Lieb-Thirring Inequalities for the Pauli Operator in Three Dimensions

Abstract

Motion of a particle with spin in a magnetic field is described by the Pauli operator, that is by the operator

$$\begin{array}{*{20}{c}} {{\mathbb{P}_0} = {{\left( {\sum \cdot ( - i\nabla - a)} \right)}^2} = {{( - i\nabla - a)}^2}\mathbb{I}{\text{ - }}\sum \cdot {\text{B}},}&{\mathbb{I}{\text{ = }}\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right)} \end{array}$$

(1.1)

acting in L ²(ℝ³) ⊕ L ²(ℝ³). Here a = (α₁α₂α₃) is a vector-potential, B = (B ¹, B ², B ³) = rot a is the magnetic field and Σ is the vector of the 2 x 2 Pauli matrices σ₁, σ₂, σ₃ (see[3]). As seen from (1.1), the operator $\mathbb{D}_0$ is non-negative. If one perturbs it by a real-valued function V (electric potential) decreasing at infinity, then the resulting operator may have some negative discrete spectrum. The main goal of the paper is to establish Lieb-Thirring type estimates for the momenta

$$ M_\gamma = \sum\limits_k {\left| {\Lambda _k } \right|} ^\gamma ,\gamma > 0, $$

((1.2))

of the negative eigenvalues Λ^k of the operator $\mathbb{P} = \mathbb{P}_0 + V\mathbb{I}$. Analogous question was studied in [16] for the Pauli operator acting on L ²(ℝ²) ⊕ L ²(ℝ²) and the present paper can be regarded as a continuation of [16]. It is well-known that without any magnetic field S _γ, satisfies the following estimate ₁:

$$ M_\gamma \leqslant C_\gamma \int {V - \left( x \right)^{\gamma + \frac{3} {2}} dx,} $$

((1.3))

which is usually referred to as the Lieb-Thirring inequality if γ > 0 and the Rosenblum-Lieb-Cwickel inequality if γ = 0. Using the diamagnetic inequality (see [1]) one can extend this estimate to the spinless operator (-i∇ - a)² + V with a ≠ 0 as well.