The fate of Landau levels under delta-interactions

0573208 - ÚJF 2024 RIV DE eng J - Journal Article
Behrndt, J. - Holzmann, M. - Lotoreichik, Vladimir - Raikov, G.
The fate of Landau levels under delta-interactions.
Journal of Spectral Theory. Roč. 12, č. 3 (2022), s. 1203-1234. ISSN 1664-039X. E-ISSN 1664-0403
R&D Projects: GA ČR(CZ) GA21-07129S
Institutional support: RVO:61389005
Keywords : Berezin-Toeplitz operators * Laguerre polynomials * Landau Hamiltonian
OECD category: Pure mathematics
Impact factor: 1, year: 2022
Method of publishing: Open access
https://doi.org/10.4171/JST/422

We consider the self-adjoint Landau Hamiltonian H-0 in L-2(R-2) whose spectrum consists of infinitely degenerate eigenvalues Lambda(q), q is an element of Z(+), and the perturbed Landau Hamiltonian H-upsilon = H-0 + upsilon delta(Gamma), where Gamma subset of R-2 is a regular Jordan C-1,C-1-curve and upsilon is an element of L-p(Gamma, R), p > 1, has a constant sign. We investigate ker(H-upsilon - Lambda(q)), q is an element of Z(+), and show that generically

0 <= dim ker(H-upsilon - Lambda(q)) - dim ker(T-q(upsilon delta(Gamma))) < infinity,

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)(R-2), and p(q) is the orthogonal projection onto ker(H-0 - Lambda(q)). If upsilon not equal 0 and q = 0, then we prove that ker(T-0(upsilon delta(Gamma))) = {0}. If q >= 1 and Gamma = C-r is a circle of radius r, then we show that dim ker(T-q(delta(Cr))) <= q, and the set of r is an element of (0, infinity) for which dim ker(T-q(delta(Cr))) >= 1 is infinite and discrete.
Permanent Link: https://hdl.handle.net/11104/0343660