×

Asymptotics of the spectrum of the Schrödinger operator with a nonregular homogeneous potential. (Russian) Zbl 0583.35083

The Schrödinger operator \(A=-\Delta +Q\) in \(L^ 2({\mathbb{R}}^ m)\), \(m\geq 3\), is considered. It is supposed that Q(x) is continuous, Q(x)\(\geq 0\) and \(Q(tx)=t^ aQ(x)\), \(t\geq 0\), \(a>0\). If \(Q(x)>0\) for \(x\in {\mathbb{S}}^{m-1}\), then the spectrum \(\{\lambda_ n\}\) of A is discrete and its asymptotics is given by the Weyl’s formula. The purpose of the paper is to study the asymptotics of \(\{\lambda_ n\}\) in case \(Q(x)=0\) for some \(K\subset {\mathbb{S}}^{m-1}\) so that the Weyl’s formula is violated. It is found out that the asymptotics of \(\{\lambda_ n\}\) is determined by the structure of \(K\) and by the behavior of Q in its neighbourhood.
Reviewer: D.Yafaev

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
35P15 Estimates of eigenvalues in context of PDEs
PDFBibTeX XMLCite
Full Text: EuDML