We consider the Landau Hamiltonian $H_0$, self-adjoint in $L^2({\mathbb R^2})$, whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues $\Lambda_q$, $q \in {\mathbb Z}_+$. We perturb $H_0$ by a non-local potential written as a bounded pseudo-differential operator ${\rm Op}^{\rm w}({\mathcal V})$ with real-valued Weyl symbol ${\mathcal V}$, such that ${\rm Op}^{\rm w}({\mathcal V}) H_0^{-1}$ is compact. We study the spectral properties of the perturbed operator $H_{\mathcal V} = H_0 + {\rm Op}^{\rm w}({\mathcal V})$. First, we construct symbols ${\mathcal V}$, possessing a suitable symmetry, such that the operator $H_{\mathcal V}$ admits an explicit eigenbasis in $L^2({\mathbb R^2})$, and calculate the corresponding eigenvalues. Moreover, for ${\mathcal V}$ which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of $H_{\mathcal V}$ adjoining any given $\Lambda_q$. We find that the effective Hamiltonian in this context is the Toeplitz operator ${\mathcal T}_q({\mathcal V}) = p_q {\rm Op}^{\rm w}({\mathcal V}) p_q$, where $p_q$ is the orthogonal projection onto ${\rm Ker}(H_0 - \Lambda_q I)$, and investigate its spectral asymptotics.

中文翻译：

---

具有非局域势的朗道哈密顿量的谱特性

我们考虑朗道哈密顿量 $H_0$，在 $L^2({\mathbb R^2})$ 中自伴随，其频谱由无限退化正特征值 $\Lambda_q$, $q \in { 的等差级数组成\mathbb Z}_+$。我们通过写为有界伪微分算子 ${\rm Op}^{\rm w}({\mathcal V})$ 的非局部势扰动 $H_0$，并使用实值 Weyl 符号 ${\mathcal V }$，使得 ${\rm Op}^{\rm w}({\mathcal V}) H_0^{-1}$ 是紧凑的。我们研究了扰动算子 $H_{\mathcal V} = H_0 + {\rm Op}^{\rm w}({\mathcal V})$ 的频谱特性。首先，我们构造符号 ${\mathcal V}$，具有合适的对称性，使得算子 $H_{\mathcal V}$ 承认 $L^2({\mathbb R^2})$ 中的显式特征基，并计算相应的特征值。此外，对于不应该具有这种对称性的 ${\mathcal V}$，我们研究了与任何给定的 $\Lambda_q$ 相邻的 $H_{\mathcal V}$ 的特征值的渐近分布。我们发现在这种情况下有效的哈密顿量是托普利兹算子 ${\mathcal T}_q({\mathcal V}) = p_q {\rm Op}^{\rm w}({\mathcal V}) p_q$，其中 $p_q$ 是 ${\rm Ker}(H_0 - \Lambda_q I)$ 上的正交投影，并研究其谱渐近性。