The fate of Landau levels under $\delta$-interactions

Abstract

We consider the self-adjoint Landau Hamiltonian $H_0$ in $L^2(\mathbb{R})$ whose spectrum consists of infinitely degenerate eigenvalues ${\Lambda }_q$, $q\in {\mathbb{Z}}_{+}$, and the perturbed Landau Hamiltonian $H_{\upsilon }=H_0+\upsilon {\delta }_{\Gamma }$, where $\Gamma \subset \mathbb{R}$ is a regular Jordan $C^{1,1}$-curve and $\upsilon \in L^p(\Gamma ;\mathbb{R})$, $p>1$, has a constant sign. We investigate $\ker (H_{\upsilon }-{\Lambda }_q)$, $q\in {\mathbb{Z}}_{+}$, and show that generically

where $T_q(\upsilon {\delta }_{\Gamma })=p_q(\upsilon {\delta }_{\Gamma })p_q$, is an operator of Berezin–Toeplitz type, acting in $p_q{L}^2(\mathbb{R})$, and $p_q$ is the orthogonal projection onto $\ker (H_0-{\Lambda }_q)$. If $\upsilon \ne 0$ and $q=0$, then we prove that $\ker (T_0(\upsilon {\delta }_{\Gamma }))=0$. If $q\ge 1$ and $\Gamma ={\mathcal{C}}_r$ is a circle of radius $r$, then we show that $dim\; ker(T_q({\delta }_{{\mathcal{C}}_r}))\le q$, and the set of $r\in (0,\infty )$ for which $dim\; ker(T_q({\delta }_{{\mathcal{C}}_r}))\ge 1$ is infinite and discrete.