The fate of Landau levels under $δ$-interactions

Abstract

We consider the self-adjoint Landau Hamiltonian H 0 in L 2 (R 2 ) whose spectrum consists of infinitely degenerate eigenvalues Λ q , q ∈ Z + , and the perturbed operator H υ = H 0 + υδ Γ , where Γ ⊂ R 2 is a regular Jordan C 1,1 -curve and υ ∈ L p (Γ; R), p > 1, has a constant sign. We investigate ker(H υ − Λ q ), q ∈ Z + , and show that genericallywhere T q (υδ Γ ) = p q (υδ Γ )p q , is an operator of Berezin-Toeplitz type, acting in p q L 2 (R 2 ), and p q is the orthogonal projection on ker (H 0 − Λ q ). If υ = 0 and q = 0, we prove that ker (T 0 (υδ Γ )) = {0}. If q ≥ 1 and Γ = C r is a circle of radius r, we show that dim ker(T q (δ Cr )) ≤ q, and the set of r ∈ (0, ∞) for which dim ker(T q (δ Cr )) ≥ 1 is infinite and discrete.