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

We consider the self-adjoint Landau Hamiltonian $H_0$ in $L^2({\mathbb{R}}^2)$ whose spectrum consists of infinitely degenerate eigenvalues ${\mathrm{\Lambda }}_q$, $q \in $\mathbb{Z}$+$,andtheperturbedoperator$H\upsilon = H_0 + \upsilon\delta_\Gamma$,where$\Gamma \subset $\mathbb{R}$^2$isaregularJordan$C^{1,1}$-curve,and$\upsilon \in L^p(\Gamma;$\mathbb{R}$)$,$p>1$, has a constant sign. We investigate $\mathrm{K}\mathrm{e}\mathrm{r}(H_{\upsilon }-{\mathrm{\Lambda }}_q)$, $q \in $\mathbb{Z}$+$,andshowthatgenerically$$0 \leq {\rm dim , Ker}(H\upsilon -\Lambda_q) - {\rm dim , Ker}(T_q(\upsilon \delta_\Gamma)) < \infty,$$where$T_q(\upsilon \delta_\Gamma) = p_q (\upsilon \delta_\Gamma)p_q$, is an operator of Berezin-Toeplitz type, acting in $p_q{L}^2({\mathbb{R}}^2)$, and $p_q$ is the orthogonal projection on $\mathrm{K}\mathrm{e}\mathrm{r}(H_0-{\mathrm{\Lambda }}_q)$. If $\upsilon \mathrm{\ne }0$ and $q=0$, we prove that $\mathrm{K}\mathrm{e}\mathrm{r}(T_0(\upsilon {\delta }_{\mathrm{\Gamma }}))=\{0\}$. If $q\ge 1$, and $\mathrm{\Gamma }={\mathcal{C}}_r$ is a circle of radius $r$, we show that $\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{K}\mathrm{e}\mathrm{r}(T_q({\delta }_{{\mathcal{C}}_r}))\le q$, and the set of $r \in (0,\infty)$forwhich${\rm dim , Ker}(T_q(\delta_{$\mathcal{C}$_r})) \geq 1$, is infinite and discrete.