On negative eigenvalues of two‐dimensional Schrödinger operators

Abstract

The paper presents estimates for the number of negative eigenvalues of a two-dimensional Schrödinger operator in terms of L log L-type Orlicz norms of the potential and proves a conjecture by

On estimates for the number of negative eigenvalues of two-dimensional Schrödinger operators with potentials supported by Lipschitz curves

“…In the case α = 2, we get the same estimate as in [40,Theorem 6.1], which is stronger than most other known estimates that use isotropic norms. (Anisotropic norms like the ones used in [40,Section 7] and [26] are not available in the case α < 2 and hence are not treated here.) In the case α = 1, our Theorem 3.1 and Corollary 3.2 are stronger than the results obtained in [21] and [39] as we are now able to cover Ahlfors regular curves rather than just Lipschitz ones.…”

“…where Ξ is a combination of certain norms, Ξ(λV ) = O(λ) as λ → +∞, and, most importantly, N − (E λV,R 2 ) = O (λ) as λ → +∞ (5) implies that Ξ(V ) < ∞. Unfortunately, even the strongest known estimates for d = 2 are not optimal in this sense (see [40]). Finding an optimal estimate of type (4) seems to be a difficult problem.…”

“…2) The Birman-Laptev-Solomyak method (see Section 4) used in this paper (and in [41], [40]) splits the problem into the radial and non-radial parts. The former is essentially a one-dimensional problem and is usually easier to handle than the latter.…”

On spectral estimates for the Schrödinger operators in global dimension 2

“…The estimate formulated below, the sharpest one known up to now, was obtained recently by Shargorodsky [16], and is a refinement of earlier estimates in [17,5,7]. Several upper bounds for N − 2 H are known.…”

100 years of Weyl’s law

“…For d = 2, the estimate is marginally worse (see [25] for the most general statement for arbitrary orders of operators and dimensions and [32] for the most general results for the Schrödinger operator in dimension 2). It occurs that we can split our domain into an overlapping regular zone {x : ρ(x)γ (x) ≥ h} and a singular zone {x : ρ(x)γ (x) ≤ 3h}, then evaluate the contribution of the regular zone using the rescaling technique and the contribution of the singular zone by the variational estimate as if on the inner boundary of this zone (i.e.…”