An estimate for the number of bound states of the Schrödinger operator in two dimensions

Abstract

For the Schrödinger operator − Δ + V -\Delta + V on R 2 \mathbb R ^2 let N ( V ) N(V) be the number of bound states. One obtains the following estimate: \[ N ( V ) ≤   1   + ∫ R 2 ∫ R 2 | V ( x ) |   | V ( y ) |   | C 1 ln ⁡ | x − y | + C 2 | 2   d x d y N(V) \leq \ 1 \ + \int _{\mathbb R ^2} \int _{\mathbb R ^2} |V(x)| \ |V(y)| \ |C_1 \ln |x-y| + C_2|^2 \ dx\,dy \] where C 1 = − 1 2 π C_1 = -\frac {1}{2\pi } and C 2 = ln ⁡ 2 − γ 2 π C_2 = \frac {\ln 2 - \gamma }{2 \pi } ( γ \gamma is the Euler constant). This estimate holds for all potentials for which the previous integral is finite.