New estimates for eigenvalues of non-self-adjoint multi-dimensional Schrodinger operators are obtained in terms of L (p) -norms of the potentials. The results cover and improve those known previously, in particular, due to Frank (Bull Lond Math Soc 43(4):745-750, 2011), Safronov (Proc Am Math Soc 138(6):2107-2112, 2010), Laptev and Safronov (Commun Math Phys 292(1):29-54, 2009). We mention the estimations of the eigenvalues situated in the strip around the real axis (in particular, the essential spectrum). The method applied for this case involves the unitary group generated by the Laplacian. The results are extended to the more general case of polyharmonic operators. Schrodinger operators with slowly decaying potentials and belonging to weak Lebesgues classes are also considered.

Estimates for eigenvalues of Schrodinger operators on the half-line with complex-valued potentials are established. Schrodinger operators with potentials belonging to weak Lebesques classes are also considered. The results cover those known previously due to R. L. Frank, A. Laptev and R. Seiringer

Estimates for the eigenvalues of non-self-adjoint 1D Dirac operators considered on the half-line are obtained in terms of the L p -norms of the potentials. The proofs are based on the resolvent estimates established for the free Dirac operator.

General Hardy-Carleman type inequalities for Dirac operators are proved. New inequalities are derived involving particular traditionally used weight functions. In particular, a version of the Agmon inequality and Treve type inequalities is established. The case of a Dirac particle in a (potential) magnetic field is also considered. The methods used are direct and based on quadratic form techniques. (C) 2015 AIP Publishing LLC.