Let us consider the one dimensional Schr{\"o}dinger operator $H=-D^2+V$. It is well known that $H$  has  no positive eigenvalues  if $V (x)$= o(x^{−1})$.   More generally, if $\limsup |xV (x)| = C$, Eastham-Kalf  show any eigenvalue must  be euqal or less than  $C^2$. On the other hand, Naboko and Simon have constructed $V (x)$ decaying arbitrarily slower than  $x^{-1}$ with dense eigenvalues.  

It is conjectured by Barry Simon that if  $\limsup |xV (x)| <\infty$, then $\sum \sqrt{\lambda_n}<\infty$, where $\{\lambda_n\}$  are the positive eigenvalues of $H$.

In this seminar, I will present a result of Kiselev-Last-Simon, which states that if $\limsup |xV (x)| = C$,  then $\sum \lambda_n\leq \frac{C^2}{2}$.