A note on eigenvalue bounds for Schrödinger operators

Abstract

We obtain a new bound on the location of eigenvalues for a nonself-adjoint Schrödinger operator with complex-valued potentials by obtaining a weighted L 2 estimate for the resolvent of the Laplacian.

Improved Eigenvalue Bounds for Schrödinger Operators with Slowly Decaying Potentials

“…This follows from the explicit formula for the kernel K z,ζ and standard Bessel function estimates, see e.g. (2.21)-(2.27) in [20] or the proof of (2.5) in the appendix of [22] (where the estimate is in fact proved for the larger range Re ζ ∈ [(d − 1)/2, (d + 1)/2], but this will not be needed here). In both references the (second) exponential factor in (2.2) is simply estimated by one.…”

A note on the Schrödinger smoothing effect

“…The quantity was already appeared in and concerning the unique continuation for the Schrödinger equation and eigenvalue bounds for the Schrödinger operator, respectively. We also refer the reader to for some weighted L 2 estimates in which a time‐dependent weight is involved.…”

Counterexample to the Laptev--Safronov conjecture

“…Some refinements for singular potentials are known, see e.g. [12], [28], [29], [5], but we focus here on the long-range aspects of the potential. This is reflected by the fact that the construction of our counterexample is local in Fourier space, similar to the examples for embedded eigenvalues in [7], where a connection between the aforementioned Ionescu-Jerison example and the "Knapp example" in Fourier restriction theory (see e.g.…”