Sufficient Conditions for an Attractive Potential to Possess Bound States

Simple conditions on the potential are given, which are sufficient to secure the existence of at least one bound state for each angular momentum l ≤ L. One such condition is given by the inequality $$-{\displaystyle {\int }_0^R}\mathrm{dr\hspace{0.17em}rV(r)}{\left[\frac{r}{R}\right]}^{\text{2L+1}}-{\displaystyle {\int }_R^{\infty }}\mathrm{dr\hspace{0.17em}rV(r)}{\left[\frac{R}{r}\right]}^{\text{2L+1}}\text{≥2L+1}$$ , where R is an arbitrary radius and V(r) an everywhere attractive potential. Upper bounds on the energy of the lower bound state for each angular momentum are also given.