Estimates for Eigenvalues of Schrodinger Operators with Complex-Valued Potentials

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