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Low energy resolvent estimates for slowly decaying attractive potentials

Kenichi Ito, Tomoya Tagawa

TL;DR

The paper develops uniform low-energy resolvent estimates for Schrödinger operators with slowly decaying attractive potentials by employing the Ito–Skibsted commutator method, avoiding microlocal machinery. It proves Rellich’s theorem, the limiting absorption principle, radiation condition bounds, and Sommerfeld’s uniqueness under $C^2$ regularity and spherical symmetry, with a short-range perturbation, via Agmon– Hörmander spaces. A central theme is the construction of an effective time via the eikonal phase $S(\lambda,x)$ and the associated distorted commutator estimates, which yield a priori decay, absence of positive eigenvalues, and robust LAP bounds at low energy. The results have implications for spectral and scattering theory in slowly decaying regimes and demonstrate a simpler route to sharp low-energy estimates without microlocal tools.

Abstract

We discuss the low energy resolvent estimates for the Schrödinger operator with slowly decaying attractive potential. The main results are Rellich's theorem, the limiting absorption principle and Sommerfeld's uniqueness theorem. For the proofs we employ an elementary commutator method due to Ito--Skibsted, for which neither of microlocal or functional-analytic techniques is required.

Low energy resolvent estimates for slowly decaying attractive potentials

TL;DR

The paper develops uniform low-energy resolvent estimates for Schrödinger operators with slowly decaying attractive potentials by employing the Ito–Skibsted commutator method, avoiding microlocal machinery. It proves Rellich’s theorem, the limiting absorption principle, radiation condition bounds, and Sommerfeld’s uniqueness under regularity and spherical symmetry, with a short-range perturbation, via Agmon– Hörmander spaces. A central theme is the construction of an effective time via the eikonal phase and the associated distorted commutator estimates, which yield a priori decay, absence of positive eigenvalues, and robust LAP bounds at low energy. The results have implications for spectral and scattering theory in slowly decaying regimes and demonstrate a simpler route to sharp low-energy estimates without microlocal tools.

Abstract

We discuss the low energy resolvent estimates for the Schrödinger operator with slowly decaying attractive potential. The main results are Rellich's theorem, the limiting absorption principle and Sommerfeld's uniqueness theorem. For the proofs we employ an elementary commutator method due to Ito--Skibsted, for which neither of microlocal or functional-analytic techniques is required.
Paper Structure (29 sections, 23 theorems, 186 equations)

This paper contains 29 sections, 23 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Basic settings
  3. Slowly decaying attractive potential
  4. Agmon-Hörmander spaces
  5. Main results
  6. Rellich's theorem
  7. LAP bounds
  8. Radiation condition bounds and applications
  9. Classical mechanics
  10. Effective time
  11. Rellich's theorem
  12. Main propositions
  13. A priori super-exponential decay estimate
  14. Absence of super-exponentially decaying eigenfunction
  15. LAP bounds
  16. ...and 14 more sections

Key Result

Theorem 1.3

Suppose Assumption cond:230806. If $\lambda\ge 0$ and $\phi\in \mathcal{B}^*_0(\lambda)$ satisfy then $\phi\equiv 0$. In particular, the self-adjoint realization of $H$ on $\mathcal{H}$ does not have non-negative eigenvalues: $\sigma_{\mathrm{pp}}(H)\cap [0,\infty)=\emptyset$.

Theorems & Definitions (49)

  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Corollary 1.10
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • ...and 39 more