The authors mostly consider, on \(R^ d\), the operator \(A(\alpha V)=- \Delta-\alpha V(x)\), where \(\Delta\) is the Laplacian, \(\alpha\) is a positive constant, and \(V(x)\) is a real potential. They are interested in estimates of the number \(N_ -(\alpha V)\) of negative eigenvalues of \(A(\alpha V)\), in the case \(V\notin L_{d/2}(R^ d)\). The main part (sections 1 to 6) of the paper is devoted to preparatory material, an operator and function theory. In sections 7 and 8, estimates for \(N_ - (\alpha V)\) are discussed, the conditions on \(V\) being stated in terms of \(V/\psi\), where \(\psi\) is a Hardy-weight. In the last section, estimates, for the higher order operator \((-\Delta)^ k-V\), are considered.
Related results were obtained by Yu. Egorov and V. A. Kondrat’ev (cf. the following review).