Improved Eigenvalue Bounds for Schrödinger Operators with Slowly Decaying Potentials

Abstract

We extend a result of Davies and Nath [5] on the location of eigenvalues of Schrödinger operators with slowly decaying complex-valued potentials to higher dimensions. In this context, we also discuss various examples related to the Laptev-Safronov conjecture [21].

Counterexample to the Laptev--Safronov conjecture

“…Moreover, the right hand side of (3) may be much smaller than that of (2). There are estimates similar to (3) for any q ∈ (d/2, (d + 1)/2], see [8]. We do not state these here but remark that our counterexample also shows that an analog of [8, Th.…”

“…Observe that if s < 2d/(d + 1) (this condition appears in [6,22]), then one is always in the long-range case (ii) since (d + 1)/2 < q s . The proof of (ii), (iii) could be obtained by closely following the arguments in [13] and [8], respectively. However, our main point here is to show that all the statements of Theorem 4 follow from the general results of Propositions 20 and 24 below in the special case h 0 (ξ) = |ξ| s .…”

Bounds for Schrödinger Operators on the Half-Line Perturbed by Dissipative Barriers

“…Stepanenko IEOT [1,Theorem 4] for Schrödinger operator on L 2 (R) with a potential V ∈ L 1 (R)∩L 2 (R). Such magnitude bounds were later generalised to include more general potentials, higher dimensions and more general geometries [6,8,[10][11][12]15,17,20,21,29]. The work most relevant to this paper was undertook by Frank, Laptev and Seiringer [14], where they show that any eigenvalue λ of a Schrödinger operator − d 2 / dx 2 + V on L 2 (R + ), endowed with a Dirichlet boundary condition at 0, satisfies…”

On Compressed Resolvents of Schrödinger Operators with Complex Potentials

“…We also mention that the analysis and spectral theory of non-self-adjoint Schrödinger operators has attracted a lot of attention in the recent past. In particular, eigenvalue bounds, Lieb-Thirring inequalities, and other spectral properties of Schrödinger operators with complex potentials were derived in, e.g., [5,6,9,10,[15][16][17][18][19]26]. The resolvent formula in Theorem 3.2 below and the compressed resolvent (1.2)-(1.3) are intimately connected with the spectral analysis of Schrödinger operators as in (1.1).…”