On the Negative Spectrum of the Two-Dimensional Schrödinger Operator with Radial Potential

Abstract

Abstract. For a two-dimensional Schrödinger operator H αV = −∆ − αV with the radial potential V (x) = F (|x|), F (r) ≥ 0, we study the behavior of the number N − (H αV ) of its negative eigenvalues, as the coupling parameter α tends to infinity. We obtain the necessary and sufficient conditions for the semi-classical growth N − (H αV ) = O(α) and for the validity of the Weyl asymptotic law.

On negative eigenvalues of two‐dimensional Schrödinger operators

“…Then it follows from Lemma 2.2 and (18), (20) that (4)), which implies (19) with C 4 = 2C 5 (see (3)).…”

“…

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 (see [11]). Upper estimates for N − (V ) in the case d = 2 can be found in [4,6,7,10,17,20,21,25,26,29,30,32] and in the references therein. Following the pioneering paper [29], we obtain upper estimates involving L log L type Orlicz norms of the potential (Theorems 3.1, 6.1 anf 7.1) and prove (Theorem 4.3) a conjecture by N.N.

…”

Bound on the number of negative eigenvalues of two-dimensional Schrödinger operators on domains

“…As is well-known, the two-dimensional case is a borderline case and is still not as well understood as the case of three and higher dimensions and the one-dimensional case. Recently, there have been several results on the twodimensional case [4,8,13,3,6,7]. Solomyak's pioneering paper [15] had a profound influence on these developments and we would like to dedicate our results here, which are also a variation on the theme in [15], to his memory.…”

“…This discussion raises the question of characterizing all V ∈ L 1 (R 2 ) (or all 0 ≥ V ∈ L 2 (R 2 )) such that either lim sup α→∞ α −1 N(−∆ + αV ) < ∞ or such that (1.4) holds. This problem was solved in the radial case in [6], but is still open in general. The eigenvalue bounds in [15,4,13,3,7] can be understood as sufficient conditions for an asymptotically linear bound.…”

On spectral estimates for the Schrödinger operators in global dimension 2

“…It was established in [6] that for such potentials this estimate gives not only sufficient, but also necessary condition for the semiclassical behavior. It was established in [6] that for such potentials this estimate gives not only sufficient, but also necessary condition for the semiclassical behavior.…”