Mihai Stoiciu
An estimate for the number of bound states of the Schrodinger operator in two dimensions
(26K, AMSTeX)
ABSTRACT. For the Schrodinger operator $-\Delta + V$ on $\R^2$ let $N(V)$ be the number of bound states. One obtains the following estimate:
$$
N(V) \leq 1 + \int_{\R^2}\int_{\R^2} |V(x)| |V(y)| |C_1 \ln |x-y| + C_2|^2 dxdy
$$
where $C_1 = -\frac{1}{2\pi}$ and $C_2 = \frac{\ln 2 - \gamma}{2 \pi}$ ($\gamma$ is the Euler constant). This estimate holds for all potentials for which the previous integral is finite.