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].

MSC:

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

Keywords:

number of negative eigenvalues