On the negative eigenvalues of a class of Schrödinger operators. (English) Zbl 0941.35055

Buslaev, V. (ed.) et al., Differential operators and spectral theory. M. Sh. Birman’s 70th anniversary collection. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 189(41), 173-186 (1999).
The number of negative eigenvalues of the operator generated by the quadratic form \(\int(|\nabla^\ell u|^2+|x|^{-2\ell}|u|^2)dx-\int|u|^2d\mu\), \(2\ell=d\), in \(L_2(\mathbb{R}^d)\), where \(\mu\) is a positive measure, is estimated if the (Borel) measure \(\mu\) has a compact support \(F\) and for any \(x\in\mathbb{R}^d\) and \(r\) small enough \((r<{\text{diam} F\over \gamma_2})\), \(\gamma_2>1\). The measures of balls \(B(x,2)\) are estimated as follows: \[ \mu\bigl\{B(x, \gamma_2)\bigr\}\geq 2\mu\bigl\{ B(x,r)\bigr\}, \quad\gamma_1>1. \] The number of negative eigenvalues is estimated (Theorem 1) by \(N(\mu)\leq C(\gamma_1,\gamma_2, d)\mu(\mathbb{R}^d)\).
For the entire collection see [Zbl 0911.00011].


35P15 Estimates of eigenvalues in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J60 Nonlinear elliptic equations