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

Abstract

We consider the self-adjoint Landau Hamiltonian $H_0$ in $L^2({\mathbb{R}}^2)$ whose spectrum consists of infinitely degenerate eigenvalues ${\Lambda }_q$, $q\in {\mathbb{Z}}_{+}$, and the perturbed operator $H_{\upsilon }=H_0+\upsilon {\delta }_{\Gamma }$, where $\Gamma \subset {\mathbb{R}}^2$ is a regular Jordan $C^{1,1}$-curve, and $\upsilon \in L^p(\Gamma ;\mathbb{R})$, $p>1$, has a constant sign. We investigate $\mathrm{Ker}(H_{\upsilon }-{\Lambda }_q)$, $q\in {\mathbb{Z}}_{+}$, and show that generically $$0\le \dim \mathrm{Ker}(H_{\upsilon }-{\Lambda }_q)-\dim \mathrm{Ker}(T_q(\upsilon {\delta }_{\Gamma }))<\infty ,$$ 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 $\mathrm{Ker}(H_0-{\Lambda }_q)$. If $\upsilon \ne 0$ and $q=0$, we prove that $\mathrm{Ker}(T_0(\upsilon {\delta }_{\Gamma }))=\{0\}$. If $q\ge 1$, and $\Gamma ={\mathcal{C}}_r$ is a circle of radius $r$, we show that $\dim \mathrm{Ker}(T_q({\delta }_{{\mathcal{C}}_r}))\le q$, and the set of $r\in (0,\infty )$ for which $\dim \mathrm{Ker}(T_q({\delta }_{{\mathcal{C}}_r}))\ge 1$, is infinite and discrete.

arXiv comment28 pages; to appear in Journal of Spectral Theory