  We study to what extent Lieb--Thirring inequalities are extendable from
self-adjoint to general (possibly non-self-adjoint) Jacobi and Schr\"{o}dinger
operators. Namely, we prove the conjecture of Hansmann and Katriel from
[Complex Anal. Oper. Theory 5, No. 1 (2011), 197-218] and answer another open
question raised therein. The results are obtained by means of asymptotic
analysis of eigenvalues of discrete Schr\"{o}dinger operators with rectangular
barrier potential and complex coupling. Applying the ideas in the continuous
setting, we also solve a similar open problem for one-dimensional
Schr\"{o}dinger operators with complex-valued potentials published by Demuth,
Hansmann, and Katriel in [Integral Equations Operator Theory 75, No. 1 (2013),
1-5].
