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Cwikel-Lieb-Rozenblum type estimates for the Pauli and magnetic Schrödinger operator in dimension two

Matthias Baur, Hynek Kovarik

TL;DR

This work establishes Cwikel-Lieb-Rozenblum type bounds for the number of negative eigenvalues of two-dimensional Pauli and magnetic Schrödinger operators, capturing both weak and strong coupling regimes. The main novelty is the explicit α-dependent counting bound that separates the influence of spin-orbit coupling (through the flux α) and the presence of the spinor structure, producing a finite offset m(α) in the weak coupling limit and sharp, weight-informed bounds for V in various L^p-based norms with local or global logarithmic corrections. The authors also derive strong-coupling asymptotics and long-range potential refinements, including a framework that accommodates slowly decaying potentials via a new [V]_a functional. A corollary for magnetic Schrödinger operators follows from Pauli bounds, underlining the role of spin-orbit coupling and magnetic flux in shaping the spectral counting. Overall, the paper advances CLR-type theory in 2D by integrating spin, magnetic effects, and long-range potentials within a unified, sharp analytic framework.

Abstract

We prove a Cwikel-Lieb-Rozenblum type inequality for the number of negative eigenvalues of Pauli operators in dimension two. The resulting upper bound is sharp both in the weak as well as in the strong coupling limit. We also derive different upper bounds for magnetic Schrödinger operators. The nature of the two estimates depends on whether or not the spin-orbit coupling is taken into account.

Cwikel-Lieb-Rozenblum type estimates for the Pauli and magnetic Schrödinger operator in dimension two

TL;DR

This work establishes Cwikel-Lieb-Rozenblum type bounds for the number of negative eigenvalues of two-dimensional Pauli and magnetic Schrödinger operators, capturing both weak and strong coupling regimes. The main novelty is the explicit α-dependent counting bound that separates the influence of spin-orbit coupling (through the flux α) and the presence of the spinor structure, producing a finite offset m(α) in the weak coupling limit and sharp, weight-informed bounds for V in various L^p-based norms with local or global logarithmic corrections. The authors also derive strong-coupling asymptotics and long-range potential refinements, including a framework that accommodates slowly decaying potentials via a new [V]_a functional. A corollary for magnetic Schrödinger operators follows from Pauli bounds, underlining the role of spin-orbit coupling and magnetic flux in shaping the spectral counting. Overall, the paper advances CLR-type theory in 2D by integrating spin, magnetic effects, and long-range potentials within a unified, sharp analytic framework.

Abstract

We prove a Cwikel-Lieb-Rozenblum type inequality for the number of negative eigenvalues of Pauli operators in dimension two. The resulting upper bound is sharp both in the weak as well as in the strong coupling limit. We also derive different upper bounds for magnetic Schrödinger operators. The nature of the two estimates depends on whether or not the spin-orbit coupling is taken into account.
Paper Structure (15 sections, 27 theorems, 231 equations)

This paper contains 15 sections, 27 theorems, 231 equations.

Table of Contents

  1. Introduction and main results
  2. Motivation
  3. Assumptions and main results
  4. Discussion
  5. Related results
  6. Notation
  7. Strategy of the proof
  8. Preliminaries
  9. Upper bound on $N( H_{-} - V)$
  10. Upper bound on $N( H_{+} - V)$
  11. Strong coupling asymptotic
  12. Long range potentials
  13. Magnetic Schrödinger operators
  14. Positive definiteness of kernels
  15. One-dimensional weighted Hardy inequalities

Key Result

Theorem 1.1

Let $B$ satisfy Assumption ass-B, and recall that $\alpha$ is given by flux.

Theorems & Definitions (54)

  • Theorem 1.1: Pauli operators
  • Corollary 1.2: Radial potentials
  • Corollary 1.3: magnetic Schrödinger operators
  • Remark 1.4: Strong coupling
  • Remark 1.5: Condition $p>1$
  • Remark 1.6: Weak coupling
  • Remark 1.7: Long range potentials
  • Remark 1.8: Condition $\alpha\neq 0$
  • Theorem 1.9: Laptev-Netrusov
  • Proposition 3.1
  • ...and 44 more