arXiv 1801.063982018-01math-phmath.MP

分数阶Pauli算子的Hardy-Lieb-Thirring不等式

Hardy-Lieb-Thirring Inequalities for Fractional Pauli Operators

Gonzalo A. Bley, Søren Fournais

DOI：: 10.1007/s00220-018-3204-y

摘要：

我们提供了下界的总和负特征值的运营商$|σ\cdot p_A|^{2s} - C_s/|x|^{2s} + V$在三维，其中$s\in (0, 1]$，涵盖有趣的物理情况$s = 1$和$s = 1/2$。这里$σ$是泡利矩阵的矢量，$p_A = p - A$，$p = -i\nabla$是三维动量算符，$A$是给定的磁矢势，$C_s$是临界哈代常数，即哈代不等式$|p|^{2s} \geq C_s/|x|^{2s}$中的最佳常数。如果忽略自旋，这种类型的结果在文献中称为Hardy-Lieb-Thirring不等式，对于正的常数$M_s$，该不等式将负特征值之和从下面限制为$-M_s\int V_{-}^{1 + 3/(2s)}$。在这种情况下，磁场的包含来自于非磁性的情况，即抗磁性。然而，旋转的增加带来了额外的挑战，使结果更加难以捉摸。这篇文章的目的是解决这个问题，通过提供简单的界限的总和负特征值的运营商的问题。特别地，对于$1/2 \leq s \leq 1$，我们能够纯粹地用磁场能量$\|B\|_2^2$和$V$的负部分的幂的积分来表示束缚。

英文摘要：

We provide lower bounds for the sum of the negative eigenvalues of the operator $|σ\cdot p_A|^{2s} - C_s/|x|^{2s} + V$ in three dimensions, where $s\in (0, 1]$, covering the interesting physical cases $s = 1$ and $s = 1/2$. Here $σ$ is the vector of Pauli matrices, $p_A = p - A$, with $p = -i\nabla$ the three-dimensional momentum operator and $A$ a given magnetic vector potential, and $C_s$ is the critical Hardy constant, that is, the optimal constant in the Hardy inequality $|p|^{2s} \geq C_s/|x|^{2s}$. If spin is neglected, results of this type are known in the literature as Hardy-Lieb-Thirring inequalities, which bound the sum of negative eigenvalues from below by $-M_s\int V_{-}^{1 + 3/(2s)}$, for a positive constant $M_s$. The inclusion of magnetic fields in this case follows from the non-magnetic case by diamagnetism. The addition of spin, however, offers extra challenges that make the result more elusive. It is the purpose of this article to resolve this problem by providing simple bounds for the sum of the negative eigenvalues of the operator in question. In particular, for $1/2 \leq s \leq 1$ we are able to express the bound purely in terms of the magnetic field energy $\|B\|_2^2$ and integrals of powers of the negative part of $V$.

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英文摘要：

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