Improved Berezin—Li—Yau inequalities with magnetic field

Abstract

In this paper we study the eigenvalue sums of Dirichlet Laplacians on bounded domains. Among our results we establish an improvement of the Berezin bound and of the Li—Yau bound in the presence of a constant magnetic field previously obtained by Erdős et al. and Melas.

Melas-type bounds for the Heisenberg Laplacian on bounded domains

“…In particular, our main result improves inequality (1.6) in a similar way in which [9] improves inequality (1.9). However, the method that we employ in the present paper is different from the one used in [9] since it does not rely on a Hardy inequality involving the distance to the boundary. In fact, as far as we know an analog of such an inequality for the Heisenberg-Laplacian with explicit constants is not known.…”

Semiclassical bounds in magnetic bottles

“…3. It has to be added that the question of semiclassical spectral bounds for such systems has been addressed before, in particular, another version of the magnetic Berezin inequality was derived by two of us [KW13]. In final part of Sec.…”

Spectral estimates for the Heisenberg Laplacian on cylinders

“…[6]. In addition, |ω β | β is a decreasing function of β on (0, R(ω)], see [9,Lemma 4.2]. Hence we compute l(ω) = |Ω| R(ω) and simplify the constant in Theorem 2.3.…”

“…The lower order term. To obtain a suitable lower bound on the second term in (4.4) we use the same technique as in [9]. The key point of this approach is to estimate the quantity…”