On negative eigenvalues of generalized Laplace operators

The negative eigenvalues problem for the generalized Laplace operator -<(<D elta>)over tilde> = -Delta(+) over tilde alphaT, alpha < 0, where T is a po sitive operator singular in L-2 and acting from the Sobolev space W-2(1) to its dual W-2(-1), is investigated. The question, whether the number of neg ative eigenvalues N-(-<(<Delta>)over tilde>) is finite or infinite is answe red. Under the assumption that the not necessarily compact operator T (I - Delta)T-1 in W-2(1) has a discrete spectrum, different conditions leading t o N-(-<(<Delta>)over tilde>) = infinity as well as to N-(-<(<Delta>)over ti lde>) < infinity are found and the corresponding examples are given.