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The distribution of negative eigenvalues of Schrödinger operators on asymptotically hyperbolic manifolds

Antônio Sá Barreto, Yiran Wang

TL;DR

This work analyzes the distribution of negative eigenvalues for Schrödinger operators $H=\Delta_g-\frac{n^2}{4}+V$ on asymptotically hyperbolic manifolds, focusing on when the discrete spectrum is finite versus infinite and the precise growth of $N_E(H)$ as $E\downarrow 0$. By combining Dirichlet--Neumann bracketing with Sturm oscillation theory and a reduction to model end-operators, the authors reduce the problem to a direct sum of 1D radial problems along the conformal boundary and derive sharp asymptotics for the counting function. Depending on the decay rate of $V$ near infinity (captured by $V_0$ and $V_1$ and their threshold variants), the paper proves finite spectra in subcritical cases and iterated-logarithmic growth in critical and supercritical cases, with explicit formulas such as $\log\log N_E(H)=\frac{1}{2-\delta}\log E^{-1}+O(1)$ or its iterated counterparts. These results extend spectral distribution analysis from Euclidean to hyperbolic-like geometries and provide a framework for understanding threshold phenomena in geometric scattering on AH manifolds.

Abstract

We study the asymptotic behavior of the counting function of negative eigenvalues of Schrödinger operators with real valued potentials on asymptotically hyperbolic manifolds. We establish conditions on the potential that determine if there are finitely or infinitely many negative eigenvalues. In the latter case, they may only accumulate at zero and we obtain the asymptotic behavior of the counting function of eigenvalues in an interval $(-\infty,-E)$ as $E\rightarrow 0$.

The distribution of negative eigenvalues of Schrödinger operators on asymptotically hyperbolic manifolds

TL;DR

This work analyzes the distribution of negative eigenvalues for Schrödinger operators on asymptotically hyperbolic manifolds, focusing on when the discrete spectrum is finite versus infinite and the precise growth of as . By combining Dirichlet--Neumann bracketing with Sturm oscillation theory and a reduction to model end-operators, the authors reduce the problem to a direct sum of 1D radial problems along the conformal boundary and derive sharp asymptotics for the counting function. Depending on the decay rate of near infinity (captured by and and their threshold variants), the paper proves finite spectra in subcritical cases and iterated-logarithmic growth in critical and supercritical cases, with explicit formulas such as or its iterated counterparts. These results extend spectral distribution analysis from Euclidean to hyperbolic-like geometries and provide a framework for understanding threshold phenomena in geometric scattering on AH manifolds.

Abstract

We study the asymptotic behavior of the counting function of negative eigenvalues of Schrödinger operators with real valued potentials on asymptotically hyperbolic manifolds. We establish conditions on the potential that determine if there are finitely or infinitely many negative eigenvalues. In the latter case, they may only accumulate at zero and we obtain the asymptotic behavior of the counting function of eigenvalues in an interval as .
Paper Structure (12 sections, 20 theorems, 269 equations)

This paper contains 12 sections, 20 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. The Strategy of the Proofs
  3. The Spectrum of $H$
  4. Dirichlet-Neumann Bracketing
  5. Model Operators on $X_\infty$
  6. The Asymptotic Behavior of $N_E({\mathcal{M}}_\pm^\bullet),$ $\bullet=D,N$ as $E\rightarrow 0$
  7. The Set up of the Problems
  8. Proof of Item T.\ref{['itT1']} of Proposition \ref{['asym-M']}
  9. The Proof of Item T.\ref{['itT2']} of Proposition \ref{['asym-M']}
  10. The Proof of Item T.\ref{['itT3']} of Proposition \ref{['asym-M']}
  11. The spectrum of ${\mathcal{M}}_j$
  12. The Proof of Proposition \ref{['lmzero']}

Key Result

Theorem 1.1

Suppose that in some interval $\rho\in(\rho_0,\infty),$ there exists a constant $c>0$ such that If $\delta<0,$ then $N_0(H)<\infty,$ but if $\delta\in (0,2),$ then $N_0(H)=\infty$ and

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • ...and 18 more