0. Introduction Let us consider a Schr5dinger operator in L2(Rd), -A+V, (0.1) where V is a real-valued function. Lieb and Thirring [231 proved that if'y>max(0, 1-ld), then there exist universal constants LT, d satisfying(1) tr (-A+V) ~ -<~ L%d [ V~_+d/2(x) dx. JR d (o.2) In the critical case d>~3 and "~=0, the bound (0.2) is known as the Cwikel LiebRozenblum (CLR) inequality, see [8], [20], [25] and also [7], [19]. For the remaining case d=l and .y=l, the estimate (0.2) has been verified in [27], see also [14]. On the other hand, it is known that (0.2) fails for 7=0 if d=2, and for 0~<7< 1 if d=l. If VCL~+d/2(Rd), then the inequalities (0.2) are accompanied by the Weyl-type asymptotic formula lim 1 1 //~ cz-++cxD OL~'+d/2 tr (-A+c~V) ~ = lim (I~I2~_oLV)~. dx cl~ -~+~ a~+d/2 d• (2~) d L~l'd /a ~/~+d/2, v _ ax, d (0.3)(1) Here and below we use the notion 2x_ := Ixl-x for the negative part of variables, functions, Hermitian matrices or self-adjoint operators.

We obtain here some inequalities for the eigenvalues of Dirichlet and Neumann value problems for general classes of operators (or system of operators) acting in L1997 Academic Press

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Abstract. We discuss properties of eigenvalues of non-self-adjoint Schrödinger operators with complex-valued potential V . Among our results are estimates of the sum of powers of imaginary parts of eigenvalues by the L p -norm of V .

Inequalities are derived for power sums of the real part and the modulus of the eigenvalues of a Schrödinger operator with a complexvalued potential.