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Dispersive estimates for Schrödinger operators with negative Coulomb-like potentials in one dimension

Akitoshi Hoshiya, Kouichi Taira

Abstract

In this paper, we consider the dispersive estimates for Schrödinger operators with Coulomb-like decaying potentials, such as $V(x)=-c|x|^{-μ}$ for $|x|\gg 1$ with $0<μ<2$, in one dimension. As an application, we establish both the standard and orthonormal Strichartz estimates for this model. One of the difficulties here is that perturbation arguments, which are typically applicable to rapidly decaying potentials, are not available. To overcome this, we derive a WKB expression for the spectral density and use a variant of the degenerate stationary phase formula to exploit its oscillatory behavior in the low-energy regime.

Dispersive estimates for Schrödinger operators with negative Coulomb-like potentials in one dimension

Abstract

In this paper, we consider the dispersive estimates for Schrödinger operators with Coulomb-like decaying potentials, such as for with , in one dimension. As an application, we establish both the standard and orthonormal Strichartz estimates for this model. One of the difficulties here is that perturbation arguments, which are typically applicable to rapidly decaying potentials, are not available. To overcome this, we derive a WKB expression for the spectral density and use a variant of the degenerate stationary phase formula to exploit its oscillatory behavior in the low-energy regime.

Paper Structure

This paper contains 23 sections, 23 theorems, 234 equations.

Table of Contents

  1. Introduction and main results
  2. Background
  3. Main result
  4. Idea of the proof
  5. Further observation
  6. Notations
  7. Preliminary: Quantitative estimates of oscillatory integrals
  8. Partition of unity
  9. Estimates for oscillatory integrals
  10. Asymptotic expansions of the spectral projection
  11. ODEs with non-oscillating solutions
  12. ODEs with simple oscillating solutions
  13. Construction of the Jost functions
  14. WKB expressions of the outgoing resolvent and the spectral projection
  15. Estimates of oscillatory integrals associated with the spectral projection
  16. ...and 8 more sections

Key Result

Theorem 1.3

Under Assumption assump:V, the dispersive estimate eq:dispersive holds (for $d=1$).

Theorems & Definitions (59)

  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Lemma 2.1: Partition of unity
  • Lemma 2.2: Quantitative stationary phase
  • Remark 2.3
  • proof
  • Lemma 2.4: Quantitative degenerate stationary phase
  • Remark 2.5
  • ...and 49 more