On the Growth of the Number of Bound States with Increase in Potential Strength

Abstract

For a wide class of potentials, it is shown that N(λ), the number of bound states (including multiplicity) of −Δ + λV, obeys the conditions Aλ32 < N(λ) < Bλ32for λ sufficiently large. A and B are positive finite numbers. In the centrally symmetric cases, a related growth condition on lmax(λ), the largest l channel with bound states, is also obtained, namely, aλ12 < lmax(λ) < bλ12. Finally, we discuss analogous results for a larger class of central potentials and for the many-body case.