| Summary: | Laptev and Safronov conjectured that any non-positive eigenvalue of a Schrödinger operator -Δ+V in L^2 (R^ν) with complex potential has absolute value at most a constant times ||V||^(γ+ν/2)/γ)_(γ+ν/2) for 0 < γ ≤ ν/2 in dimension ν ≥ 2. We prove this conjecture for radial potentials if 0 < γ < ν/2 and we 'almost disprove' it for general potentials if 1/2 < γ < ν/2. In addition, we prove various bounds that hold, in particular, for positive eigenvalues. © 2017 European Mathematical Society. Received April 5, 2015; revised May 26, 2015. Published online: 2017-09-28. Work partially supported by U.S. National Science Foundation grants PHY-1347399, DMS-1363432 (R. L. Frank), and DMS-1265592 (B. Simon). Submitted - 1504.01144.pdf
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