Multi-dimensional Schrödinger Operators with Some Negative Spectrum

In this paper we consider Schrödinger operators −∆+ V (x), V ∈ L∞(Rd) acting in the space L(R). If V = 0 then the operator has purely absolutely continuous spectrum on (0,+∞). We find conditions on V which guarantee that the absolutely continuous spectrum of both operators H+ = −∆ + V and H− = −∆− V is essentially supported by [0,∞). This means that the spectral projection associated to any subset of positive Lebesgue measure is not zero. Our main result is the following theorem (compare with [6]): Theorem 1.1. Let V ∈ L∞(Rd) be a real function. Assume that the negative spectrum of the operators H+ = −∆+ V and H− = −∆− V consists only of eigenvalues, denoted by λn(V ) and λn(−V ), which satisfy the condition ∑