On negative eigenvalues of generalized laplace operators

The negative eigenvalues problem for the generalized Laplace operator −Δ = −Δ++αT, α < 0, where T is a positive operator singular in L2 and acting from the Sobolev space W12 to its dual W−12, is investigated. The question, whether the number of negative eigenvalues N_(−Δ) is finite or infinite is answered. Under the assumption that the not necessarily compact operator T = (I − Δ)−1T in W12 has a discrete spectrum, different conditions leading to N_(−Δ) = ∞, as well as to N_(−Δ) < ∞ are found and the corresponding examples are given.