zbMATH — the first resource for mathematics

“Non-standard” spectral asymptotics for a two-dimensional Schrödinger operator. (English) Zbl 0910.35086
Greiner, Peter C. (ed.) et al., Partial differential equations and their applications. Lectures given at the 1995 annual seminar of the Canadian Mathematical Society, Toronto, Canada, June 12–23, 1995. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 12, 9-16 (1997).
Consider the Schrödinger operator \[ H(\alpha)=-\Delta-\alpha V(x), \quad \alpha>0,\;x\in\mathbb R^d, \] and for \(\lambda=-\gamma^2\), \(\gamma\geq 0\), denote by \(N(\alpha,-\gamma^2,V)\) the number of eigenvalues lying on the left of \(\lambda\). The asymptotic formulae for \(N(\alpha,-\gamma^2,V)\) as \(\alpha\to\infty\) are known in the case \(d\geq 3\), in particular, if \(V\in L_{d/2}(\mathbb R^n)\) (the Weyl case). The main aim of the paper is to show that some of the spectral properties of the Schrödinger operator \(H(\alpha)\) valid for \(d\geq 3\) do not hold in the two-dimensional case. The authors note that in order to avoid complicated formulations they formulated the results in terms of upper and lower limits for power type asymptotics for the counting function of an auxiliary operator on a semiaxis although they could give more general statements.
For the entire collection see [Zbl 0878.00060].

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
35P15 Estimates of eigenvalues in context of PDEs