NEW BOUNDS ON THE NUMBER OF BOUND-STATES FOR SCHRODINGER-OPERATORS

We consider the Schrodinger operator H = -Delta + V(\x\) on R(3). Let n(l) denote the number of bound states with angular momentum l (not co unting the 2l + 1 degeneracy). We prove the following bounds on n(l). Let V less than or equal to 0 and d/dr r(1-2p)(-V)(1-p) less than or e qual to 0 for some p is an element of [1/2, 1) then n(l) less than or equal to p(1 - p)(p-1)(2l + 1)(1-2p) integral(0)(infinity)(-r(2)V)(p) dr/r. This bound closes the gap between the celebrated bounds by Calog ero (p = 1/2) and Bargmann (p = 1).