Schrödinger operator. Estimates for number of bound states as function- theoretical problem. (English) Zbl 0756.35057
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).
Related results were obtained by Yu. Egorov and V. A. Kondrat’ev (cf. the following review).
Reviewer: D.Huet (Nancy)
MSC:
35P15 | Estimates of eigenvalues in context of PDEs |
47F05 | General theory of partial differential operators |
35J10 | Schrödinger operator, Schrödinger equation |