On negative eigenvalues of two-dimensional Schrödinger operators with singular potentials

Abstract

We present upper estimates for the number of negative eigenvalues of two-dimensional Schrödinger operators with potentials generated by Ahlfors regular measures of arbitrary fractional dimension α ∈ (0, 2]. The estimates are given in terms of integrals of the potential with a logarithmic weight and of its L log L type Orlicz norms. In the case α = 1, our results are stronger than the known ones about Schrödinger operators with potentials supported by Lipschitz curves.

Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case

“…Proof. The proof is similar to that of [9,Lemma 2.13]. It is sufficient to prove the lemma for compactly supported measures as the general case then follows easily from the assumption that is -finite.…”

“…Theorem 2.1 follows from a spectral estimate in a more general setting extending the considerations in [9]. Definition 2.2.…”

“…Recently, the interest to the critical case has revived, due to some new applications, see, e.g., [18], [12]. So, a class of singular measures was considered in [9]. For a singular measure = in R 2 , where is Ahlfors -regular, ∈ (0, 2), and belongs to a certain Orlicz class, an estimate for the eigenvalues of the problem − Δ = was obtained, | | ≤ ( , ) −1 .…”

“…In the usual way, the estimates and asymptotics of eigenvalues of the weighted problem, by means of the Birman-Schwinger principle, produce similar results for the number of negative eigenvalues of the Schrödiger-like operator as the coupling constant grows. This relation was explored in [9]. Some of the results of the paper were presented in the short note [14] without proofs.…”

Calogero type bounds in two dimensions

“…In [BL] the authors gave examples of different non-Weyl law formulae under the condition V P L 1 pR 2 q. Some upper estimates for NpH 0 ´V q in the two-dimensional case were obtained in papers [GN,KS,LS1,LS2,MV,MW,St]. In [Sh] the author gives 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 N. N. Khuri, A. Martin and T. T. Wu [KMW] (see also [CKMW]).…”