtain long-time wave asymptotics for the Aharonov--Bohm Hamiltonian with multiple poles. An interesting phenomenon is that if the total flux is an integer then the scattering resembles even-dimensional Euclidean scattering, while if it is half an odd integer then it resembles odd-dimensional Euclidean scattering. The behavior for other values of total flux thus provides an `interpolation' between these.

Low energy resolvent asymptotics of the multipole Aharonov--Bohm Hamiltonian

TL;DR

This work derives low-energy resolvent expansions for the Aharonov–Bohm Hamiltonian with multiple poles, revealing that the total flux dictates distinct analytic structures. When is an integer, the operator is conjugate to a compact perturbation of the Laplacian, yielding a three-term resolvent expansion with logarithmic terms; for non-integer , a model resolvent governs the near-zero behavior, and the resolvent exhibits a mixed expansion with powers determined by and . The resolvent has meromorphic continuation to the logarithmic cover (and to for rational ), and these expansions underpin the long-time wave asymptotics, showing an interpolation between even- and odd-dimensional scattering depending on . The results extend the low-energy scattering theory to multi-pole Aharonov–Bohm systems and provide explicit leading-order terms and structural insights into the zero-energy regime. This has implications for understanding wave propagation and decay in magnetic systems with multiple topological singularities.

Abstract

We compute low energy asymptotics for the resolvent of the Aharonov--Bohm Hamiltonian with multiple poles for both integer and non-integer total fluxes. For integral total flux we reduce to prior results in black-box scattering while for non-integral total flux we build on the corresponding techniques using an appropriately chosen model resolvent. The resolvent expansion can be used to obtain long-time wave asymptotics for the Aharonov--Bohm Hamiltonian with multiple poles. An interesting phenomenon is that if the total flux is an integer then the scattering resembles even-dimensional Euclidean scattering, while if it is half an odd integer then it resembles odd-dimensional Euclidean scattering. The behavior for other values of total flux thus provides an `interpolation' between these.
Paper Structure (21 sections, 14 theorems, 69 equations, 2 figures)

This paper contains 21 sections, 14 theorems, 69 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Wave asymptotics
  3. Low energy resolvent expansions
  4. Resolvent for integer total flux
  5. Resolvent for non-integer total flux
  6. Meromorphic continuation of the resolvent
  7. Background and context
  8. Plan of the paper
  9. Preliminaries
  10. Operators and domains
  11. Conjugated operators
  12. Integer flux
  13. Non-integer flux
  14. Resolvent expansion for integer total flux
  15. Proof of Theorem \ref{['t:resint']}
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\chi \in C_c^\infty(\mathbb{R}^2)$, and $f_1,\;u$ be as in eq:waveIntro. Suppose no three elements of $S$ are colinear. Then, as $t\rightarrow \infty$,

Figures (2)

  • Figure 1: In this example, $\tilde{\alpha} = \alpha_j$ on $\Gamma_j$ for $j = 2$ and $j=5$, $\tilde{\alpha} = \alpha_3 + \alpha_4$ on $\Gamma_3$, and $\tilde{\alpha} = \alpha_4$ on $\Gamma_4 \setminus \Gamma_3$.
  • Figure 2: Examples for which we can compute $G$ and $C_{\vec{A}}$.

Theorems & Definitions (24)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • ...and 14 more